LMFDB - Embedded newform 4006.2.a.g.1.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4006,2,Mod(1,4006)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4006, 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("4006.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4006 = 2 \cdot 2003 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4006.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:	$31.9880710497$
Analytic rank:	$1$
Dimension:	$40$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Character	$\chi$	$=$	4006.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -0.0866100 q^{3} +1.00000 q^{4} +2.47499 q^{5} +0.0866100 q^{6} +0.719708 q^{7} -1.00000 q^{8} -2.99250 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -0.0866100 q^{3} +1.00000 q^{4} +2.47499 q^{5} +0.0866100 q^{6} +0.719708 q^{7} -1.00000 q^{8} -2.99250 q^{9} -2.47499 q^{10} -4.02218 q^{11} -0.0866100 q^{12} -1.17365 q^{13} -0.719708 q^{14} -0.214358 q^{15} +1.00000 q^{16} +0.882057 q^{17} +2.99250 q^{18} +0.0145269 q^{19} +2.47499 q^{20} -0.0623338 q^{21} +4.02218 q^{22} -5.06055 q^{23} +0.0866100 q^{24} +1.12556 q^{25} +1.17365 q^{26} +0.519010 q^{27} +0.719708 q^{28} +5.12559 q^{29} +0.214358 q^{30} +7.34428 q^{31} -1.00000 q^{32} +0.348361 q^{33} -0.882057 q^{34} +1.78127 q^{35} -2.99250 q^{36} +5.66296 q^{37} -0.0145269 q^{38} +0.101650 q^{39} -2.47499 q^{40} +7.20437 q^{41} +0.0623338 q^{42} +2.79702 q^{43} -4.02218 q^{44} -7.40639 q^{45} +5.06055 q^{46} +6.06598 q^{47} -0.0866100 q^{48} -6.48202 q^{49} -1.12556 q^{50} -0.0763950 q^{51} -1.17365 q^{52} -13.1686 q^{53} -0.519010 q^{54} -9.95484 q^{55} -0.719708 q^{56} -0.00125817 q^{57} -5.12559 q^{58} -15.0453 q^{59} -0.214358 q^{60} -11.4828 q^{61} -7.34428 q^{62} -2.15372 q^{63} +1.00000 q^{64} -2.90477 q^{65} -0.348361 q^{66} +9.75512 q^{67} +0.882057 q^{68} +0.438294 q^{69} -1.78127 q^{70} +14.9650 q^{71} +2.99250 q^{72} -11.9589 q^{73} -5.66296 q^{74} -0.0974845 q^{75} +0.0145269 q^{76} -2.89479 q^{77} -0.101650 q^{78} -8.14549 q^{79} +2.47499 q^{80} +8.93254 q^{81} -7.20437 q^{82} -12.2879 q^{83} -0.0623338 q^{84} +2.18308 q^{85} -2.79702 q^{86} -0.443927 q^{87} +4.02218 q^{88} -17.7820 q^{89} +7.40639 q^{90} -0.844686 q^{91} -5.06055 q^{92} -0.636088 q^{93} -6.06598 q^{94} +0.0359538 q^{95} +0.0866100 q^{96} +0.575278 q^{97} +6.48202 q^{98} +12.0364 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 40 q^{2} - q^{3} + 40 q^{4} - 23 q^{5} + q^{6} + 12 q^{7} - 40 q^{8} + 29 q^{9}+O(q^{10}) $ 40 * q - 40 * q^2 - q^3 + 40 * q^4 - 23 * q^5 + q^6 + 12 * q^7 - 40 * q^8 + 29 * q^9	\( 40 q - 40 q^{2} - q^{3} + 40 q^{4} - 23 q^{5} + q^{6} + 12 q^{7} - 40 q^{8} + 29 q^{9} + 23 q^{10} - 28 q^{11} - q^{12} - 12 q^{13} - 12 q^{14} - 14 q^{15} + 40 q^{16} - 10 q^{17} - 29 q^{18} + 3 q^{19} - 23 q^{20} - 40 q^{21} + 28 q^{22} - 10 q^{23} + q^{24} + 43 q^{25} + 12 q^{26} - 7 q^{27} + 12 q^{28} - 37 q^{29} + 14 q^{30} - 16 q^{31} - 40 q^{32} - 11 q^{33} + 10 q^{34} - 24 q^{35} + 29 q^{36} - 17 q^{37} - 3 q^{38} - 10 q^{39} + 23 q^{40} - 58 q^{41} + 40 q^{42} + 37 q^{43} - 28 q^{44} - 66 q^{45} + 10 q^{46} - 34 q^{47} - q^{48} + 28 q^{49} - 43 q^{50} - 7 q^{51} - 12 q^{52} - 43 q^{53} + 7 q^{54} + 28 q^{55} - 12 q^{56} - 25 q^{57} + 37 q^{58} - 92 q^{59} - 14 q^{60} - 37 q^{61} + 16 q^{62} + 14 q^{63} + 40 q^{64} - 54 q^{65} + 11 q^{66} - 10 q^{67} - 10 q^{68} - 49 q^{69} + 24 q^{70} - 87 q^{71} - 29 q^{72} + 12 q^{73} + 17 q^{74} - 31 q^{75} + 3 q^{76} - 53 q^{77} + 10 q^{78} + 14 q^{79} - 23 q^{80} - 4 q^{81} + 58 q^{82} - 42 q^{83} - 40 q^{84} - 24 q^{85} - 37 q^{86} + 24 q^{87} + 28 q^{88} - 118 q^{89} + 66 q^{90} - 35 q^{91} - 10 q^{92} - 22 q^{93} + 34 q^{94} - 18 q^{95} + q^{96} + 2 q^{97} - 28 q^{98} - 48 q^{99}+O(q^{100}) \) 40 * q - 40 * q^2 - q^3 + 40 * q^4 - 23 * q^5 + q^6 + 12 * q^7 - 40 * q^8 + 29 * q^9 + 23 * q^10 - 28 * q^11 - q^12 - 12 * q^13 - 12 * q^14 - 14 * q^15 + 40 * q^16 - 10 * q^17 - 29 * q^18 + 3 * q^19 - 23 * q^20 - 40 * q^21 + 28 * q^22 - 10 * q^23 + q^24 + 43 * q^25 + 12 * q^26 - 7 * q^27 + 12 * q^28 - 37 * q^29 + 14 * q^30 - 16 * q^31 - 40 * q^32 - 11 * q^33 + 10 * q^34 - 24 * q^35 + 29 * q^36 - 17 * q^37 - 3 * q^38 - 10 * q^39 + 23 * q^40 - 58 * q^41 + 40 * q^42 + 37 * q^43 - 28 * q^44 - 66 * q^45 + 10 * q^46 - 34 * q^47 - q^48 + 28 * q^49 - 43 * q^50 - 7 * q^51 - 12 * q^52 - 43 * q^53 + 7 * q^54 + 28 * q^55 - 12 * q^56 - 25 * q^57 + 37 * q^58 - 92 * q^59 - 14 * q^60 - 37 * q^61 + 16 * q^62 + 14 * q^63 + 40 * q^64 - 54 * q^65 + 11 * q^66 - 10 * q^67 - 10 * q^68 - 49 * q^69 + 24 * q^70 - 87 * q^71 - 29 * q^72 + 12 * q^73 + 17 * q^74 - 31 * q^75 + 3 * q^76 - 53 * q^77 + 10 * q^78 + 14 * q^79 - 23 * q^80 - 4 * q^81 + 58 * q^82 - 42 * q^83 - 40 * q^84 - 24 * q^85 - 37 * q^86 + 24 * q^87 + 28 * q^88 - 118 * q^89 + 66 * q^90 - 35 * q^91 - 10 * q^92 - 22 * q^93 + 34 * q^94 - 18 * q^95 + q^96 + 2 * q^97 - 28 * q^98 - 48 * 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
$3$	−0.0866100	−0.0500043	−0.0250021	−	0.999687i	$-0.507959\pi$
$3$	−0.0866100	−0.0500043	−0.0250021	+	0.999687i	$0.507959\pi$
$4$	1.00000	0.500000
$5$	2.47499	1.10685	0.553424	−	0.832900i	$-0.313321\pi$
$5$	2.47499	1.10685	0.553424	+	0.832900i	$0.313321\pi$
$6$	0.0866100	0.0353584
$7$	0.719708	0.272024	0.136012	−	0.990707i	$-0.456571\pi$
$7$	0.719708	0.272024	0.136012	+	0.990707i	$0.456571\pi$
$8$	−1.00000	−0.353553
$9$	−2.99250	−0.997500
$10$	−2.47499	−0.782659
$11$	−4.02218	−1.21273	−0.606366	−	0.795185i	$-0.707373\pi$
$11$	−4.02218	−1.21273	−0.606366	+	0.795185i	$0.707373\pi$
$12$	−0.0866100	−0.0250021
$13$	−1.17365	−0.325512	−0.162756	−	0.986666i	$-0.552038\pi$
$13$	−1.17365	−0.325512	−0.162756	+	0.986666i	$0.552038\pi$
$14$	−0.719708	−0.192350
$15$	−0.214358	−0.0553471
$16$	1.00000	0.250000
$17$	0.882057	0.213930	0.106965	−	0.994263i	$-0.465887\pi$
$17$	0.882057	0.213930	0.106965	+	0.994263i	$0.465887\pi$
$18$	2.99250	0.705339
$19$	0.0145269	0.00333269	0.00166635	−	0.999999i	$-0.499470\pi$
$19$	0.0145269	0.00333269	0.00166635	+	0.999999i	$0.499470\pi$
$20$	2.47499	0.553424
$21$	−0.0623338	−0.0136024
$22$	4.02218	0.857532
$23$	−5.06055	−1.05520	−0.527599	−	0.849493i	$-0.676908\pi$
$23$	−5.06055	−1.05520	−0.527599	+	0.849493i	$0.676908\pi$
$24$	0.0866100	0.0176792
$25$	1.12556	0.225111
$26$	1.17365	0.230172
$27$	0.519010	0.0998835
$28$	0.719708	0.136012
$29$	5.12559	0.951799	0.475899	−	0.879500i	$-0.342123\pi$
$29$	5.12559	0.951799	0.475899	+	0.879500i	$0.342123\pi$
$30$	0.214358	0.0391363
$31$	7.34428	1.31907	0.659536	−	0.751673i	$-0.270753\pi$
$31$	7.34428	1.31907	0.659536	+	0.751673i	$0.270753\pi$
$32$	−1.00000	−0.176777
$33$	0.348361	0.0606418
$34$	−0.882057	−0.151272
$35$	1.78127	0.301089
$36$	−2.99250	−0.498750
$37$	5.66296	0.930985	0.465492	−	0.885052i	$-0.345877\pi$
$37$	5.66296	0.930985	0.465492	+	0.885052i	$0.345877\pi$
$38$	−0.0145269	−0.00235657
$39$	0.101650	0.0162770
$40$	−2.47499	−0.391330
$41$	7.20437	1.12513	0.562567	−	0.826752i	$-0.309814\pi$
$41$	7.20437	1.12513	0.562567	+	0.826752i	$0.309814\pi$
$42$	0.0623338	0.00961832
$43$	2.79702	0.426541	0.213271	−	0.976993i	$-0.431588\pi$
$43$	2.79702	0.426541	0.213271	+	0.976993i	$0.431588\pi$
$44$	−4.02218	−0.606366
$45$	−7.40639	−1.10408
$46$	5.06055	0.746138
$47$	6.06598	0.884814	0.442407	−	0.896814i	$-0.354125\pi$
$47$	6.06598	0.884814	0.442407	+	0.896814i	$0.354125\pi$
$48$	−0.0866100	−0.0125011
$49$	−6.48202	−0.926003
$50$	−1.12556	−0.159178
$51$	−0.0763950	−0.0106974
$52$	−1.17365	−0.162756
$53$	−13.1686	−1.80885	−0.904424	−	0.426635i	$-0.859699\pi$
$53$	−13.1686	−1.80885	−0.904424	+	0.426635i	$0.859699\pi$
$54$	−0.519010	−0.0706283
$55$	−9.95484	−1.34231
$56$	−0.719708	−0.0961750
$57$	−0.00125817	−0.000166649 0
$58$	−5.12559	−0.673023
$59$	−15.0453	−1.95873	−0.979363	−	0.202107i	$-0.935221\pi$
$59$	−15.0453	−1.95873	−0.979363	+	0.202107i	$0.935221\pi$
$60$	−0.214358	−0.0276736
$61$	−11.4828	−1.47022	−0.735110	−	0.677948i	$-0.762869\pi$
$61$	−11.4828	−1.47022	−0.735110	+	0.677948i	$0.762869\pi$
$62$	−7.34428	−0.932725
$63$	−2.15372	−0.271344
$64$	1.00000	0.125000
$65$	−2.90477	−0.360293
$66$	−0.348361	−0.0428803
$67$	9.75512	1.19178	0.595889	−	0.803067i	$-0.296800\pi$
$67$	9.75512	1.19178	0.595889	+	0.803067i	$0.296800\pi$
$68$	0.882057	0.106965
$69$	0.438294	0.0527644
$70$	−1.78127	−0.212902
$71$	14.9650	1.77601	0.888007	−	0.459831i	$-0.152090\pi$
$71$	14.9650	1.77601	0.888007	+	0.459831i	$0.152090\pi$
$72$	2.99250	0.352669
$73$	−11.9589	−1.39969	−0.699845	−	0.714295i	$-0.746747\pi$
$73$	−11.9589	−1.39969	−0.699845	+	0.714295i	$0.746747\pi$
$74$	−5.66296	−0.658306
$75$	−0.0974845	−0.0112565
$76$	0.0145269	0.00166635
$77$	−2.89479	−0.329892
$78$	−0.101650	−0.0115096
$79$	−8.14549	−0.916439	−0.458220	−	0.888839i	$-0.651513\pi$
$79$	−8.14549	−0.916439	−0.458220	+	0.888839i	$0.651513\pi$
$80$	2.47499	0.276712
$81$	8.93254	0.992505
$82$	−7.20437	−0.795589
$83$	−12.2879	−1.34877	−0.674386	−	0.738379i	$-0.735591\pi$
$83$	−12.2879	−1.34877	−0.674386	+	0.738379i	$0.735591\pi$
$84$	−0.0623338	−0.00680118
$85$	2.18308	0.236788
$86$	−2.79702	−0.301610
$87$	−0.443927	−0.0475940
$88$	4.02218	0.428766
$89$	−17.7820	−1.88489	−0.942445	−	0.334360i	$-0.891480\pi$
$89$	−17.7820	−1.88489	−0.942445	+	0.334360i	$0.891480\pi$
$90$	7.40639	0.780702
$91$	−0.844686	−0.0885471
$92$	−5.06055	−0.527599
$93$	−0.636088	−0.0659593
$94$	−6.06598	−0.625658
$95$	0.0359538	0.00368878
$96$	0.0866100	0.00883959
$97$	0.575278	0.0584106	0.0292053	−	0.999573i	$-0.490702\pi$
$97$	0.575278	0.0584106	0.0292053	+	0.999573i	$0.490702\pi$
$98$	6.48202	0.654783
$99$	12.0364	1.20970
$100$	1.12556	0.112556
$101$	−11.5157	−1.14585	−0.572926	−	0.819607i	$-0.694192\pi$
$101$	−11.5157	−1.14585	−0.572926	+	0.819607i	$0.694192\pi$
$102$	0.0763950	0.00756423
$103$	0.875520	0.0862675	0.0431338	−	0.999069i	$-0.486266\pi$
$103$	0.875520	0.0862675	0.0431338	+	0.999069i	$0.486266\pi$
$104$	1.17365	0.115086
$105$	−0.154275	−0.0150557
$106$	13.1686	1.27905
$107$	−2.63094	−0.254343	−0.127171	−	0.991881i	$-0.540590\pi$
$107$	−2.63094	−0.254343	−0.127171	+	0.991881i	$0.540590\pi$
$108$	0.519010	0.0499418
$109$	−10.4638	−1.00225	−0.501123	−	0.865376i	$-0.667080\pi$
$109$	−10.4638	−1.00225	−0.501123	+	0.865376i	$0.667080\pi$
$110$	9.95484	0.949157
$111$	−0.490469	−0.0465532
$112$	0.719708	0.0680060
$113$	−7.93666	−0.746618	−0.373309	−	0.927707i	$-0.621777\pi$
$113$	−7.93666	−0.746618	−0.373309	+	0.927707i	$0.621777\pi$
$114$	0.00125817	0.000117839 0
$115$	−12.5248	−1.16794
$116$	5.12559	0.475899
$117$	3.51215	0.324698
$118$	15.0453	1.38503
$119$	0.634823	0.0581942
$120$	0.214358	0.0195682
$121$	5.17793	0.470721
$122$	11.4828	1.03960
$123$	−0.623970	−0.0562615
$124$	7.34428	0.659536
$125$	−9.58919	−0.857683
$126$	2.15372	0.191869
$127$	10.7592	0.954727	0.477363	−	0.878706i	$-0.341593\pi$
$127$	10.7592	0.954727	0.477363	+	0.878706i	$0.341593\pi$
$128$	−1.00000	−0.0883883
$129$	−0.242250	−0.0213289
$130$	2.90477	0.254765
$131$	−5.75133	−0.502496	−0.251248	−	0.967923i	$-0.580841\pi$
$131$	−5.75133	−0.502496	−0.251248	+	0.967923i	$0.580841\pi$
$132$	0.348361	0.0303209
$133$	0.0104551	0.000906572 0
$134$	−9.75512	−0.842714
$135$	1.28454	0.110556
$136$	−0.882057	−0.0756358
$137$	−7.35468	−0.628353	−0.314176	−	0.949365i	$-0.601728\pi$
$137$	−7.35468	−0.628353	−0.314176	+	0.949365i	$0.601728\pi$
$138$	−0.438294	−0.0373101
$139$	−8.55859	−0.725930	−0.362965	−	0.931803i	$-0.618236\pi$
$139$	−8.55859	−0.725930	−0.362965	+	0.931803i	$0.618236\pi$
$140$	1.78127	0.150544
$141$	−0.525374	−0.0442445
$142$	−14.9650	−1.25583
$143$	4.72064	0.394760
$144$	−2.99250	−0.249375
$145$	12.6858	1.05350
$146$	11.9589	0.989730
$147$	0.561408	0.0463041
$148$	5.66296	0.465492
$149$	3.19817	0.262004	0.131002	−	0.991382i	$-0.458181\pi$
$149$	3.19817	0.262004	0.131002	+	0.991382i	$0.458181\pi$
$150$	0.0974845	0.00795957
$151$	−4.40206	−0.358234	−0.179117	−	0.983828i	$-0.557324\pi$
$151$	−4.40206	−0.358234	−0.179117	+	0.983828i	$0.557324\pi$
$152$	−0.0145269	−0.00117828
$153$	−2.63956	−0.213395
$154$	2.89479	0.233269
$155$	18.1770	1.46001
$156$	0.101650	0.00813851
$157$	−22.5756	−1.80173	−0.900866	−	0.434097i	$-0.857067\pi$
$157$	−22.5756	−1.80173	−0.900866	+	0.434097i	$0.857067\pi$
$158$	8.14549	0.648020
$159$	1.14053	0.0904501
$160$	−2.47499	−0.195665
$161$	−3.64212	−0.287039
$162$	−8.93254	−0.701807
$163$	−7.40401	−0.579927	−0.289964	−	0.957038i	$-0.593643\pi$
$163$	−7.40401	−0.579927	−0.289964	+	0.957038i	$0.593643\pi$
$164$	7.20437	0.562567
$165$	0.862188	0.0671213
$166$	12.2879	0.953725
$167$	−21.9499	−1.69853	−0.849267	−	0.527963i	$-0.822956\pi$
$167$	−21.9499	−1.69853	−0.849267	+	0.527963i	$0.822956\pi$
$168$	0.0623338	0.00480916
$169$	−11.6225	−0.894042
$170$	−2.18308	−0.167435
$171$	−0.0434716	−0.00332436
$172$	2.79702	0.213271
$173$	−4.99367	−0.379661	−0.189831	−	0.981817i	$-0.560794\pi$
$173$	−4.99367	−0.379661	−0.189831	+	0.981817i	$0.560794\pi$
$174$	0.443927	0.0336541
$175$	0.810072	0.0612357
$176$	−4.02218	−0.303183
$177$	1.30307	0.0979447
$178$	17.7820	1.33282
$179$	5.25556	0.392819	0.196410	−	0.980522i	$-0.437072\pi$
$179$	5.25556	0.392819	0.196410	+	0.980522i	$0.437072\pi$
$180$	−7.40639	−0.552040
$181$	17.6280	1.31028	0.655139	−	0.755508i	$-0.272610\pi$
$181$	17.6280	1.31028	0.655139	+	0.755508i	$0.272610\pi$
$182$	0.844686	0.0626123
$183$	0.994523	0.0735173
$184$	5.06055	0.373069
$185$	14.0157	1.03046
$186$	0.636088	0.0466403
$187$	−3.54779	−0.259440
$188$	6.06598	0.442407
$189$	0.373535	0.0271707
$190$	−0.0359538	−0.00260836
$191$	−0.288421	−0.0208694	−0.0104347	−	0.999946i	$-0.503322\pi$
$191$	−0.288421	−0.0208694	−0.0104347	+	0.999946i	$0.503322\pi$
$192$	−0.0866100	−0.00625054
$193$	−2.30616	−0.166001	−0.0830006	−	0.996549i	$-0.526450\pi$
$193$	−2.30616	−0.166001	−0.0830006	+	0.996549i	$0.526450\pi$
$194$	−0.575278	−0.0413025
$195$	0.251582	0.0180162
$196$	−6.48202	−0.463002
$197$	16.6893	1.18906	0.594530	−	0.804073i	$-0.297338\pi$
$197$	16.6893	1.18906	0.594530	+	0.804073i	$0.297338\pi$
$198$	−12.0364	−0.855387
$199$	13.6845	0.970067	0.485034	−	0.874496i	$-0.338807\pi$
$199$	13.6845	0.970067	0.485034	+	0.874496i	$0.338807\pi$
$200$	−1.12556	−0.0795889
$201$	−0.844890	−0.0595940
$202$	11.5157	0.810240
$203$	3.68893	0.258912
$204$	−0.0763950	−0.00534872
$205$	17.8307	1.24535
$206$	−0.875520	−0.0610003
$207$	15.1437	1.05256
$208$	−1.17365	−0.0813781
$209$	−0.0584297	−0.00404166
$210$	0.154275	0.0106460
$211$	−0.344938	−0.0237465	−0.0118732	−	0.999930i	$-0.503779\pi$
$211$	−0.344938	−0.0237465	−0.0118732	+	0.999930i	$0.503779\pi$
$212$	−13.1686	−0.904424
$213$	−1.29611	−0.0888083
$214$	2.63094	0.179847
$215$	6.92258	0.472116
$216$	−0.519010	−0.0353142
$217$	5.28574	0.358819
$218$	10.4638	0.708695
$219$	1.03576	0.0699904
$220$	−9.95484	−0.671155
$221$	−1.03523	−0.0696370
$222$	0.490469	0.0329181
$223$	8.24741	0.552288	0.276144	−	0.961116i	$-0.410943\pi$
$223$	8.24741	0.552288	0.276144	+	0.961116i	$0.410943\pi$
$224$	−0.719708	−0.0480875
$225$	−3.36823	−0.224549
$226$	7.93666	0.527939
$227$	15.9491	1.05858	0.529291	−	0.848441i	$-0.322458\pi$
$227$	15.9491	1.05858	0.529291	+	0.848441i	$0.322458\pi$
$228$	−0.00125817	−8.33244e−5 0
$229$	17.6910	1.16906	0.584528	−	0.811374i	$-0.301280\pi$
$229$	17.6910	1.16906	0.584528	+	0.811374i	$0.301280\pi$
$230$	12.5248	0.825861
$231$	0.250718	0.0164960
$232$	−5.12559	−0.336512
$233$	4.52282	0.296300	0.148150	−	0.988965i	$-0.452668\pi$
$233$	4.52282	0.296300	0.148150	+	0.988965i	$0.452668\pi$
$234$	−3.51215	−0.229596
$235$	15.0132	0.979354
$236$	−15.0453	−0.979363
$237$	0.705481	0.0458259
$238$	−0.634823	−0.0411495
$239$	10.2411	0.662440	0.331220	−	0.943554i	$-0.392540\pi$
$239$	10.2411	0.662440	0.331220	+	0.943554i	$0.392540\pi$
$240$	−0.214358	−0.0138368
$241$	13.4817	0.868430	0.434215	−	0.900809i	$-0.357026\pi$
$241$	13.4817	0.868430	0.434215	+	0.900809i	$0.357026\pi$
$242$	−5.17793	−0.332850
$243$	−2.33068	−0.149513
$244$	−11.4828	−0.735110
$245$	−16.0429	−1.02494
$246$	0.623970	0.0397829
$247$	−0.0170495	−0.00108483
$248$	−7.34428	−0.466363
$249$	1.06425	0.0674443
$250$	9.58919	0.606474
$251$	−13.3357	−0.841743	−0.420871	−	0.907120i	$-0.638276\pi$
$251$	−13.3357	−0.841743	−0.420871	+	0.907120i	$0.638276\pi$
$252$	−2.15372	−0.135672
$253$	20.3545	1.27967
$254$	−10.7592	−0.675094
$255$	−0.189076	−0.0118404
$256$	1.00000	0.0625000
$257$	−23.0432	−1.43739	−0.718697	−	0.695324i	$-0.755261\pi$
$257$	−23.0432	−1.43739	−0.718697	+	0.695324i	$0.755261\pi$
$258$	0.242250	0.0150818
$259$	4.07567	0.253250
$260$	−2.90477	−0.180146
$261$	−15.3383	−0.949419
$262$	5.75133	0.355318
$263$	−17.2883	−1.06604	−0.533021	−	0.846102i	$-0.678943\pi$
$263$	−17.2883	−1.06604	−0.533021	+	0.846102i	$0.678943\pi$
$264$	−0.348361	−0.0214401
$265$	−32.5921	−2.00212
$266$	−0.0104551	−0.000641043 0
$267$	1.54010	0.0942526
$268$	9.75512	0.595889
$269$	−10.7977	−0.658346	−0.329173	−	0.944270i	$-0.606770\pi$
$269$	−10.7977	−0.658346	−0.329173	+	0.944270i	$0.606770\pi$
$270$	−1.28454	−0.0781748
$271$	−8.91559	−0.541584	−0.270792	−	0.962638i	$-0.587285\pi$
$271$	−8.91559	−0.541584	−0.270792	+	0.962638i	$0.587285\pi$
$272$	0.882057	0.0534826
$273$	0.0731582	0.00442774
$274$	7.35468	0.444312
$275$	−4.52719	−0.273000
$276$	0.438294	0.0263822
$277$	4.51833	0.271480	0.135740	−	0.990745i	$-0.456659\pi$
$277$	4.51833	0.271480	0.135740	+	0.990745i	$0.456659\pi$
$278$	8.55859	0.513310
$279$	−21.9778	−1.31577
$280$	−1.78127	−0.106451
$281$	14.9366	0.891040	0.445520	−	0.895272i	$-0.353019\pi$
$281$	14.9366	0.891040	0.445520	+	0.895272i	$0.353019\pi$
$282$	0.525374	0.0312856
$283$	−15.6586	−0.930805	−0.465402	−	0.885099i	$-0.654090\pi$
$283$	−15.6586	−0.930805	−0.465402	+	0.885099i	$0.654090\pi$
$284$	14.9650	0.888007
$285$	−0.00311396	−0.000184455 0
$286$	−4.72064	−0.279137
$287$	5.18504	0.306063
$288$	2.99250	0.176335
$289$	−16.2220	−0.954234
$290$	−12.6858	−0.744934
$291$	−0.0498248	−0.00292078
$292$	−11.9589	−0.699845
$293$	23.7676	1.38852	0.694258	−	0.719726i	$-0.255733\pi$
$293$	23.7676	1.38852	0.694258	+	0.719726i	$0.255733\pi$
$294$	−0.561408	−0.0327420
$295$	−37.2368	−2.16801
$296$	−5.66296	−0.329153
$297$	−2.08755	−0.121132
$298$	−3.19817	−0.185265
$299$	5.93933	0.343480
$300$	−0.0974845	−0.00562827
$301$	2.01304	0.116029
$302$	4.40206	0.253310
$303$	0.997372	0.0572975
$304$	0.0145269	0.000833173 0
$305$	−28.4197	−1.62731
$306$	2.63956	0.150893
$307$	−8.32429	−0.475092	−0.237546	−	0.971376i	$-0.576343\pi$
$307$	−8.32429	−0.475092	−0.237546	+	0.971376i	$0.576343\pi$
$308$	−2.89479	−0.164946
$309$	−0.0758287	−0.00431374
$310$	−18.1770	−1.03238
$311$	−11.8177	−0.670119	−0.335059	−	0.942197i	$-0.608756\pi$
$311$	−11.8177	−0.670119	−0.335059	+	0.942197i	$0.608756\pi$
$312$	−0.101650	−0.00575479
$313$	−12.3763	−0.699551	−0.349775	−	0.936834i	$-0.613742\pi$
$313$	−12.3763	−0.699551	−0.349775	+	0.936834i	$0.613742\pi$
$314$	22.5756	1.27402
$315$	−5.33044	−0.300336
$316$	−8.14549	−0.458220
$317$	0.620603	0.0348566	0.0174283	−	0.999848i	$-0.494452\pi$
$317$	0.620603	0.0348566	0.0174283	+	0.999848i	$0.494452\pi$
$318$	−1.14053	−0.0639579
$319$	−20.6161	−1.15428
$320$	2.47499	0.138356
$321$	0.227866	0.0127182
$322$	3.64212	0.202967
$323$	0.0128135	0.000712964 0
$324$	8.93254	0.496252
$325$	−1.32101	−0.0732766
$326$	7.40401	0.410070
$327$	0.906266	0.0501166
$328$	−7.20437	−0.397795
$329$	4.36573	0.240691
$330$	−0.862188	−0.0474619
$331$	−9.37988	−0.515565	−0.257783	−	0.966203i	$-0.582992\pi$
$331$	−9.37988	−0.515565	−0.257783	+	0.966203i	$0.582992\pi$
$332$	−12.2879	−0.674386
$333$	−16.9464	−0.928657
$334$	21.9499	1.20105
$335$	24.1438	1.31912
$336$	−0.0623338	−0.00340059
$337$	2.50685	0.136557	0.0682784	−	0.997666i	$-0.478249\pi$
$337$	2.50685	0.136557	0.0682784	+	0.997666i	$0.478249\pi$
$338$	11.6225	0.632183
$339$	0.687394	0.0373341
$340$	2.18308	0.118394
$341$	−29.5400	−1.59968
$342$	0.0434716	0.00235068
$343$	−9.70311	−0.523919
$344$	−2.79702	−0.150805
$345$	1.08477	0.0584022
$346$	4.99367	0.268461
$347$	−0.0754958	−0.00405283	−0.00202641	−	0.999998i	$-0.500645\pi$
$347$	−0.0754958	−0.00405283	−0.00202641	+	0.999998i	$0.500645\pi$
$348$	−0.443927	−0.0237970
$349$	24.9751	1.33689	0.668444	−	0.743763i	$-0.266961\pi$
$349$	24.9751	1.33689	0.668444	+	0.743763i	$0.266961\pi$
$350$	−0.810072	−0.0433002
$351$	−0.609137	−0.0325133
$352$	4.02218	0.214383
$353$	8.13318	0.432886	0.216443	−	0.976295i	$-0.430555\pi$
$353$	8.13318	0.432886	0.216443	+	0.976295i	$0.430555\pi$
$354$	−1.30307	−0.0692574
$355$	37.0381	1.96578
$356$	−17.7820	−0.942445
$357$	−0.0549820	−0.00290996
$358$	−5.25556	−0.277765
$359$	28.6150	1.51024	0.755122	−	0.655584i	$-0.227578\pi$
$359$	28.6150	1.51024	0.755122	+	0.655584i	$0.227578\pi$
$360$	7.40639	0.390351
$361$	−18.9998	−0.999989
$362$	−17.6280	−0.926507
$363$	−0.448460	−0.0235380
$364$	−0.844686	−0.0442736
$365$	−29.5982	−1.54924
$366$	−0.994523	−0.0519846
$367$	−9.09315	−0.474658	−0.237329	−	0.971429i	$-0.576272\pi$
$367$	−9.09315	−0.474658	−0.237329	+	0.971429i	$0.576272\pi$
$368$	−5.06055	−0.263800
$369$	−21.5591	−1.12232
$370$	−14.0157	−0.728644
$371$	−9.47755	−0.492050
$372$	−0.636088	−0.0329796
$373$	−36.6472	−1.89752	−0.948760	−	0.315998i	$-0.897661\pi$
$373$	−36.6472	−1.89752	−0.948760	+	0.315998i	$0.897661\pi$
$374$	3.54779	0.183452
$375$	0.830520	0.0428878
$376$	−6.06598	−0.312829
$377$	−6.01566	−0.309822
$378$	−0.373535	−0.0192126
$379$	−18.0899	−0.929218	−0.464609	−	0.885516i	$-0.653805\pi$
$379$	−18.0899	−0.929218	−0.464609	+	0.885516i	$0.653805\pi$
$380$	0.0359538	0.00184439
$381$	−0.931856	−0.0477404
$382$	0.288421	0.0147569
$383$	27.9297	1.42714	0.713570	−	0.700584i	$-0.247077\pi$
$383$	27.9297	1.42714	0.713570	+	0.700584i	$0.247077\pi$
$384$	0.0866100	0.00441980
$385$	−7.16457	−0.365140
$386$	2.30616	0.117381
$387$	−8.37007	−0.425475
$388$	0.575278	0.0292053
$389$	4.12087	0.208937	0.104468	−	0.994528i	$-0.466686\pi$
$389$	4.12087	0.208937	0.104468	+	0.994528i	$0.466686\pi$
$390$	−0.251582	−0.0127394
$391$	−4.46370	−0.225739
$392$	6.48202	0.327392
$393$	0.498123	0.0251270
$394$	−16.6893	−0.840793
$395$	−20.1600	−1.01436
$396$	12.0364	0.604850
$397$	16.3734	0.821756	0.410878	−	0.911690i	$-0.365222\pi$
$397$	16.3734	0.821756	0.410878	+	0.911690i	$0.365222\pi$
$398$	−13.6845	−0.685941
$399$	−0.000905515 0	−4.53325e−5 0
$400$	1.12556	0.0562779
$401$	−22.5009	−1.12364	−0.561820	−	0.827260i	$-0.689899\pi$
$401$	−22.5009	−1.12364	−0.561820	+	0.827260i	$0.689899\pi$
$402$	0.844890	0.0421393
$403$	−8.61963	−0.429374
$404$	−11.5157	−0.572926
$405$	22.1079	1.09855
$406$	−3.68893	−0.183078
$407$	−22.7774	−1.12904
$408$	0.0763950	0.00378211
$409$	14.9244	0.737966	0.368983	−	0.929436i	$-0.379706\pi$
$409$	14.9244	0.737966	0.368983	+	0.929436i	$0.379706\pi$
$410$	−17.8307	−0.880596
$411$	0.636988	0.0314203
$412$	0.875520	0.0431338
$413$	−10.8282	−0.532820
$414$	−15.1437	−0.744272
$415$	−30.4124	−1.49288
$416$	1.17365	0.0575430
$417$	0.741259	0.0362996
$418$	0.0584297	0.00285789
$419$	−28.6308	−1.39871	−0.699353	−	0.714776i	$-0.746528\pi$
$419$	−28.6308	−1.39871	−0.699353	+	0.714776i	$0.746528\pi$
$420$	−0.154275	−0.00752787
$421$	15.6243	0.761480	0.380740	−	0.924682i	$-0.375669\pi$
$421$	15.6243	0.761480	0.380740	+	0.924682i	$0.375669\pi$
$422$	0.344938	0.0167913
$423$	−18.1524	−0.882602
$424$	13.1686	0.639524
$425$	0.992806	0.0481582
$426$	1.29611	0.0627969
$427$	−8.26425	−0.399935
$428$	−2.63094	−0.127171
$429$	−0.408854	−0.0197397
$430$	−6.92258	−0.333837
$431$	22.3042	1.07436	0.537178	−	0.843469i	$-0.319490\pi$
$431$	22.3042	1.07436	0.537178	+	0.843469i	$0.319490\pi$
$432$	0.519010	0.0249709
$433$	35.6307	1.71230	0.856152	−	0.516725i	$-0.172849\pi$
$433$	35.6307	1.71230	0.856152	+	0.516725i	$0.172849\pi$
$434$	−5.28574	−0.253724
$435$	−1.09871	−0.0526793
$436$	−10.4638	−0.501123
$437$	−0.0735140	−0.00351665
$438$	−1.03576	−0.0494907
$439$	37.7141	1.80000	0.899999	−	0.435893i	$-0.143567\pi$
$439$	37.7141	1.80000	0.899999	+	0.435893i	$0.143567\pi$
$440$	9.95484	0.474578
$441$	19.3974	0.923688
$442$	1.03523	0.0492408
$443$	18.8613	0.896128	0.448064	−	0.894001i	$-0.352114\pi$
$443$	18.8613	0.896128	0.448064	+	0.894001i	$0.352114\pi$
$444$	−0.490469	−0.0232766
$445$	−44.0103	−2.08629
$446$	−8.24741	−0.390526
$447$	−0.276993	−0.0131013
$448$	0.719708	0.0340030
$449$	−33.1802	−1.56587	−0.782935	−	0.622103i	$-0.786279\pi$
$449$	−33.1802	−1.56587	−0.782935	+	0.622103i	$0.786279\pi$
$450$	3.36823	0.158780
$451$	−28.9773	−1.36449
$452$	−7.93666	−0.373309
$453$	0.381262	0.0179133
$454$	−15.9491	−0.748530
$455$	−2.09059	−0.0980082
$456$	0.00125817	5.89193e−5 0
$457$	2.62314	0.122705	0.0613526	−	0.998116i	$-0.480459\pi$
$457$	2.62314	0.122705	0.0613526	+	0.998116i	$0.480459\pi$
$458$	−17.6910	−0.826647
$459$	0.457797	0.0213681
$460$	−12.5248	−0.583972
$461$	−12.1494	−0.565852	−0.282926	−	0.959142i	$-0.591305\pi$
$461$	−12.1494	−0.565852	−0.282926	+	0.959142i	$0.591305\pi$
$462$	−0.250718	−0.0116645
$463$	9.59647	0.445986	0.222993	−	0.974820i	$-0.428417\pi$
$463$	9.59647	0.445986	0.222993	+	0.974820i	$0.428417\pi$
$464$	5.12559	0.237950
$465$	−1.57431	−0.0730069
$466$	−4.52282	−0.209516
$467$	−29.1248	−1.34773	−0.673867	−	0.738853i	$-0.735368\pi$
$467$	−29.1248	−1.34773	−0.673867	+	0.738853i	$0.735368\pi$
$468$	3.51215	0.162349
$469$	7.02083	0.324192
$470$	−15.0132	−0.692508
$471$	1.95528	0.0900943
$472$	15.0453	0.692514
$473$	−11.2501	−0.517281
$474$	−0.705481	−0.0324038
$475$	0.0163508	0.000750227 0
$476$	0.634823	0.0290971
$477$	39.4070	1.80432
$478$	−10.2411	−0.468416
$479$	−9.68670	−0.442597	−0.221298	−	0.975206i	$-0.571029\pi$
$479$	−9.68670	−0.442597	−0.221298	+	0.975206i	$0.571029\pi$
$480$	0.214358	0.00978408
$481$	−6.64634	−0.303047
$482$	−13.4817	−0.614073
$483$	0.315444	0.0143532
$484$	5.17793	0.235360
$485$	1.42380	0.0646516
$486$	2.33068	0.105722
$487$	36.6179	1.65931	0.829657	−	0.558274i	$-0.188536\pi$
$487$	36.6179	1.65931	0.829657	+	0.558274i	$0.188536\pi$
$488$	11.4828	0.519801
$489$	0.641261	0.0289988
$490$	16.0429	0.724745
$491$	19.3351	0.872581	0.436291	−	0.899806i	$-0.356292\pi$
$491$	19.3351	0.872581	0.436291	+	0.899806i	$0.356292\pi$
$492$	−0.623970	−0.0281307
$493$	4.52107	0.203619
$494$	0.0170495	0.000767092 0
$495$	29.7898	1.33895
$496$	7.34428	0.329768
$497$	10.7704	0.483118
$498$	−1.06425	−0.0476903
$499$	36.7305	1.64428	0.822141	−	0.569284i	$-0.192779\pi$
$499$	36.7305	1.64428	0.822141	+	0.569284i	$0.192779\pi$
$500$	−9.58919	−0.428842
$501$	1.90108	0.0849340
$502$	13.3357	0.595202
$503$	11.9816	0.534235	0.267117	−	0.963664i	$-0.413929\pi$
$503$	11.9816	0.534235	0.267117	+	0.963664i	$0.413929\pi$
$504$	2.15372	0.0959345
$505$	−28.5011	−1.26828
$506$	−20.3545	−0.904866
$507$	1.00663	0.0447059
$508$	10.7592	0.477363
$509$	5.65869	0.250817	0.125409	−	0.992105i	$-0.459976\pi$
$509$	5.65869	0.250817	0.125409	+	0.992105i	$0.459976\pi$
$510$	0.189076	0.00837245
$511$	−8.60695	−0.380749
$512$	−1.00000	−0.0441942
$513$	0.00753959	0.000332881 0
$514$	23.0432	1.01639
$515$	2.16690	0.0954850
$516$	−0.242250	−0.0106644
$517$	−24.3985	−1.07304
$518$	−4.07567	−0.179075
$519$	0.432501	0.0189847
$520$	2.90477	0.127383
$521$	29.3943	1.28779	0.643894	−	0.765114i	$-0.277318\pi$
$521$	29.3943	1.28779	0.643894	+	0.765114i	$0.277318\pi$
$522$	15.3383	0.671341
$523$	33.7248	1.47468	0.737342	−	0.675520i	$-0.236081\pi$
$523$	33.7248	1.47468	0.737342	+	0.675520i	$0.236081\pi$
$524$	−5.75133	−0.251248
$525$	−0.0701603	−0.00306205
$526$	17.2883	0.753805
$527$	6.47808	0.282190
$528$	0.348361	0.0151605
$529$	2.60920	0.113443
$530$	32.5921	1.41571
$531$	45.0229	1.95383
$532$	0.0104551	0.000453286 0
$533$	−8.45542	−0.366245
$534$	−1.54010	−0.0666467
$535$	−6.51155	−0.281519
$536$	−9.75512	−0.421357
$537$	−0.455184	−0.0196426
$538$	10.7977	0.465521
$539$	26.0719	1.12299
$540$	1.28454	0.0552779
$541$	12.0531	0.518201	0.259101	−	0.965850i	$-0.416574\pi$
$541$	12.0531	0.518201	0.259101	+	0.965850i	$0.416574\pi$
$542$	8.91559	0.382957
$543$	−1.52676	−0.0655195
$544$	−0.882057	−0.0378179
$545$	−25.8977	−1.10933
$546$	−0.0731582	−0.00313088
$547$	−8.05868	−0.344564	−0.172282	−	0.985048i	$-0.555114\pi$
$547$	−8.05868	−0.344564	−0.172282	+	0.985048i	$0.555114\pi$
$548$	−7.35468	−0.314176
$549$	34.3622	1.46654
$550$	4.52719	0.193040
$551$	0.0744588	0.00317205
$552$	−0.438294	−0.0186550
$553$	−5.86237	−0.249293
$554$	−4.51833	−0.191965
$555$	−1.21390	−0.0515273
$556$	−8.55859	−0.362965
$557$	−33.2645	−1.40946	−0.704731	−	0.709475i	$-0.748932\pi$
$557$	−33.2645	−1.40946	−0.704731	+	0.709475i	$0.748932\pi$
$558$	21.9778	0.930393
$559$	−3.28272	−0.138844
$560$	1.78127	0.0752722
$561$	0.307274	0.0129731
$562$	−14.9366	−0.630060
$563$	−19.7919	−0.834131	−0.417065	−	0.908876i	$-0.636941\pi$
$563$	−19.7919	−0.834131	−0.417065	+	0.908876i	$0.636941\pi$
$564$	−0.525374	−0.0221222
$565$	−19.6431	−0.826392
$566$	15.6586	0.658178
$567$	6.42882	0.269985
$568$	−14.9650	−0.627916
$569$	−12.6053	−0.528443	−0.264222	−	0.964462i	$-0.585115\pi$
$569$	−12.6053	−0.528443	−0.264222	+	0.964462i	$0.585115\pi$
$570$	0.00311396	0.000130429 0
$571$	−14.3011	−0.598482	−0.299241	−	0.954178i	$-0.596733\pi$
$571$	−14.3011	−0.598482	−0.299241	+	0.954178i	$0.596733\pi$
$572$	4.72064	0.197380
$573$	0.0249801	0.00104356
$574$	−5.18504	−0.216419
$575$	−5.69594	−0.237537
$576$	−2.99250	−0.124687
$577$	−4.21658	−0.175539	−0.0877693	−	0.996141i	$-0.527974\pi$
$577$	−4.21658	−0.175539	−0.0877693	+	0.996141i	$0.527974\pi$
$578$	16.2220	0.674745
$579$	0.199737	0.00830077
$580$	12.6858	0.526748
$581$	−8.84369	−0.366898
$582$	0.0498248	0.00206530
$583$	52.9665	2.19365
$584$	11.9589	0.494865
$585$	8.69253	0.359392
$586$	−23.7676	−0.981829
$587$	2.92899	0.120892	0.0604462	−	0.998171i	$-0.480748\pi$
$587$	2.92899	0.120892	0.0604462	+	0.998171i	$0.480748\pi$
$588$	0.561408	0.0231521
$589$	0.106689	0.00439606
$590$	37.2368	1.53302
$591$	−1.44546	−0.0594581
$592$	5.66296	0.232746
$593$	−33.9955	−1.39603	−0.698013	−	0.716085i	$-0.745932\pi$
$593$	−33.9955	−1.39603	−0.698013	+	0.716085i	$0.745932\pi$
$594$	2.08755	0.0856533
$595$	1.57118	0.0644121
$596$	3.19817	0.131002
$597$	−1.18521	−0.0485075
$598$	−5.93933	−0.242877
$599$	−20.2876	−0.828928	−0.414464	−	0.910066i	$-0.636031\pi$
$599$	−20.2876	−0.828928	−0.414464	+	0.910066i	$0.636031\pi$
$600$	0.0974845	0.00397979
$601$	−23.9482	−0.976868	−0.488434	−	0.872601i	$-0.662432\pi$
$601$	−23.9482	−0.976868	−0.488434	+	0.872601i	$0.662432\pi$
$602$	−2.01304	−0.0820452
$603$	−29.1922	−1.18880
$604$	−4.40206	−0.179117
$605$	12.8153	0.521016
$606$	−0.997372	−0.0405155
$607$	11.6635	0.473409	0.236704	−	0.971582i	$-0.423933\pi$
$607$	11.6635	0.473409	0.236704	+	0.971582i	$0.423933\pi$
$608$	−0.0145269	−0.000589142 0
$609$	−0.319498	−0.0129467
$610$	28.4197	1.15068
$611$	−7.11935	−0.288018
$612$	−2.63956	−0.106698
$613$	−10.2199	−0.412779	−0.206389	−	0.978470i	$-0.566171\pi$
$613$	−10.2199	−0.412779	−0.206389	+	0.978470i	$0.566171\pi$
$614$	8.32429	0.335941
$615$	−1.54432	−0.0622729
$616$	2.89479	0.116635
$617$	−45.4631	−1.83028	−0.915138	−	0.403140i	$-0.867919\pi$
$617$	−45.4631	−1.83028	−0.915138	+	0.403140i	$0.867919\pi$
$618$	0.0758287	0.00305028
$619$	−13.1784	−0.529683	−0.264842	−	0.964292i	$-0.585320\pi$
$619$	−13.1784	−0.529683	−0.264842	+	0.964292i	$0.585320\pi$
$620$	18.1770	0.730006
$621$	−2.62648	−0.105397
$622$	11.8177	0.473845
$623$	−12.7979	−0.512735
$624$	0.101650	0.00406925
$625$	−29.3609	−1.17444
$626$	12.3763	0.494657
$627$	0.00506059	0.000202101 0
$628$	−22.5756	−0.900866
$629$	4.99506	0.199166
$630$	5.33044	0.212370
$631$	−6.51397	−0.259317	−0.129658	−	0.991559i	$-0.541388\pi$
$631$	−6.51397	−0.259317	−0.129658	+	0.991559i	$0.541388\pi$
$632$	8.14549	0.324010
$633$	0.0298750	0.00118743
$634$	−0.620603	−0.0246473
$635$	26.6289	1.05674
$636$	1.14053	0.0452251
$637$	7.60763	0.301425
$638$	20.6161	0.816197
$639$	−44.7826	−1.77157
$640$	−2.47499	−0.0978324
$641$	18.2644	0.721402	0.360701	−	0.932682i	$-0.382538\pi$
$641$	18.2644	0.721402	0.360701	+	0.932682i	$0.382538\pi$
$642$	−0.227866	−0.00899314
$643$	27.6039	1.08859	0.544296	−	0.838893i	$-0.316797\pi$
$643$	27.6039	1.08859	0.544296	+	0.838893i	$0.316797\pi$
$644$	−3.64212	−0.143520
$645$	−0.599565	−0.0236078
$646$	−0.0128135	−0.000504142 0
$647$	31.8749	1.25313	0.626567	−	0.779368i	$-0.284460\pi$
$647$	31.8749	1.25313	0.626567	+	0.779368i	$0.284460\pi$
$648$	−8.93254	−0.350903
$649$	60.5148	2.37541
$650$	1.32101	0.0518144
$651$	−0.457797	−0.0179425
$652$	−7.40401	−0.289964
$653$	−17.2519	−0.675119	−0.337560	−	0.941304i	$-0.609601\pi$
$653$	−17.2519	−0.675119	−0.337560	+	0.941304i	$0.609601\pi$
$654$	−0.906266	−0.0354378
$655$	−14.2345	−0.556187
$656$	7.20437	0.281283
$657$	35.7871	1.39619
$658$	−4.36573	−0.170194
$659$	−1.95689	−0.0762295	−0.0381148	−	0.999273i	$-0.512135\pi$
$659$	−1.95689	−0.0762295	−0.0381148	+	0.999273i	$0.512135\pi$
$660$	0.862188	0.0335606
$661$	40.5283	1.57637	0.788185	−	0.615439i	$-0.211021\pi$
$661$	40.5283	1.57637	0.788185	+	0.615439i	$0.211021\pi$
$662$	9.37988	0.364560
$663$	0.0896611	0.00348215
$664$	12.2879	0.476863
$665$	0.0258762	0.00100344
$666$	16.9464	0.656660
$667$	−25.9383	−1.00434
$668$	−21.9499	−0.849267
$669$	−0.714308	−0.0276167
$670$	−24.1438	−0.932756
$671$	46.1858	1.78298
$672$	0.0623338	0.00240458
$673$	42.4721	1.63718	0.818590	−	0.574378i	$-0.194756\pi$
$673$	42.4721	1.63718	0.818590	+	0.574378i	$0.194756\pi$
$674$	−2.50685	−0.0965602
$675$	0.584176	0.0224849
$676$	−11.6225	−0.447021
$677$	−5.57011	−0.214077	−0.107038	−	0.994255i	$-0.534137\pi$
$677$	−5.57011	−0.214077	−0.107038	+	0.994255i	$0.534137\pi$
$678$	−0.687394	−0.0263992
$679$	0.414032	0.0158891
$680$	−2.18308	−0.0837173
$681$	−1.38135	−0.0529336
$682$	29.5400	1.13115
$683$	−5.21261	−0.199455	−0.0997276	−	0.995015i	$-0.531797\pi$
$683$	−5.21261	−0.199455	−0.0997276	+	0.995015i	$0.531797\pi$
$684$	−0.0434716	−0.00166218
$685$	−18.2027	−0.695490
$686$	9.70311	0.370467
$687$	−1.53222	−0.0584578
$688$	2.79702	0.106635
$689$	15.4554	0.588802
$690$	−1.08477	−0.0412966
$691$	45.8844	1.74552	0.872762	−	0.488147i	$-0.162327\pi$
$691$	45.8844	1.74552	0.872762	+	0.488147i	$0.162327\pi$
$692$	−4.99367	−0.189831
$693$	8.66266	0.329067
$694$	0.0754958	0.00286578
$695$	−21.1824	−0.803494
$696$	0.443927	0.0168270
$697$	6.35467	0.240700
$698$	−24.9751	−0.945322
$699$	−0.391721	−0.0148163
$700$	0.810072	0.0306178
$701$	−17.4338	−0.658465	−0.329232	−	0.944249i	$-0.606790\pi$
$701$	−17.4338	−0.658465	−0.329232	+	0.944249i	$0.606790\pi$
$702$	0.609137	0.0229904
$703$	0.0822651	0.00310269
$704$	−4.02218	−0.151592
$705$	−1.30029	−0.0489719
$706$	−8.13318	−0.306096
$707$	−8.28792	−0.311699
$708$	1.30307	0.0489724
$709$	2.47271	0.0928646	0.0464323	−	0.998921i	$-0.485215\pi$
$709$	2.47271	0.0928646	0.0464323	+	0.998921i	$0.485215\pi$
$710$	−37.0381	−1.39001
$711$	24.3754	0.914148
$712$	17.7820	0.666409
$713$	−37.1661	−1.39188
$714$	0.0549820	0.00205765
$715$	11.6835	0.436939
$716$	5.25556	0.196410
$717$	−0.886979	−0.0331248
$718$	−28.6150	−1.06790
$719$	9.37847	0.349758	0.174879	−	0.984590i	$-0.444047\pi$
$719$	9.37847	0.349758	0.174879	+	0.984590i	$0.444047\pi$
$720$	−7.40639	−0.276020
$721$	0.630118	0.0234668
$722$	18.9998	0.707099
$723$	−1.16765	−0.0434252
$724$	17.6280	0.655139
$725$	5.76915	0.214261
$726$	0.448460	0.0166439
$727$	−28.2806	−1.04887	−0.524435	−	0.851450i	$-0.675723\pi$
$727$	−28.2806	−1.04887	−0.524435	+	0.851450i	$0.675723\pi$
$728$	0.844686	0.0313061
$729$	−26.5958	−0.985029
$730$	29.5982	1.09548
$731$	2.46713	0.0912501
$732$	0.994523	0.0367586
$733$	53.3296	1.96977	0.984887	−	0.173197i	$-0.0554096\pi$
$733$	53.3296	1.96977	0.984887	+	0.173197i	$0.0554096\pi$
$734$	9.09315	0.335634
$735$	1.38948	0.0512516
$736$	5.06055	0.186534
$737$	−39.2368	−1.44531
$738$	21.5591	0.793600
$739$	−42.0566	−1.54708	−0.773539	−	0.633748i	$-0.781515\pi$
$739$	−42.0566	−1.54708	−0.773539	+	0.633748i	$0.781515\pi$
$740$	14.0157	0.515229
$741$	0.00147665	5.42463e−5 0
$742$	9.47755	0.347932
$743$	13.0948	0.480400	0.240200	−	0.970723i	$-0.422787\pi$
$743$	13.0948	0.480400	0.240200	+	0.970723i	$0.422787\pi$
$744$	0.636088	0.0233201
$745$	7.91542	0.289998
$746$	36.6472	1.34175
$747$	36.7715	1.34540
$748$	−3.54779	−0.129720
$749$	−1.89351	−0.0691873
$750$	−0.830520	−0.0303263
$751$	9.10727	0.332329	0.166165	−	0.986098i	$-0.446862\pi$
$751$	9.10727	0.332329	0.166165	+	0.986098i	$0.446862\pi$
$752$	6.06598	0.221204
$753$	1.15501	0.0420908
$754$	6.01566	0.219077
$755$	−10.8950	−0.396511
$756$	0.373535	0.0135854
$757$	52.9668	1.92511	0.962556	−	0.271083i	$-0.0873819\pi$
$757$	52.9668	1.92511	0.962556	+	0.271083i	$0.0873819\pi$
$758$	18.0899	0.657057
$759$	−1.76290	−0.0639892
$760$	−0.0359538	−0.00130418
$761$	−28.3104	−1.02625	−0.513126	−	0.858313i	$-0.671513\pi$
$761$	−28.3104	−1.02625	−0.513126	+	0.858313i	$0.671513\pi$
$762$	0.931856	0.0337576
$763$	−7.53085	−0.272635
$764$	−0.288421	−0.0104347
$765$	−6.53286	−0.236196
$766$	−27.9297	−1.00914
$767$	17.6579	0.637590
$768$	−0.0866100	−0.00312527
$769$	−51.9878	−1.87473	−0.937365	−	0.348350i	$-0.886742\pi$
$769$	−51.9878	−1.87473	−0.937365	+	0.348350i	$0.886742\pi$
$770$	7.16457	0.258193
$771$	1.99577	0.0718758
$772$	−2.30616	−0.0830006
$773$	13.3517	0.480229	0.240115	−	0.970745i	$-0.422815\pi$
$773$	13.3517	0.480229	0.240115	+	0.970745i	$0.422815\pi$
$774$	8.37007	0.300856
$775$	8.26641	0.296938
$776$	−0.575278	−0.0206513
$777$	−0.352994	−0.0126636
$778$	−4.12087	−0.147741
$779$	0.104657	0.00374972
$780$	0.251582	0.00900809
$781$	−60.1917	−2.15383
$782$	4.46370	0.159622
$783$	2.66023	0.0950690
$784$	−6.48202	−0.231501
$785$	−55.8744	−1.99424
$786$	−0.498123	−0.0177674
$787$	47.2718	1.68506	0.842529	−	0.538651i	$-0.181066\pi$
$787$	47.2718	1.68506	0.842529	+	0.538651i	$0.181066\pi$
$788$	16.6893	0.594530
$789$	1.49734	0.0533066
$790$	20.1600	0.717260
$791$	−5.71207	−0.203098
$792$	−12.0364	−0.427694
$793$	13.4768	0.478575
$794$	−16.3734	−0.581069
$795$	2.82280	0.100115
$796$	13.6845	0.485034
$797$	18.6392	0.660233	0.330117	−	0.943940i	$-0.392912\pi$
$797$	18.6392	0.660233	0.330117	+	0.943940i	$0.392912\pi$
$798$	0.000905515 0	3.20549e−5 0
$799$	5.35054	0.189289
$800$	−1.12556	−0.0397945
$801$	53.2127	1.88018
$802$	22.5009	0.794533
$803$	48.1010	1.69745
$804$	−0.844890	−0.0297970
$805$	−9.01419	−0.317709
$806$	8.61963	0.303614
$807$	0.935186	0.0329201
$808$	11.5157	0.405120
$809$	6.00277	0.211046	0.105523	−	0.994417i	$-0.466348\pi$
$809$	6.00277	0.211046	0.105523	+	0.994417i	$0.466348\pi$
$810$	−22.1079	−0.776793
$811$	33.4766	1.17552	0.587762	−	0.809034i	$-0.300009\pi$
$811$	33.4766	1.17552	0.587762	+	0.809034i	$0.300009\pi$
$812$	3.68893	0.129456
$813$	0.772179	0.0270815
$814$	22.7774	0.798349
$815$	−18.3248	−0.641891
$816$	−0.0763950	−0.00267436
$817$	0.0406319	0.00142153
$818$	−14.9244	−0.521821
$819$	2.52772	0.0883257
$820$	17.8307	0.622676
$821$	−3.13609	−0.109450	−0.0547251	−	0.998501i	$-0.517428\pi$
$821$	−3.13609	−0.109450	−0.0547251	+	0.998501i	$0.517428\pi$
$822$	−0.636988	−0.0222175
$823$	53.9959	1.88218	0.941090	−	0.338156i	$-0.109803\pi$
$823$	53.9959	1.88218	0.941090	+	0.338156i	$0.109803\pi$
$824$	−0.875520	−0.0305002
$825$	0.392100	0.0136512
$826$	10.8282	0.376761
$827$	14.1570	0.492287	0.246143	−	0.969233i	$-0.420837\pi$
$827$	14.1570	0.492287	0.246143	+	0.969233i	$0.420837\pi$
$828$	15.1437	0.526280
$829$	11.2896	0.392105	0.196052	−	0.980593i	$-0.437188\pi$
$829$	11.2896	0.392105	0.196052	+	0.980593i	$0.437188\pi$
$830$	30.4124	1.05563
$831$	−0.391332	−0.0135752
$832$	−1.17365	−0.0406890
$833$	−5.71751	−0.198100
$834$	−0.741259	−0.0256677
$835$	−54.3257	−1.88002
$836$	−0.0584297	−0.00202083
$837$	3.81176	0.131754
$838$	28.6308	0.989035
$839$	25.2091	0.870316	0.435158	−	0.900354i	$-0.356693\pi$
$839$	25.2091	0.870316	0.435158	+	0.900354i	$0.356693\pi$
$840$	0.154275	0.00532301
$841$	−2.72829	−0.0940791
$842$	−15.6243	−0.538448
$843$	−1.29365	−0.0445558
$844$	−0.344938	−0.0118732
$845$	−28.7656	−0.989568
$846$	18.1524	0.624094
$847$	3.72659	0.128047
$848$	−13.1686	−0.452212
$849$	1.35619	0.0465442
$850$	−0.992806	−0.0340530
$851$	−28.6577	−0.982374
$852$	−1.29611	−0.0444041
$853$	37.2095	1.27403	0.637014	−	0.770852i	$-0.280169\pi$
$853$	37.2095	1.27403	0.637014	+	0.770852i	$0.280169\pi$
$854$	8.26425	0.282797
$855$	−0.107592	−0.00367956
$856$	2.63094	0.0899237
$857$	−36.3804	−1.24273	−0.621365	−	0.783521i	$-0.713422\pi$
$857$	−36.3804	−1.24273	−0.621365	+	0.783521i	$0.713422\pi$
$858$	0.408854	0.0139581
$859$	−30.8665	−1.05315	−0.526575	−	0.850128i	$-0.676524\pi$
$859$	−30.8665	−1.05315	−0.526575	+	0.850128i	$0.676524\pi$
$860$	6.92258	0.236058
$861$	−0.449076	−0.0153045
$862$	−22.3042	−0.759684
$863$	5.45571	0.185714	0.0928572	−	0.995679i	$-0.470400\pi$
$863$	5.45571	0.185714	0.0928572	+	0.995679i	$0.470400\pi$
$864$	−0.519010	−0.0176571
$865$	−12.3593	−0.420227
$866$	−35.6307	−1.21078
$867$	1.40498	0.0477158
$868$	5.28574	0.179410
$869$	32.7626	1.11140
$870$	1.09871	0.0372499
$871$	−11.4491	−0.387938
$872$	10.4638	0.354348
$873$	−1.72152	−0.0582645
$874$	0.0735140	0.00248665
$875$	−6.90141	−0.233310
$876$	1.03576	0.0349952
$877$	−43.4652	−1.46772	−0.733858	−	0.679303i	$-0.762282\pi$
$877$	−43.4652	−1.46772	−0.733858	+	0.679303i	$0.762282\pi$
$878$	−37.7141	−1.27279
$879$	−2.05851	−0.0694318
$880$	−9.95484	−0.335578
$881$	−50.4024	−1.69810	−0.849050	−	0.528312i	$-0.822825\pi$
$881$	−50.4024	−1.69810	−0.849050	+	0.528312i	$0.822825\pi$
$882$	−19.3974	−0.653146
$883$	−50.0310	−1.68368	−0.841838	−	0.539730i	$-0.818526\pi$
$883$	−50.0310	−1.68368	−0.841838	+	0.539730i	$0.818526\pi$
$884$	−1.03523	−0.0348185
$885$	3.22508	0.108410
$886$	−18.8613	−0.633658
$887$	55.8349	1.87475	0.937376	−	0.348318i	$-0.113247\pi$
$887$	55.8349	1.87475	0.937376	+	0.348318i	$0.113247\pi$
$888$	0.490469	0.0164591
$889$	7.74349	0.259708
$890$	44.0103	1.47523
$891$	−35.9283	−1.20364
$892$	8.24741	0.276144
$893$	0.0881197	0.00294881
$894$	0.276993	0.00926403
$895$	13.0074	0.434791
$896$	−0.719708	−0.0240437
$897$	−0.514405	−0.0171755
$898$	33.1802	1.10724
$899$	37.6438	1.25549
$900$	−3.36823	−0.112274
$901$	−11.6155	−0.386967
$902$	28.9773	0.964837
$903$	−0.174349	−0.00580197
$904$	7.93666	0.263969
$905$	43.6290	1.45028
$906$	−0.381262	−0.0126666
$907$	20.9035	0.694089	0.347044	−	0.937849i	$-0.387185\pi$
$907$	20.9035	0.694089	0.347044	+	0.937849i	$0.387185\pi$
$908$	15.9491	0.529291
$909$	34.4606	1.14299
$910$	2.09059	0.0693023
$911$	45.1890	1.49718	0.748589	−	0.663034i	$-0.230732\pi$
$911$	45.1890	1.49718	0.748589	+	0.663034i	$0.230732\pi$
$912$	−0.00125817	−4.16622e−5 0
$913$	49.4241	1.63570
$914$	−2.62314	−0.0867657
$915$	2.46143	0.0813724
$916$	17.6910	0.584528
$917$	−4.13928	−0.136691
$918$	−0.457797	−0.0151095
$919$	−28.3254	−0.934369	−0.467185	−	0.884160i	$-0.654732\pi$
$919$	−28.3254	−0.934369	−0.467185	+	0.884160i	$0.654732\pi$
$920$	12.5248	0.412930
$921$	0.720966	0.0237566
$922$	12.1494	0.400118
$923$	−17.5636	−0.578114
$924$	0.250718	0.00824801
$925$	6.37399	0.209575
$926$	−9.59647	−0.315359
$927$	−2.61999	−0.0860518
$928$	−5.12559	−0.168256
$929$	6.47936	0.212581	0.106290	−	0.994335i	$-0.466103\pi$
$929$	6.47936	0.212581	0.106290	+	0.994335i	$0.466103\pi$
$930$	1.57431	0.0516236
$931$	−0.0941635	−0.00308608
$932$	4.52282	0.148150
$933$	1.02353	0.0335088
$934$	29.1248	0.952992
$935$	−8.78074	−0.287161
$936$	−3.51215	−0.114798
$937$	4.60177	0.150333	0.0751666	−	0.997171i	$-0.476051\pi$
$937$	4.60177	0.150333	0.0751666	+	0.997171i	$0.476051\pi$
$938$	−7.02083	−0.229238
$939$	1.07191	0.0349805
$940$	15.0132	0.489677
$941$	−6.91735	−0.225499	−0.112750	−	0.993623i	$-0.535966\pi$
$941$	−6.91735	−0.225499	−0.112750	+	0.993623i	$0.535966\pi$
$942$	−1.95528	−0.0637063
$943$	−36.4581	−1.18724
$944$	−15.0453	−0.489682
$945$	0.924495	0.0300738
$946$	11.2501	0.365773
$947$	−18.6923	−0.607418	−0.303709	−	0.952765i	$-0.598225\pi$
$947$	−18.6923	−0.607418	−0.303709	+	0.952765i	$0.598225\pi$
$948$	0.705481	0.0229129
$949$	14.0356	0.455616
$950$	−0.0163508	−0.000530491 0
$951$	−0.0537504	−0.00174298
$952$	−0.634823	−0.0205747
$953$	14.4918	0.469436	0.234718	−	0.972064i	$-0.424583\pi$
$953$	14.4918	0.469436	0.234718	+	0.972064i	$0.424583\pi$
$954$	−39.4070	−1.27585
$955$	−0.713837	−0.0230992
$956$	10.2411	0.331220
$957$	1.78556	0.0577188
$958$	9.68670	0.312963
$959$	−5.29322	−0.170927
$960$	−0.214358	−0.00691839
$961$	22.9385	0.739952
$962$	6.64634	0.214287
$963$	7.87309	0.253707
$964$	13.4817	0.434215
$965$	−5.70772	−0.183738
$966$	−0.315444	−0.0101492
$967$	11.9258	0.383508	0.191754	−	0.981443i	$-0.438582\pi$
$967$	11.9258	0.383508	0.191754	+	0.981443i	$0.438582\pi$
$968$	−5.17793	−0.166425
$969$	−0.00110978	−3.56513e−5 0
$970$	−1.42380	−0.0457156
$971$	−10.6936	−0.343175	−0.171587	−	0.985169i	$-0.554890\pi$
$971$	−10.6936	−0.343175	−0.171587	+	0.985169i	$0.554890\pi$
$972$	−2.33068	−0.0747565
$973$	−6.15968	−0.197470
$974$	−36.6179	−1.17331
$975$	0.114413	0.00366414
$976$	−11.4828	−0.367555
$977$	56.5515	1.80924	0.904622	−	0.426215i	$-0.140153\pi$
$977$	56.5515	1.80924	0.904622	+	0.426215i	$0.140153\pi$
$978$	−0.641261	−0.0205053
$979$	71.5225	2.28587
$980$	−16.0429	−0.512472
$981$	31.3128	0.999741
$982$	−19.3351	−0.617008
$983$	−3.58442	−0.114325	−0.0571626	−	0.998365i	$-0.518205\pi$
$983$	−3.58442	−0.114325	−0.0571626	+	0.998365i	$0.518205\pi$
$984$	0.623970	0.0198914
$985$	41.3057	1.31611
$986$	−4.52107	−0.143980
$987$	−0.378116	−0.0120356
$988$	−0.0170495	−0.000542416 0
$989$	−14.1545	−0.450086
$990$	−29.7898	−0.946783
$991$	12.6564	0.402043	0.201022	−	0.979587i	$-0.435574\pi$
$991$	12.6564	0.402043	0.201022	+	0.979587i	$0.435574\pi$
$992$	−7.34428	−0.233181
$993$	0.812391	0.0257805
$994$	−10.7704	−0.341616
$995$	33.8689	1.07372
$996$	1.06425	0.0337222
$997$	53.6616	1.69948	0.849739	−	0.527203i	$-0.176759\pi$
$997$	53.6616	1.69948	0.849739	+	0.527203i	$0.176759\pi$
$998$	−36.7305	−1.16268
$999$	2.93913	0.0929901

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4006.2.a.g.1.20	✓	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4006.2.a.g.1.20	✓	40	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:	\( N \)	\(=\)	\( 4006 = 2 \cdot 2003 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4006.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -0.0866100 q^{3} +1.00000 q^{4} +2.47499 q^{5} +0.0866100 q^{6} +0.719708 q^{7} -1.00000 q^{8} -2.99250 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -0.0866100 q^{3} +1.00000 q^{4} +2.47499 q^{5} +0.0866100 q^{6} +0.719708 q^{7} -1.00000 q^{8} -2.99250 q^{9} -2.47499 q^{10} -4.02218 q^{11} -0.0866100 q^{12} -1.17365 q^{13} -0.719708 q^{14} -0.214358 q^{15} +1.00000 q^{16} +0.882057 q^{17} +2.99250 q^{18} +0.0145269 q^{19} +2.47499 q^{20} -0.0623338 q^{21} +4.02218 q^{22} -5.06055 q^{23} +0.0866100 q^{24} +1.12556 q^{25} +1.17365 q^{26} +0.519010 q^{27} +0.719708 q^{28} +5.12559 q^{29} +0.214358 q^{30} +7.34428 q^{31} -1.00000 q^{32} +0.348361 q^{33} -0.882057 q^{34} +1.78127 q^{35} -2.99250 q^{36} +5.66296 q^{37} -0.0145269 q^{38} +0.101650 q^{39} -2.47499 q^{40} +7.20437 q^{41} +0.0623338 q^{42} +2.79702 q^{43} -4.02218 q^{44} -7.40639 q^{45} +5.06055 q^{46} +6.06598 q^{47} -0.0866100 q^{48} -6.48202 q^{49} -1.12556 q^{50} -0.0763950 q^{51} -1.17365 q^{52} -13.1686 q^{53} -0.519010 q^{54} -9.95484 q^{55} -0.719708 q^{56} -0.00125817 q^{57} -5.12559 q^{58} -15.0453 q^{59} -0.214358 q^{60} -11.4828 q^{61} -7.34428 q^{62} -2.15372 q^{63} +1.00000 q^{64} -2.90477 q^{65} -0.348361 q^{66} +9.75512 q^{67} +0.882057 q^{68} +0.438294 q^{69} -1.78127 q^{70} +14.9650 q^{71} +2.99250 q^{72} -11.9589 q^{73} -5.66296 q^{74} -0.0974845 q^{75} +0.0145269 q^{76} -2.89479 q^{77} -0.101650 q^{78} -8.14549 q^{79} +2.47499 q^{80} +8.93254 q^{81} -7.20437 q^{82} -12.2879 q^{83} -0.0623338 q^{84} +2.18308 q^{85} -2.79702 q^{86} -0.443927 q^{87} +4.02218 q^{88} -17.7820 q^{89} +7.40639 q^{90} -0.844686 q^{91} -5.06055 q^{92} -0.636088 q^{93} -6.06598 q^{94} +0.0359538 q^{95} +0.0866100 q^{96} +0.575278 q^{97} +6.48202 q^{98} +12.0364 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 40 q^{2} - q^{3} + 40 q^{4} - 23 q^{5} + q^{6} + 12 q^{7} - 40 q^{8} + 29 q^{9}+O(q^{10}) \) 40 * q - 40 * q^2 - q^3 + 40 * q^4 - 23 * q^5 + q^6 + 12 * q^7 - 40 * q^8 + 29 * q^9	\( 40 q - 40 q^{2} - q^{3} + 40 q^{4} - 23 q^{5} + q^{6} + 12 q^{7} - 40 q^{8} + 29 q^{9} + 23 q^{10} - 28 q^{11} - q^{12} - 12 q^{13} - 12 q^{14} - 14 q^{15} + 40 q^{16} - 10 q^{17} - 29 q^{18} + 3 q^{19} - 23 q^{20} - 40 q^{21} + 28 q^{22} - 10 q^{23} + q^{24} + 43 q^{25} + 12 q^{26} - 7 q^{27} + 12 q^{28} - 37 q^{29} + 14 q^{30} - 16 q^{31} - 40 q^{32} - 11 q^{33} + 10 q^{34} - 24 q^{35} + 29 q^{36} - 17 q^{37} - 3 q^{38} - 10 q^{39} + 23 q^{40} - 58 q^{41} + 40 q^{42} + 37 q^{43} - 28 q^{44} - 66 q^{45} + 10 q^{46} - 34 q^{47} - q^{48} + 28 q^{49} - 43 q^{50} - 7 q^{51} - 12 q^{52} - 43 q^{53} + 7 q^{54} + 28 q^{55} - 12 q^{56} - 25 q^{57} + 37 q^{58} - 92 q^{59} - 14 q^{60} - 37 q^{61} + 16 q^{62} + 14 q^{63} + 40 q^{64} - 54 q^{65} + 11 q^{66} - 10 q^{67} - 10 q^{68} - 49 q^{69} + 24 q^{70} - 87 q^{71} - 29 q^{72} + 12 q^{73} + 17 q^{74} - 31 q^{75} + 3 q^{76} - 53 q^{77} + 10 q^{78} + 14 q^{79} - 23 q^{80} - 4 q^{81} + 58 q^{82} - 42 q^{83} - 40 q^{84} - 24 q^{85} - 37 q^{86} + 24 q^{87} + 28 q^{88} - 118 q^{89} + 66 q^{90} - 35 q^{91} - 10 q^{92} - 22 q^{93} + 34 q^{94} - 18 q^{95} + q^{96} + 2 q^{97} - 28 q^{98} - 48 q^{99}+O(q^{100}) \) 40 * q - 40 * q^2 - q^3 + 40 * q^4 - 23 * q^5 + q^6 + 12 * q^7 - 40 * q^8 + 29 * q^9 + 23 * q^10 - 28 * q^11 - q^12 - 12 * q^13 - 12 * q^14 - 14 * q^15 + 40 * q^16 - 10 * q^17 - 29 * q^18 + 3 * q^19 - 23 * q^20 - 40 * q^21 + 28 * q^22 - 10 * q^23 + q^24 + 43 * q^25 + 12 * q^26 - 7 * q^27 + 12 * q^28 - 37 * q^29 + 14 * q^30 - 16 * q^31 - 40 * q^32 - 11 * q^33 + 10 * q^34 - 24 * q^35 + 29 * q^36 - 17 * q^37 - 3 * q^38 - 10 * q^39 + 23 * q^40 - 58 * q^41 + 40 * q^42 + 37 * q^43 - 28 * q^44 - 66 * q^45 + 10 * q^46 - 34 * q^47 - q^48 + 28 * q^49 - 43 * q^50 - 7 * q^51 - 12 * q^52 - 43 * q^53 + 7 * q^54 + 28 * q^55 - 12 * q^56 - 25 * q^57 + 37 * q^58 - 92 * q^59 - 14 * q^60 - 37 * q^61 + 16 * q^62 + 14 * q^63 + 40 * q^64 - 54 * q^65 + 11 * q^66 - 10 * q^67 - 10 * q^68 - 49 * q^69 + 24 * q^70 - 87 * q^71 - 29 * q^72 + 12 * q^73 + 17 * q^74 - 31 * q^75 + 3 * q^76 - 53 * q^77 + 10 * q^78 + 14 * q^79 - 23 * q^80 - 4 * q^81 + 58 * q^82 - 42 * q^83 - 40 * q^84 - 24 * q^85 - 37 * q^86 + 24 * q^87 + 28 * q^88 - 118 * q^89 + 66 * q^90 - 35 * q^91 - 10 * q^92 - 22 * q^93 + 34 * q^94 - 18 * q^95 + q^96 + 2 * q^97 - 28 * q^98 - 48 * q^99

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