LMFDB - Embedded newform 1690.2.a.v.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1690,2,Mod(1,1690)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1690, 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("1690.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1690 = 2 \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1690.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:	$13.4947179416$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.20439713.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 15x^{4} + 22x^{3} + 70x^{2} - 56x - 91 $ x^6 - 2x^5 - 15x^4 + 22x^3 + 70x^2 - 56*x - 91
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			1.5
Root			$-2.23727$ of defining polynomial
Character	$\chi$	$=$	1690.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +2.23727 q^{3} +1.00000 q^{4} -1.00000 q^{5} -2.23727 q^{6} -1.43294 q^{7} -1.00000 q^{8} +2.00539 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +2.23727 q^{3} +1.00000 q^{4} -1.00000 q^{5} -2.23727 q^{6} -1.43294 q^{7} -1.00000 q^{8} +2.00539 q^{9} +1.00000 q^{10} -0.315763 q^{11} +2.23727 q^{12} +1.43294 q^{14} -2.23727 q^{15} +1.00000 q^{16} -1.69981 q^{17} -2.00539 q^{18} -5.83768 q^{19} -1.00000 q^{20} -3.20587 q^{21} +0.315763 q^{22} -9.32839 q^{23} -2.23727 q^{24} +1.00000 q^{25} -2.22522 q^{27} -1.43294 q^{28} +7.09544 q^{29} +2.23727 q^{30} -4.69438 q^{31} -1.00000 q^{32} -0.706448 q^{33} +1.69981 q^{34} +1.43294 q^{35} +2.00539 q^{36} +11.6581 q^{37} +5.83768 q^{38} +1.00000 q^{40} -2.79223 q^{41} +3.20587 q^{42} -4.77478 q^{43} -0.315763 q^{44} -2.00539 q^{45} +9.32839 q^{46} -10.4362 q^{47} +2.23727 q^{48} -4.94669 q^{49} -1.00000 q^{50} -3.80293 q^{51} -1.48663 q^{53} +2.22522 q^{54} +0.315763 q^{55} +1.43294 q^{56} -13.0605 q^{57} -7.09544 q^{58} +3.93789 q^{59} -2.23727 q^{60} +0.971828 q^{61} +4.69438 q^{62} -2.87359 q^{63} +1.00000 q^{64} +0.706448 q^{66} +12.7810 q^{67} -1.69981 q^{68} -20.8701 q^{69} -1.43294 q^{70} -4.17629 q^{71} -2.00539 q^{72} -8.83875 q^{73} -11.6581 q^{74} +2.23727 q^{75} -5.83768 q^{76} +0.452469 q^{77} -6.80085 q^{79} -1.00000 q^{80} -10.9946 q^{81} +2.79223 q^{82} -11.1336 q^{83} -3.20587 q^{84} +1.69981 q^{85} +4.77478 q^{86} +15.8744 q^{87} +0.315763 q^{88} -1.48615 q^{89} +2.00539 q^{90} -9.32839 q^{92} -10.5026 q^{93} +10.4362 q^{94} +5.83768 q^{95} -2.23727 q^{96} +11.9669 q^{97} +4.94669 q^{98} -0.633227 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{2} - 2 q^{3} + 6 q^{4} - 6 q^{5} + 2 q^{6} - 3 q^{7} - 6 q^{8} + 16 q^{9}+O(q^{10}) $ 6 * q - 6 * q^2 - 2 * q^3 + 6 * q^4 - 6 * q^5 + 2 * q^6 - 3 * q^7 - 6 * q^8 + 16 * q^9	$ 6 q - 6 q^{2} - 2 q^{3} + 6 q^{4} - 6 q^{5} + 2 q^{6} - 3 q^{7} - 6 q^{8} + 16 q^{9} + 6 q^{10} - 15 q^{11} - 2 q^{12} + 3 q^{14} + 2 q^{15} + 6 q^{16} - 3 q^{17} - 16 q^{18} - q^{19} - 6 q^{20} + 2 q^{21} + 15 q^{22} - 3 q^{23} + 2 q^{24} + 6 q^{25} - 20 q^{27} - 3 q^{28} + 7 q^{29} - 2 q^{30} - 6 q^{32} - 4 q^{33} + 3 q^{34} + 3 q^{35} + 16 q^{36} - 6 q^{37} + q^{38} + 6 q^{40} - 2 q^{41} - 2 q^{42} - 22 q^{43} - 15 q^{44} - 16 q^{45} + 3 q^{46} - 7 q^{47} - 2 q^{48} + 31 q^{49} - 6 q^{50} - 22 q^{51} - 16 q^{53} + 20 q^{54} + 15 q^{55} + 3 q^{56} - 2 q^{57} - 7 q^{58} - 15 q^{59} + 2 q^{60} + 33 q^{61} - 25 q^{63} + 6 q^{64} + 4 q^{66} + 8 q^{67} - 3 q^{68} - 6 q^{69} - 3 q^{70} - 40 q^{71} - 16 q^{72} - 21 q^{73} + 6 q^{74} - 2 q^{75} - q^{76} - 34 q^{77} + 20 q^{79} - 6 q^{80} - 2 q^{81} + 2 q^{82} - 22 q^{83} + 2 q^{84} + 3 q^{85} + 22 q^{86} - 39 q^{87} + 15 q^{88} - 20 q^{89} + 16 q^{90} - 3 q^{92} - 48 q^{93} + 7 q^{94} + q^{95} + 2 q^{96} + 7 q^{97} - 31 q^{98} - 55 q^{99}+O(q^{100}) $ 6 * q - 6 * q^2 - 2 * q^3 + 6 * q^4 - 6 * q^5 + 2 * q^6 - 3 * q^7 - 6 * q^8 + 16 * q^9 + 6 * q^10 - 15 * q^11 - 2 * q^12 + 3 * q^14 + 2 * q^15 + 6 * q^16 - 3 * q^17 - 16 * q^18 - q^19 - 6 * q^20 + 2 * q^21 + 15 * q^22 - 3 * q^23 + 2 * q^24 + 6 * q^25 - 20 * q^27 - 3 * q^28 + 7 * q^29 - 2 * q^30 - 6 * q^32 - 4 * q^33 + 3 * q^34 + 3 * q^35 + 16 * q^36 - 6 * q^37 + q^38 + 6 * q^40 - 2 * q^41 - 2 * q^42 - 22 * q^43 - 15 * q^44 - 16 * q^45 + 3 * q^46 - 7 * q^47 - 2 * q^48 + 31 * q^49 - 6 * q^50 - 22 * q^51 - 16 * q^53 + 20 * q^54 + 15 * q^55 + 3 * q^56 - 2 * q^57 - 7 * q^58 - 15 * q^59 + 2 * q^60 + 33 * q^61 - 25 * q^63 + 6 * q^64 + 4 * q^66 + 8 * q^67 - 3 * q^68 - 6 * q^69 - 3 * q^70 - 40 * q^71 - 16 * q^72 - 21 * q^73 + 6 * q^74 - 2 * q^75 - q^76 - 34 * q^77 + 20 * q^79 - 6 * q^80 - 2 * q^81 + 2 * q^82 - 22 * q^83 + 2 * q^84 + 3 * q^85 + 22 * q^86 - 39 * q^87 + 15 * q^88 - 20 * q^89 + 16 * q^90 - 3 * q^92 - 48 * q^93 + 7 * q^94 + q^95 + 2 * q^96 + 7 * q^97 - 31 * q^98 - 55 * 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
$3$	2.23727	1.29169	0.645845	−	0.763469i	$-0.276505\pi$
$3$	2.23727	1.29169	0.645845	+	0.763469i	$0.276505\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−2.23727	−0.913363
$7$	−1.43294	−0.541599	−0.270800	−	0.962636i	$-0.587288\pi$
$7$	−1.43294	−0.541599	−0.270800	+	0.962636i	$0.587288\pi$
$8$	−1.00000	−0.353553
$9$	2.00539	0.668462
$10$	1.00000	0.316228
$11$	−0.315763	−0.0952062	−0.0476031	−	0.998866i	$-0.515158\pi$
$11$	−0.315763	−0.0952062	−0.0476031	+	0.998866i	$0.515158\pi$
$12$	2.23727	0.645845
$13$	0	0
$13$	0	0
$14$	1.43294	0.382969
$15$	−2.23727	−0.577661
$16$	1.00000	0.250000
$17$	−1.69981	−0.412264	−0.206132	−	0.978524i	$-0.566088\pi$
$17$	−1.69981	−0.412264	−0.206132	+	0.978524i	$0.566088\pi$
$18$	−2.00539	−0.472674
$19$	−5.83768	−1.33926	−0.669628	−	0.742696i	$-0.733547\pi$
$19$	−5.83768	−1.33926	−0.669628	+	0.742696i	$0.733547\pi$
$20$	−1.00000	−0.223607
$21$	−3.20587	−0.699578
$22$	0.315763	0.0673209
$23$	−9.32839	−1.94510	−0.972552	−	0.232686i	$-0.925248\pi$
$23$	−9.32839	−1.94510	−0.972552	+	0.232686i	$0.925248\pi$
$24$	−2.23727	−0.456681
$25$	1.00000	0.200000
$26$	0	0
$27$	−2.22522	−0.428244
$28$	−1.43294	−0.270800
$29$	7.09544	1.31759	0.658795	−	0.752323i	$-0.271067\pi$
$29$	7.09544	1.31759	0.658795	+	0.752323i	$0.271067\pi$
$30$	2.23727	0.408468
$31$	−4.69438	−0.843135	−0.421568	−	0.906797i	$-0.638520\pi$
$31$	−4.69438	−0.843135	−0.421568	+	0.906797i	$0.638520\pi$
$32$	−1.00000	−0.176777
$33$	−0.706448	−0.122977
$34$	1.69981	0.291514
$35$	1.43294	0.242211
$36$	2.00539	0.334231
$37$	11.6581	1.91658	0.958289	−	0.285802i	$-0.0922599\pi$
$37$	11.6581	1.91658	0.958289	+	0.285802i	$0.0922599\pi$
$38$	5.83768	0.946997
$39$	0	0
$40$	1.00000	0.158114
$41$	−2.79223	−0.436073	−0.218037	−	0.975941i	$-0.569965\pi$
$41$	−2.79223	−0.436073	−0.218037	+	0.975941i	$0.569965\pi$
$42$	3.20587	0.494677
$43$	−4.77478	−0.728147	−0.364074	−	0.931370i	$-0.618614\pi$
$43$	−4.77478	−0.728147	−0.364074	+	0.931370i	$0.618614\pi$
$44$	−0.315763	−0.0476031
$45$	−2.00539	−0.298945
$46$	9.32839	1.37540
$47$	−10.4362	−1.52227	−0.761136	−	0.648592i	$-0.775358\pi$
$47$	−10.4362	−1.52227	−0.761136	+	0.648592i	$0.775358\pi$
$48$	2.23727	0.322922
$49$	−4.94669	−0.706670
$50$	−1.00000	−0.141421
$51$	−3.80293	−0.532517
$52$	0	0
$53$	−1.48663	−0.204204	−0.102102	−	0.994774i	$-0.532557\pi$
$53$	−1.48663	−0.204204	−0.102102	+	0.994774i	$0.532557\pi$
$54$	2.22522	0.302814
$55$	0.315763	0.0425775
$56$	1.43294	0.191484
$57$	−13.0605	−1.72990
$58$	−7.09544	−0.931677
$59$	3.93789	0.512670	0.256335	−	0.966588i	$-0.417485\pi$
$59$	3.93789	0.512670	0.256335	+	0.966588i	$0.417485\pi$
$60$	−2.23727	−0.288831
$61$	0.971828	0.124430	0.0622149	−	0.998063i	$-0.480184\pi$
$61$	0.971828	0.124430	0.0622149	+	0.998063i	$0.480184\pi$
$62$	4.69438	0.596186
$63$	−2.87359	−0.362039
$64$	1.00000	0.125000
$65$	0	0
$66$	0.706448	0.0869578
$67$	12.7810	1.56144	0.780722	−	0.624879i	$-0.214852\pi$
$67$	12.7810	1.56144	0.780722	+	0.624879i	$0.214852\pi$
$68$	−1.69981	−0.206132
$69$	−20.8701	−2.51247
$70$	−1.43294	−0.171269
$71$	−4.17629	−0.495635	−0.247817	−	0.968807i	$-0.579713\pi$
$71$	−4.17629	−0.495635	−0.247817	+	0.968807i	$0.579713\pi$
$72$	−2.00539	−0.236337
$73$	−8.83875	−1.03450	−0.517249	−	0.855835i	$-0.673044\pi$
$73$	−8.83875	−1.03450	−0.517249	+	0.855835i	$0.673044\pi$
$74$	−11.6581	−1.35522
$75$	2.23727	0.258338
$76$	−5.83768	−0.669628
$77$	0.452469	0.0515636
$78$	0	0
$79$	−6.80085	−0.765156	−0.382578	−	0.923923i	$-0.624964\pi$
$79$	−6.80085	−0.765156	−0.382578	+	0.923923i	$0.624964\pi$
$80$	−1.00000	−0.111803
$81$	−10.9946	−1.22162
$82$	2.79223	0.308350
$83$	−11.1336	−1.22207	−0.611037	−	0.791602i	$-0.709247\pi$
$83$	−11.1336	−1.22207	−0.611037	+	0.791602i	$0.709247\pi$
$84$	−3.20587	−0.349789
$85$	1.69981	0.184370
$86$	4.77478	0.514878
$87$	15.8744	1.70192
$88$	0.315763	0.0336605
$89$	−1.48615	−0.157532	−0.0787659	−	0.996893i	$-0.525098\pi$
$89$	−1.48615	−0.157532	−0.0787659	+	0.996893i	$0.525098\pi$
$90$	2.00539	0.211386
$91$	0	0
$92$	−9.32839	−0.972552
$93$	−10.5026	−1.08907
$94$	10.4362	1.07641
$95$	5.83768	0.598934
$96$	−2.23727	−0.228341
$97$	11.9669	1.21506	0.607528	−	0.794298i	$-0.292161\pi$
$97$	11.9669	1.21506	0.607528	+	0.794298i	$0.292161\pi$
$98$	4.94669	0.499691
$99$	−0.633227	−0.0636418
$100$	1.00000	0.100000
$101$	16.6774	1.65947	0.829734	−	0.558159i	$-0.188492\pi$
$101$	16.6774	1.65947	0.829734	+	0.558159i	$0.188492\pi$
$102$	3.80293	0.376546
$103$	−7.91612	−0.779999	−0.389999	−	0.920815i	$-0.627525\pi$
$103$	−7.91612	−0.779999	−0.389999	+	0.920815i	$0.627525\pi$
$104$	0	0
$105$	3.20587	0.312861
$106$	1.48663	0.144394
$107$	−6.46441	−0.624939	−0.312469	−	0.949928i	$-0.601156\pi$
$107$	−6.46441	−0.624939	−0.312469	+	0.949928i	$0.601156\pi$
$108$	−2.22522	−0.214122
$109$	16.6406	1.59388	0.796939	−	0.604059i	$-0.206451\pi$
$109$	16.6406	1.59388	0.796939	+	0.604059i	$0.206451\pi$
$110$	−0.315763	−0.0301068
$111$	26.0823	2.47562
$112$	−1.43294	−0.135400
$113$	−14.2361	−1.33922	−0.669612	−	0.742711i	$-0.733540\pi$
$113$	−14.2361	−1.33922	−0.669612	+	0.742711i	$0.733540\pi$
$114$	13.0605	1.22323
$115$	9.32839	0.869877
$116$	7.09544	0.658795
$117$	0	0
$118$	−3.93789	−0.362512
$119$	2.43572	0.223282
$120$	2.23727	0.204234
$121$	−10.9003	−0.990936
$122$	−0.971828	−0.0879852
$123$	−6.24698	−0.563271
$124$	−4.69438	−0.421568
$125$	−1.00000	−0.0894427
$126$	2.87359	0.256000
$127$	16.6911	1.48109	0.740546	−	0.672006i	$-0.234567\pi$
$127$	16.6911	1.48109	0.740546	+	0.672006i	$0.234567\pi$
$128$	−1.00000	−0.0883883
$129$	−10.6825	−0.940540
$130$	0	0
$131$	5.22386	0.456411	0.228205	−	0.973613i	$-0.426714\pi$
$131$	5.22386	0.456411	0.228205	+	0.973613i	$0.426714\pi$
$132$	−0.706448	−0.0614884
$133$	8.36503	0.725340
$134$	−12.7810	−1.10411
$135$	2.22522	0.191516
$136$	1.69981	0.145757
$137$	−10.8426	−0.926345	−0.463173	−	0.886268i	$-0.653289\pi$
$137$	−10.8426	−0.926345	−0.463173	+	0.886268i	$0.653289\pi$
$138$	20.8701	1.77658
$139$	12.7506	1.08149	0.540745	−	0.841187i	$-0.318142\pi$
$139$	12.7506	1.08149	0.540745	+	0.841187i	$0.318142\pi$
$140$	1.43294	0.121105
$141$	−23.3486	−1.96630
$142$	4.17629	0.350467
$143$	0	0
$144$	2.00539	0.167116
$145$	−7.09544	−0.589244
$146$	8.83875	0.731500
$147$	−11.0671	−0.912799
$148$	11.6581	0.958289
$149$	2.68691	0.220120	0.110060	−	0.993925i	$-0.464896\pi$
$149$	2.68691	0.220120	0.110060	+	0.993925i	$0.464896\pi$
$150$	−2.23727	−0.182673
$151$	−15.3443	−1.24870	−0.624351	−	0.781144i	$-0.714637\pi$
$151$	−15.3443	−1.24870	−0.624351	+	0.781144i	$0.714637\pi$
$152$	5.83768	0.473499
$153$	−3.40877	−0.275583
$154$	−0.452469	−0.0364610
$155$	4.69438	0.377061
$156$	0	0
$157$	−5.59044	−0.446166	−0.223083	−	0.974799i	$-0.571612\pi$
$157$	−5.59044	−0.446166	−0.223083	+	0.974799i	$0.571612\pi$
$158$	6.80085	0.541047
$159$	−3.32599	−0.263768
$160$	1.00000	0.0790569
$161$	13.3670	1.05347
$162$	10.9946	0.863816
$163$	−4.33627	−0.339643	−0.169821	−	0.985475i	$-0.554319\pi$
$163$	−4.33627	−0.339643	−0.169821	+	0.985475i	$0.554319\pi$
$164$	−2.79223	−0.218037
$165$	0.706448	0.0549969
$166$	11.1336	0.864137
$167$	21.8290	1.68918	0.844590	−	0.535413i	$-0.179844\pi$
$167$	21.8290	1.68918	0.844590	+	0.535413i	$0.179844\pi$
$168$	3.20587	0.247338
$169$	0	0
$170$	−1.69981	−0.130369
$171$	−11.7068	−0.895243
$172$	−4.77478	−0.364074
$173$	−4.20426	−0.319644	−0.159822	−	0.987146i	$-0.551092\pi$
$173$	−4.20426	−0.319644	−0.159822	+	0.987146i	$0.551092\pi$
$174$	−15.8744	−1.20344
$175$	−1.43294	−0.108320
$176$	−0.315763	−0.0238015
$177$	8.81014	0.662210
$178$	1.48615	0.111392
$179$	−23.5902	−1.76322	−0.881608	−	0.471983i	$-0.843538\pi$
$179$	−23.5902	−1.76322	−0.881608	+	0.471983i	$0.843538\pi$
$180$	−2.00539	−0.149473
$181$	16.4103	1.21976	0.609882	−	0.792492i	$-0.291217\pi$
$181$	16.4103	1.21976	0.609882	+	0.792492i	$0.291217\pi$
$182$	0	0
$183$	2.17424	0.160725
$184$	9.32839	0.687698
$185$	−11.6581	−0.857120
$186$	10.5026	0.770088
$187$	0.536736	0.0392500
$188$	−10.4362	−0.761136
$189$	3.18860	0.231936
$190$	−5.83768	−0.423510
$191$	−21.8818	−1.58331	−0.791656	−	0.610967i	$-0.790781\pi$
$191$	−21.8818	−1.58331	−0.791656	+	0.610967i	$0.790781\pi$
$192$	2.23727	0.161461
$193$	0.440205	0.0316866	0.0158433	−	0.999874i	$-0.494957\pi$
$193$	0.440205	0.0316866	0.0158433	+	0.999874i	$0.494957\pi$
$194$	−11.9669	−0.859175
$195$	0	0
$196$	−4.94669	−0.353335
$197$	−13.7762	−0.981516	−0.490758	−	0.871296i	$-0.663280\pi$
$197$	−13.7762	−0.981516	−0.490758	+	0.871296i	$0.663280\pi$
$198$	0.633227	0.0450015
$199$	2.53270	0.179539	0.0897693	−	0.995963i	$-0.471387\pi$
$199$	2.53270	0.179539	0.0897693	+	0.995963i	$0.471387\pi$
$200$	−1.00000	−0.0707107
$201$	28.5945	2.01690
$202$	−16.6774	−1.17342
$203$	−10.1673	−0.713606
$204$	−3.80293	−0.266258
$205$	2.79223	0.195018
$206$	7.91612	0.551542
$207$	−18.7070	−1.30023
$208$	0	0
$209$	1.84333	0.127505
$210$	−3.20587	−0.221226
$211$	6.69030	0.460579	0.230290	−	0.973122i	$-0.426033\pi$
$211$	6.69030	0.460579	0.230290	+	0.973122i	$0.426033\pi$
$212$	−1.48663	−0.102102
$213$	−9.34350	−0.640206
$214$	6.46441	0.441898
$215$	4.77478	0.325637
$216$	2.22522	0.151407
$217$	6.72675	0.456641
$218$	−16.6406	−1.12704
$219$	−19.7747	−1.33625
$220$	0.315763	0.0212888
$221$	0	0
$222$	−26.0823	−1.75053
$223$	1.36952	0.0917097	0.0458548	−	0.998948i	$-0.485399\pi$
$223$	1.36952	0.0917097	0.0458548	+	0.998948i	$0.485399\pi$
$224$	1.43294	0.0957421
$225$	2.00539	0.133692
$226$	14.2361	0.946974
$227$	−1.46657	−0.0973398	−0.0486699	−	0.998815i	$-0.515498\pi$
$227$	−1.46657	−0.0973398	−0.0486699	+	0.998815i	$0.515498\pi$
$228$	−13.0605	−0.864952
$229$	7.36084	0.486418	0.243209	−	0.969974i	$-0.421800\pi$
$229$	7.36084	0.486418	0.243209	+	0.969974i	$0.421800\pi$
$230$	−9.32839	−0.615096
$231$	1.01230	0.0666042
$232$	−7.09544	−0.465838
$233$	15.5806	1.02072	0.510359	−	0.859961i	$-0.329512\pi$
$233$	15.5806	1.02072	0.510359	+	0.859961i	$0.329512\pi$
$234$	0	0
$235$	10.4362	0.680781
$236$	3.93789	0.256335
$237$	−15.2154	−0.988344
$238$	−2.43572	−0.157884
$239$	9.78398	0.632873	0.316436	−	0.948614i	$-0.397514\pi$
$239$	9.78398	0.632873	0.316436	+	0.948614i	$0.397514\pi$
$240$	−2.23727	−0.144415
$241$	11.7487	0.756802	0.378401	−	0.925642i	$-0.376474\pi$
$241$	11.7487	0.756802	0.378401	+	0.925642i	$0.376474\pi$
$242$	10.9003	0.700697
$243$	−17.9222	−1.14971
$244$	0.971828	0.0622149
$245$	4.94669	0.316033
$246$	6.24698	0.398293
$247$	0	0
$248$	4.69438	0.298093
$249$	−24.9089	−1.57854
$250$	1.00000	0.0632456
$251$	29.0462	1.83338	0.916689	−	0.399602i	$-0.130852\pi$
$251$	29.0462	1.83338	0.916689	+	0.399602i	$0.130852\pi$
$252$	−2.87359	−0.181019
$253$	2.94556	0.185186
$254$	−16.6911	−1.04729
$255$	3.80293	0.238149
$256$	1.00000	0.0625000
$257$	−15.6549	−0.976524	−0.488262	−	0.872697i	$-0.662369\pi$
$257$	−15.6549	−0.976524	−0.488262	+	0.872697i	$0.662369\pi$
$258$	10.6825	0.665062
$259$	−16.7053	−1.03802
$260$	0	0
$261$	14.2291	0.880759
$262$	−5.22386	−0.322731
$263$	−11.7545	−0.724813	−0.362407	−	0.932020i	$-0.618045\pi$
$263$	−11.7545	−0.724813	−0.362407	+	0.932020i	$0.618045\pi$
$264$	0.706448	0.0434789
$265$	1.48663	0.0913227
$266$	−8.36503	−0.512893
$267$	−3.32492	−0.203482
$268$	12.7810	0.780722
$269$	−0.786410	−0.0479483	−0.0239741	−	0.999713i	$-0.507632\pi$
$269$	−0.786410	−0.0479483	−0.0239741	+	0.999713i	$0.507632\pi$
$270$	−2.22522	−0.135423
$271$	−1.63501	−0.0993200	−0.0496600	−	0.998766i	$-0.515814\pi$
$271$	−1.63501	−0.0993200	−0.0496600	+	0.998766i	$0.515814\pi$
$272$	−1.69981	−0.103066
$273$	0	0
$274$	10.8426	0.655025
$275$	−0.315763	−0.0190412
$276$	−20.8701	−1.25624
$277$	0.567538	0.0341000	0.0170500	−	0.999855i	$-0.494573\pi$
$277$	0.567538	0.0341000	0.0170500	+	0.999855i	$0.494573\pi$
$278$	−12.7506	−0.764729
$279$	−9.41404	−0.563604
$280$	−1.43294	−0.0856344
$281$	−11.9390	−0.712221	−0.356111	−	0.934444i	$-0.615897\pi$
$281$	−11.9390	−0.712221	−0.356111	+	0.934444i	$0.615897\pi$
$282$	23.3486	1.39039
$283$	−14.0478	−0.835054	−0.417527	−	0.908664i	$-0.637103\pi$
$283$	−14.0478	−0.835054	−0.417527	+	0.908664i	$0.637103\pi$
$284$	−4.17629	−0.247817
$285$	13.0605	0.773636
$286$	0	0
$287$	4.00109	0.236177
$288$	−2.00539	−0.118169
$289$	−14.1107	−0.830039
$290$	7.09544	0.416658
$291$	26.7733	1.56948
$292$	−8.83875	−0.517249
$293$	−28.7352	−1.67873	−0.839364	−	0.543570i	$-0.817072\pi$
$293$	−28.7352	−1.67873	−0.839364	+	0.543570i	$0.817072\pi$
$294$	11.0671	0.645446
$295$	−3.93789	−0.229273
$296$	−11.6581	−0.677612
$297$	0.702643	0.0407715
$298$	−2.68691	−0.155649
$299$	0	0
$300$	2.23727	0.129169
$301$	6.84196	0.394364
$302$	15.3443	0.882966
$303$	37.3120	2.14352
$304$	−5.83768	−0.334814
$305$	−0.971828	−0.0556467
$306$	3.40877	0.194866
$307$	8.72162	0.497769	0.248885	−	0.968533i	$-0.419936\pi$
$307$	8.72162	0.497769	0.248885	+	0.968533i	$0.419936\pi$
$308$	0.452469	0.0257818
$309$	−17.7105	−1.00752
$310$	−4.69438	−0.266623
$311$	23.6123	1.33893	0.669466	−	0.742843i	$-0.266523\pi$
$311$	23.6123	1.33893	0.669466	+	0.742843i	$0.266523\pi$
$312$	0	0
$313$	22.3185	1.26152	0.630759	−	0.775979i	$-0.282744\pi$
$313$	22.3185	1.26152	0.630759	+	0.775979i	$0.282744\pi$
$314$	5.59044	0.315487
$315$	2.87359	0.161909
$316$	−6.80085	−0.382578
$317$	−22.3655	−1.25617	−0.628086	−	0.778144i	$-0.716161\pi$
$317$	−22.3655	−1.25617	−0.628086	+	0.778144i	$0.716161\pi$
$318$	3.32599	0.186512
$319$	−2.24048	−0.125443
$320$	−1.00000	−0.0559017
$321$	−14.4627	−0.807227
$322$	−13.3670	−0.744913
$323$	9.92293	0.552127
$324$	−10.9946	−0.610810
$325$	0	0
$326$	4.33627	0.240164
$327$	37.2295	2.05880
$328$	2.79223	0.154175
$329$	14.9544	0.824461
$330$	−0.706448	−0.0388887
$331$	−15.6598	−0.860740	−0.430370	−	0.902653i	$-0.641617\pi$
$331$	−15.6598	−0.860740	−0.430370	+	0.902653i	$0.641617\pi$
$332$	−11.1336	−0.611037
$333$	23.3790	1.28116
$334$	−21.8290	−1.19443
$335$	−12.7810	−0.698299
$336$	−3.20587	−0.174895
$337$	−28.8471	−1.57140	−0.785701	−	0.618606i	$-0.787698\pi$
$337$	−28.8471	−1.57140	−0.785701	+	0.618606i	$0.787698\pi$
$338$	0	0
$339$	−31.8501	−1.72986
$340$	1.69981	0.0921849
$341$	1.48231	0.0802717
$342$	11.7068	0.633032
$343$	17.1189	0.924331
$344$	4.77478	0.257439
$345$	20.8701	1.12361
$346$	4.20426	0.226023
$347$	−6.44079	−0.345760	−0.172880	−	0.984943i	$-0.555307\pi$
$347$	−6.44079	−0.345760	−0.172880	+	0.984943i	$0.555307\pi$
$348$	15.8744	0.850958
$349$	21.5301	1.15248	0.576239	−	0.817281i	$-0.304520\pi$
$349$	21.5301	1.15248	0.576239	+	0.817281i	$0.304520\pi$
$350$	1.43294	0.0765937
$351$	0	0
$352$	0.315763	0.0168302
$353$	−12.5441	−0.667657	−0.333828	−	0.942634i	$-0.608341\pi$
$353$	−12.5441	−0.667657	−0.333828	+	0.942634i	$0.608341\pi$
$354$	−8.81014	−0.468253
$355$	4.17629	0.221655
$356$	−1.48615	−0.0787659
$357$	5.44936	0.288411
$358$	23.5902	1.24678
$359$	9.39258	0.495721	0.247861	−	0.968796i	$-0.420272\pi$
$359$	9.39258	0.495721	0.247861	+	0.968796i	$0.420272\pi$
$360$	2.00539	0.105693
$361$	15.0785	0.793608
$362$	−16.4103	−0.862504
$363$	−24.3869	−1.27998
$364$	0	0
$365$	8.83875	0.462641
$366$	−2.17424	−0.113650
$367$	9.41908	0.491672	0.245836	−	0.969311i	$-0.420938\pi$
$367$	9.41908	0.491672	0.245836	+	0.969311i	$0.420938\pi$
$368$	−9.32839	−0.486276
$369$	−5.59950	−0.291498
$370$	11.6581	0.606075
$371$	2.13024	0.110597
$372$	−10.5026	−0.544534
$373$	11.4780	0.594310	0.297155	−	0.954829i	$-0.403962\pi$
$373$	11.4780	0.594310	0.297155	+	0.954829i	$0.403962\pi$
$374$	−0.536736	−0.0277540
$375$	−2.23727	−0.115532
$376$	10.4362	0.538204
$377$	0	0
$378$	−3.18860	−0.164004
$379$	17.6283	0.905505	0.452752	−	0.891636i	$-0.350442\pi$
$379$	17.6283	0.905505	0.452752	+	0.891636i	$0.350442\pi$
$380$	5.83768	0.299467
$381$	37.3424	1.91311
$382$	21.8818	1.11957
$383$	4.16819	0.212985	0.106492	−	0.994314i	$-0.466038\pi$
$383$	4.16819	0.212985	0.106492	+	0.994314i	$0.466038\pi$
$384$	−2.23727	−0.114170
$385$	−0.452469	−0.0230599
$386$	−0.440205	−0.0224058
$387$	−9.57528	−0.486739
$388$	11.9669	0.607528
$389$	35.7341	1.81179	0.905895	−	0.423503i	$-0.139200\pi$
$389$	35.7341	1.81179	0.905895	+	0.423503i	$0.139200\pi$
$390$	0	0
$391$	15.8565	0.801895
$392$	4.94669	0.249846
$393$	11.6872	0.589541
$394$	13.7762	0.694037
$395$	6.80085	0.342188
$396$	−0.633227	−0.0318209
$397$	−23.4645	−1.17765	−0.588824	−	0.808262i	$-0.700409\pi$
$397$	−23.4645	−1.17765	−0.588824	+	0.808262i	$0.700409\pi$
$398$	−2.53270	−0.126953
$399$	18.7149	0.936915
$400$	1.00000	0.0500000
$401$	−0.137896	−0.00688618	−0.00344309	−	0.999994i	$-0.501096\pi$
$401$	−0.137896	−0.00688618	−0.00344309	+	0.999994i	$0.501096\pi$
$402$	−28.5945	−1.42616
$403$	0	0
$404$	16.6774	0.829734
$405$	10.9946	0.546325
$406$	10.1673	0.504595
$407$	−3.68119	−0.182470
$408$	3.80293	0.188273
$409$	24.2501	1.19909	0.599546	−	0.800340i	$-0.295348\pi$
$409$	24.2501	1.19909	0.599546	+	0.800340i	$0.295348\pi$
$410$	−2.79223	−0.137898
$411$	−24.2578	−1.19655
$412$	−7.91612	−0.389999
$413$	−5.64275	−0.277662
$414$	18.7070	0.919400
$415$	11.1336	0.546528
$416$	0	0
$417$	28.5265	1.39695
$418$	−1.84333	−0.0901600
$419$	8.19191	0.400201	0.200100	−	0.979775i	$-0.435873\pi$
$419$	8.19191	0.400201	0.200100	+	0.979775i	$0.435873\pi$
$420$	3.20587	0.156430
$421$	24.7799	1.20770	0.603848	−	0.797099i	$-0.293633\pi$
$421$	24.7799	1.20770	0.603848	+	0.797099i	$0.293633\pi$
$422$	−6.69030	−0.325679
$423$	−20.9286	−1.01758
$424$	1.48663	0.0721969
$425$	−1.69981	−0.0824527
$426$	9.34350	0.452694
$427$	−1.39257	−0.0673911
$428$	−6.46441	−0.312469
$429$	0	0
$430$	−4.77478	−0.230260
$431$	1.94731	0.0937987	0.0468994	−	0.998900i	$-0.485066\pi$
$431$	1.94731	0.0937987	0.0468994	+	0.998900i	$0.485066\pi$
$432$	−2.22522	−0.107061
$433$	−23.7805	−1.14282	−0.571410	−	0.820665i	$-0.693603\pi$
$433$	−23.7805	−1.14282	−0.571410	+	0.820665i	$0.693603\pi$
$434$	−6.72675	−0.322894
$435$	−15.8744	−0.761120
$436$	16.6406	0.796939
$437$	54.4562	2.60499
$438$	19.7747	0.944871
$439$	33.4007	1.59413	0.797063	−	0.603896i	$-0.206386\pi$
$439$	33.4007	1.59413	0.797063	+	0.603896i	$0.206386\pi$
$440$	−0.315763	−0.0150534
$441$	−9.92003	−0.472382
$442$	0	0
$443$	15.5768	0.740075	0.370038	−	0.929017i	$-0.379345\pi$
$443$	15.5768	0.740075	0.370038	+	0.929017i	$0.379345\pi$
$444$	26.0823	1.23781
$445$	1.48615	0.0704503
$446$	−1.36952	−0.0648485
$447$	6.01135	0.284327
$448$	−1.43294	−0.0676999
$449$	−31.2239	−1.47355	−0.736773	−	0.676140i	$-0.763652\pi$
$449$	−31.2239	−1.47355	−0.736773	+	0.676140i	$0.763652\pi$
$450$	−2.00539	−0.0945349
$451$	0.881684	0.0415169
$452$	−14.2361	−0.669612
$453$	−34.3294	−1.61294
$454$	1.46657	0.0688296
$455$	0	0
$456$	13.0605	0.611613
$457$	22.4950	1.05227	0.526135	−	0.850401i	$-0.323641\pi$
$457$	22.4950	1.05227	0.526135	+	0.850401i	$0.323641\pi$
$458$	−7.36084	−0.343949
$459$	3.78244	0.176549
$460$	9.32839	0.434938
$461$	18.4663	0.860059	0.430030	−	0.902815i	$-0.358503\pi$
$461$	18.4663	0.860059	0.430030	+	0.902815i	$0.358503\pi$
$462$	−1.01230	−0.0470963
$463$	−20.3900	−0.947602	−0.473801	−	0.880632i	$-0.657118\pi$
$463$	−20.3900	−0.947602	−0.473801	+	0.880632i	$0.657118\pi$
$464$	7.09544	0.329397
$465$	10.5026	0.487046
$466$	−15.5806	−0.721757
$467$	−28.0498	−1.29799	−0.648996	−	0.760792i	$-0.724811\pi$
$467$	−28.0498	−1.29799	−0.648996	+	0.760792i	$0.724811\pi$
$468$	0	0
$469$	−18.3143	−0.845676
$470$	−10.4362	−0.481385
$471$	−12.5073	−0.576308
$472$	−3.93789	−0.181256
$473$	1.50770	0.0693241
$474$	15.2154	0.698865
$475$	−5.83768	−0.267851
$476$	2.43572	0.111641
$477$	−2.98126	−0.136503
$478$	−9.78398	−0.447509
$479$	8.09122	0.369697	0.184849	−	0.982767i	$-0.440820\pi$
$479$	8.09122	0.369697	0.184849	+	0.982767i	$0.440820\pi$
$480$	2.23727	0.102117
$481$	0	0
$482$	−11.7487	−0.535140
$483$	29.9056	1.36075
$484$	−10.9003	−0.495468
$485$	−11.9669	−0.543390
$486$	17.9222	0.812968
$487$	14.3897	0.652057	0.326029	−	0.945360i	$-0.394289\pi$
$487$	14.3897	0.652057	0.326029	+	0.945360i	$0.394289\pi$
$488$	−0.971828	−0.0439926
$489$	−9.70141	−0.438713
$490$	−4.94669	−0.223469
$491$	−10.2104	−0.460789	−0.230395	−	0.973097i	$-0.574002\pi$
$491$	−10.2104	−0.460789	−0.230395	+	0.973097i	$0.574002\pi$
$492$	−6.24698	−0.281636
$493$	−12.0609	−0.543194
$494$	0	0
$495$	0.633227	0.0284615
$496$	−4.69438	−0.210784
$497$	5.98436	0.268435
$498$	24.9089	1.11620
$499$	−3.13991	−0.140562	−0.0702809	−	0.997527i	$-0.522390\pi$
$499$	−3.13991	−0.140562	−0.0702809	+	0.997527i	$0.522390\pi$
$500$	−1.00000	−0.0447214
$501$	48.8375	2.18190
$502$	−29.0462	−1.29639
$503$	24.1794	1.07811	0.539054	−	0.842271i	$-0.318782\pi$
$503$	24.1794	1.07811	0.539054	+	0.842271i	$0.318782\pi$
$504$	2.87359	0.128000
$505$	−16.6774	−0.742137
$506$	−2.94556	−0.130946
$507$	0	0
$508$	16.6911	0.740546
$509$	15.4846	0.686343	0.343171	−	0.939273i	$-0.388499\pi$
$509$	15.4846	0.686343	0.343171	+	0.939273i	$0.388499\pi$
$510$	−3.80293	−0.168397
$511$	12.6654	0.560283
$512$	−1.00000	−0.0441942
$513$	12.9901	0.573528
$514$	15.6549	0.690506
$515$	7.91612	0.348826
$516$	−10.6825	−0.470270
$517$	3.29536	0.144930
$518$	16.7053	0.733989
$519$	−9.40608	−0.412881
$520$	0	0
$521$	8.76055	0.383807	0.191903	−	0.981414i	$-0.438534\pi$
$521$	8.76055	0.383807	0.191903	+	0.981414i	$0.438534\pi$
$522$	−14.2291	−0.622791
$523$	−3.75051	−0.163999	−0.0819993	−	0.996632i	$-0.526131\pi$
$523$	−3.75051	−0.163999	−0.0819993	+	0.996632i	$0.526131\pi$
$524$	5.22386	0.228205
$525$	−3.20587	−0.139916
$526$	11.7545	0.512520
$527$	7.97953	0.347594
$528$	−0.706448	−0.0307442
$529$	64.0188	2.78343
$530$	−1.48663	−0.0645749
$531$	7.89700	0.342701
$532$	8.36503	0.362670
$533$	0	0
$534$	3.32492	0.143884
$535$	6.46441	0.279481
$536$	−12.7810	−0.552053
$537$	−52.7777	−2.27753
$538$	0.786410	0.0339046
$539$	1.56198	0.0672794
$540$	2.22522	0.0957582
$541$	34.8297	1.49745	0.748723	−	0.662883i	$-0.230667\pi$
$541$	34.8297	1.49745	0.748723	+	0.662883i	$0.230667\pi$
$542$	1.63501	0.0702299
$543$	36.7142	1.57556
$544$	1.69981	0.0728786
$545$	−16.6406	−0.712804
$546$	0	0
$547$	−41.0027	−1.75315	−0.876575	−	0.481265i	$-0.840177\pi$
$547$	−41.0027	−1.75315	−0.876575	+	0.481265i	$0.840177\pi$
$548$	−10.8426	−0.463173
$549$	1.94889	0.0831766
$550$	0.315763	0.0134642
$551$	−41.4209	−1.76459
$552$	20.8701	0.888292
$553$	9.74519	0.414408
$554$	−0.567538	−0.0241124
$555$	−26.0823	−1.10713
$556$	12.7506	0.540745
$557$	−12.9939	−0.550569	−0.275285	−	0.961363i	$-0.588772\pi$
$557$	−12.9939	−0.550569	−0.275285	+	0.961363i	$0.588772\pi$
$558$	9.41404	0.398528
$559$	0	0
$560$	1.43294	0.0605526
$561$	1.20083	0.0506989
$562$	11.9390	0.503617
$563$	18.8597	0.794840	0.397420	−	0.917637i	$-0.369906\pi$
$563$	18.8597	0.794840	0.397420	+	0.917637i	$0.369906\pi$
$564$	−23.3486	−0.983152
$565$	14.2361	0.598919
$566$	14.0478	0.590472
$567$	15.7545	0.661629
$568$	4.17629	0.175233
$569$	−36.2095	−1.51798	−0.758991	−	0.651102i	$-0.774307\pi$
$569$	−36.2095	−1.51798	−0.758991	+	0.651102i	$0.774307\pi$
$570$	−13.0605	−0.547044
$571$	13.5330	0.566340	0.283170	−	0.959070i	$-0.408614\pi$
$571$	13.5330	0.566340	0.283170	+	0.959070i	$0.408614\pi$
$572$	0	0
$573$	−48.9556	−2.04515
$574$	−4.00109	−0.167002
$575$	−9.32839	−0.389021
$576$	2.00539	0.0835578
$577$	9.49114	0.395121	0.197561	−	0.980291i	$-0.436698\pi$
$577$	9.49114	0.395121	0.197561	+	0.980291i	$0.436698\pi$
$578$	14.1107	0.586926
$579$	0.984858	0.0409293
$580$	−7.09544	−0.294622
$581$	15.9538	0.661874
$582$	−26.7733	−1.10979
$583$	0.469422	0.0194415
$584$	8.83875	0.365750
$585$	0	0
$586$	28.7352	1.18704
$587$	10.2761	0.424138	0.212069	−	0.977255i	$-0.431980\pi$
$587$	10.2761	0.424138	0.212069	+	0.977255i	$0.431980\pi$
$588$	−11.0671	−0.456399
$589$	27.4043	1.12917
$590$	3.93789	0.162120
$591$	−30.8212	−1.26781
$592$	11.6581	0.479144
$593$	−36.5714	−1.50181	−0.750904	−	0.660411i	$-0.770382\pi$
$593$	−36.5714	−1.50181	−0.750904	+	0.660411i	$0.770382\pi$
$594$	−0.702643	−0.0288298
$595$	−2.43572	−0.0998546
$596$	2.68691	0.110060
$597$	5.66635	0.231908
$598$	0	0
$599$	−42.2976	−1.72823	−0.864116	−	0.503292i	$-0.832122\pi$
$599$	−42.2976	−1.72823	−0.864116	+	0.503292i	$0.832122\pi$
$600$	−2.23727	−0.0913363
$601$	17.6798	0.721173	0.360587	−	0.932726i	$-0.382576\pi$
$601$	17.6798	0.721173	0.360587	+	0.932726i	$0.382576\pi$
$602$	−6.84196	−0.278857
$603$	25.6308	1.04377
$604$	−15.3443	−0.624351
$605$	10.9003	0.443160
$606$	−37.3120	−1.51570
$607$	−40.2686	−1.63445	−0.817227	−	0.576316i	$-0.804490\pi$
$607$	−40.2686	−1.63445	−0.817227	+	0.576316i	$0.804490\pi$
$608$	5.83768	0.236749
$609$	−22.7471	−0.921757
$610$	0.971828	0.0393482
$611$	0	0
$612$	−3.40877	−0.137791
$613$	−4.11330	−0.166135	−0.0830673	−	0.996544i	$-0.526472\pi$
$613$	−4.11330	−0.166135	−0.0830673	+	0.996544i	$0.526472\pi$
$614$	−8.72162	−0.351976
$615$	6.24698	0.251903
$616$	−0.452469	−0.0182305
$617$	−28.2379	−1.13682	−0.568408	−	0.822747i	$-0.692440\pi$
$617$	−28.2379	−1.13682	−0.568408	+	0.822747i	$0.692440\pi$
$618$	17.7105	0.712422
$619$	7.31241	0.293911	0.146955	−	0.989143i	$-0.453053\pi$
$619$	7.31241	0.293911	0.146955	+	0.989143i	$0.453053\pi$
$620$	4.69438	0.188531
$621$	20.7577	0.832978
$622$	−23.6123	−0.946768
$623$	2.12956	0.0853191
$624$	0	0
$625$	1.00000	0.0400000
$626$	−22.3185	−0.892027
$627$	4.12402	0.164698
$628$	−5.59044	−0.223083
$629$	−19.8165	−0.790135
$630$	−2.87359	−0.114487
$631$	−36.0224	−1.43403	−0.717014	−	0.697059i	$-0.754492\pi$
$631$	−36.0224	−1.43403	−0.717014	+	0.697059i	$0.754492\pi$
$632$	6.80085	0.270523
$633$	14.9680	0.594926
$634$	22.3655	0.888248
$635$	−16.6911	−0.662364
$636$	−3.32599	−0.131884
$637$	0	0
$638$	2.24048	0.0887014
$639$	−8.37508	−0.331313
$640$	1.00000	0.0395285
$641$	11.3889	0.449835	0.224917	−	0.974378i	$-0.427789\pi$
$641$	11.3889	0.449835	0.224917	+	0.974378i	$0.427789\pi$
$642$	14.4627	0.570795
$643$	14.2162	0.560630	0.280315	−	0.959908i	$-0.409561\pi$
$643$	14.2162	0.560630	0.280315	+	0.959908i	$0.409561\pi$
$644$	13.3670	0.526733
$645$	10.6825	0.420622
$646$	−9.92293	−0.390412
$647$	−2.24397	−0.0882198	−0.0441099	−	0.999027i	$-0.514045\pi$
$647$	−2.24397	−0.0882198	−0.0441099	+	0.999027i	$0.514045\pi$
$648$	10.9946	0.431908
$649$	−1.24344	−0.0488093
$650$	0	0
$651$	15.0496	0.589839
$652$	−4.33627	−0.169821
$653$	21.9585	0.859301	0.429651	−	0.902995i	$-0.358637\pi$
$653$	21.9585	0.859301	0.429651	+	0.902995i	$0.358637\pi$
$654$	−37.2295	−1.45579
$655$	−5.22386	−0.204113
$656$	−2.79223	−0.109018
$657$	−17.7251	−0.691523
$658$	−14.9544	−0.582982
$659$	−29.0740	−1.13256	−0.566281	−	0.824212i	$-0.691618\pi$
$659$	−29.0740	−1.13256	−0.566281	+	0.824212i	$0.691618\pi$
$660$	0.706448	0.0274985
$661$	−35.2273	−1.37018	−0.685091	−	0.728457i	$-0.740238\pi$
$661$	−35.2273	−1.37018	−0.685091	+	0.728457i	$0.740238\pi$
$662$	15.6598	0.608635
$663$	0	0
$664$	11.1336	0.432068
$665$	−8.36503	−0.324382
$666$	−23.3790	−0.905917
$667$	−66.1890	−2.56285
$668$	21.8290	0.844590
$669$	3.06398	0.118460
$670$	12.7810	0.493772
$671$	−0.306867	−0.0118465
$672$	3.20587	0.123669
$673$	13.7113	0.528531	0.264266	−	0.964450i	$-0.414870\pi$
$673$	13.7113	0.528531	0.264266	+	0.964450i	$0.414870\pi$
$674$	28.8471	1.11115
$675$	−2.22522	−0.0856487
$676$	0	0
$677$	31.9818	1.22916	0.614581	−	0.788854i	$-0.289325\pi$
$677$	31.9818	1.22916	0.614581	+	0.788854i	$0.289325\pi$
$678$	31.8501	1.22320
$679$	−17.1478	−0.658074
$680$	−1.69981	−0.0651846
$681$	−3.28112	−0.125733
$682$	−1.48231	−0.0567606
$683$	3.53220	0.135156	0.0675779	−	0.997714i	$-0.478473\pi$
$683$	3.53220	0.135156	0.0675779	+	0.997714i	$0.478473\pi$
$684$	−11.7068	−0.447621
$685$	10.8426	0.414274
$686$	−17.1189	−0.653601
$687$	16.4682	0.628301
$688$	−4.77478	−0.182037
$689$	0	0
$690$	−20.8701	−0.794513
$691$	−39.4068	−1.49911	−0.749553	−	0.661944i	$-0.769732\pi$
$691$	−39.4068	−1.49911	−0.749553	+	0.661944i	$0.769732\pi$
$692$	−4.20426	−0.159822
$693$	0.907375	0.0344683
$694$	6.44079	0.244489
$695$	−12.7506	−0.483657
$696$	−15.8744	−0.601719
$697$	4.74625	0.179777
$698$	−21.5301	−0.814925
$699$	34.8580	1.31845
$700$	−1.43294	−0.0541599
$701$	26.5304	1.00204	0.501020	−	0.865436i	$-0.332958\pi$
$701$	26.5304	1.00204	0.501020	+	0.865436i	$0.332958\pi$
$702$	0	0
$703$	−68.0562	−2.56679
$704$	−0.315763	−0.0119008
$705$	23.3486	0.879358
$706$	12.5441	0.472105
$707$	−23.8977	−0.898767
$708$	8.81014	0.331105
$709$	−42.0988	−1.58106	−0.790528	−	0.612426i	$-0.790194\pi$
$709$	−42.0988	−1.58106	−0.790528	+	0.612426i	$0.790194\pi$
$710$	−4.17629	−0.156733
$711$	−13.6383	−0.511478
$712$	1.48615	0.0556959
$713$	43.7910	1.63998
$714$	−5.44936	−0.203937
$715$	0	0
$716$	−23.5902	−0.881608
$717$	21.8894	0.817475
$718$	−9.39258	−0.350528
$719$	21.0467	0.784911	0.392455	−	0.919771i	$-0.371626\pi$
$719$	21.0467	0.784911	0.392455	+	0.919771i	$0.371626\pi$
$720$	−2.00539	−0.0747364
$721$	11.3433	0.422447
$722$	−15.0785	−0.561165
$723$	26.2851	0.977553
$724$	16.4103	0.609882
$725$	7.09544	0.263518
$726$	24.3869	0.905084
$727$	−7.85688	−0.291396	−0.145698	−	0.989329i	$-0.546543\pi$
$727$	−7.85688	−0.291396	−0.145698	+	0.989329i	$0.546543\pi$
$728$	0	0
$729$	−7.11313	−0.263449
$730$	−8.83875	−0.327137
$731$	8.11620	0.300189
$732$	2.17424	0.0803624
$733$	−40.1151	−1.48168	−0.740842	−	0.671679i	$-0.765573\pi$
$733$	−40.1151	−1.48168	−0.740842	+	0.671679i	$0.765573\pi$
$734$	−9.41908	−0.347665
$735$	11.0671	0.408216
$736$	9.32839	0.343849
$737$	−4.03576	−0.148659
$738$	5.59950	0.206121
$739$	34.1597	1.25658	0.628292	−	0.777977i	$-0.283754\pi$
$739$	34.1597	1.25658	0.628292	+	0.777977i	$0.283754\pi$
$740$	−11.6581	−0.428560
$741$	0	0
$742$	−2.13024	−0.0782036
$743$	−33.6854	−1.23580	−0.617898	−	0.786258i	$-0.712016\pi$
$743$	−33.6854	−1.23580	−0.617898	+	0.786258i	$0.712016\pi$
$744$	10.5026	0.385044
$745$	−2.68691	−0.0984408
$746$	−11.4780	−0.420241
$747$	−22.3272	−0.816910
$748$	0.536736	0.0196250
$749$	9.26310	0.338466
$750$	2.23727	0.0816936
$751$	9.13590	0.333374	0.166687	−	0.986010i	$-0.446693\pi$
$751$	9.13590	0.333374	0.166687	+	0.986010i	$0.446693\pi$
$752$	−10.4362	−0.380568
$753$	64.9842	2.36816
$754$	0	0
$755$	15.3443	0.558437
$756$	3.18860	0.115968
$757$	47.3897	1.72241	0.861203	−	0.508261i	$-0.169711\pi$
$757$	47.3897	1.72241	0.861203	+	0.508261i	$0.169711\pi$
$758$	−17.6283	−0.640289
$759$	6.59002	0.239203
$760$	−5.83768	−0.211755
$761$	2.34689	0.0850747	0.0425374	−	0.999095i	$-0.486456\pi$
$761$	2.34689	0.0850747	0.0425374	+	0.999095i	$0.486456\pi$
$762$	−37.3424	−1.35277
$763$	−23.8449	−0.863244
$764$	−21.8818	−0.791656
$765$	3.40877	0.123244
$766$	−4.16819	−0.150603
$767$	0	0
$768$	2.23727	0.0807306
$769$	−40.3538	−1.45519	−0.727597	−	0.686004i	$-0.759363\pi$
$769$	−40.3538	−1.45519	−0.727597	+	0.686004i	$0.759363\pi$
$770$	0.452469	0.0163058
$771$	−35.0242	−1.26137
$772$	0.440205	0.0158433
$773$	−36.3971	−1.30911	−0.654556	−	0.756013i	$-0.727144\pi$
$773$	−36.3971	−1.30911	−0.654556	+	0.756013i	$0.727144\pi$
$774$	9.57528	0.344176
$775$	−4.69438	−0.168627
$776$	−11.9669	−0.429587
$777$	−37.3743	−1.34080
$778$	−35.7341	−1.28113
$779$	16.3002	0.584014
$780$	0	0
$781$	1.31872	0.0471875
$782$	−15.8565	−0.567026
$783$	−15.7889	−0.564249
$784$	−4.94669	−0.176668
$785$	5.59044	0.199531
$786$	−11.6872	−0.416869
$787$	18.5638	0.661729	0.330864	−	0.943678i	$-0.392660\pi$
$787$	18.5638	0.661729	0.330864	+	0.943678i	$0.392660\pi$
$788$	−13.7762	−0.490758
$789$	−26.2980	−0.936234
$790$	−6.80085	−0.241963
$791$	20.3995	0.725323
$792$	0.633227	0.0225008
$793$	0	0
$794$	23.4645	0.832722
$795$	3.32599	0.117961
$796$	2.53270	0.0897693
$797$	15.9864	0.566268	0.283134	−	0.959080i	$-0.408626\pi$
$797$	15.9864	0.566268	0.283134	+	0.959080i	$0.408626\pi$
$798$	−18.7149	−0.662499
$799$	17.7395	0.627577
$800$	−1.00000	−0.0353553
$801$	−2.98031	−0.105304
$802$	0.137896	0.00486927
$803$	2.79095	0.0984906
$804$	28.5945	1.00845
$805$	−13.3670	−0.471125
$806$	0	0
$807$	−1.75941	−0.0619343
$808$	−16.6774	−0.586711
$809$	6.97062	0.245074	0.122537	−	0.992464i	$-0.460897\pi$
$809$	6.97062	0.245074	0.122537	+	0.992464i	$0.460897\pi$
$810$	−10.9946	−0.386310
$811$	−24.0216	−0.843511	−0.421755	−	0.906710i	$-0.638586\pi$
$811$	−24.0216	−0.843511	−0.421755	+	0.906710i	$0.638586\pi$
$812$	−10.1673	−0.356803
$813$	−3.65797	−0.128291
$814$	3.68119	0.129026
$815$	4.33627	0.151893
$816$	−3.80293	−0.133129
$817$	27.8737	0.975176
$818$	−24.2501	−0.847886
$819$	0	0
$820$	2.79223	0.0975089
$821$	−27.7112	−0.967126	−0.483563	−	0.875309i	$-0.660658\pi$
$821$	−27.7112	−0.967126	−0.483563	+	0.875309i	$0.660658\pi$
$822$	24.2578	0.846089
$823$	−24.4911	−0.853706	−0.426853	−	0.904321i	$-0.640378\pi$
$823$	−24.4911	−0.853706	−0.426853	+	0.904321i	$0.640378\pi$
$824$	7.91612	0.275771
$825$	−0.706448	−0.0245954
$826$	5.64275	0.196336
$827$	18.0841	0.628846	0.314423	−	0.949283i	$-0.398189\pi$
$827$	18.0841	0.628846	0.314423	+	0.949283i	$0.398189\pi$
$828$	−18.7070	−0.650114
$829$	11.4496	0.397661	0.198831	−	0.980034i	$-0.436286\pi$
$829$	11.4496	0.397661	0.198831	+	0.980034i	$0.436286\pi$
$830$	−11.1336	−0.386454
$831$	1.26974	0.0440467
$832$	0	0
$833$	8.40842	0.291334
$834$	−28.5265	−0.987792
$835$	−21.8290	−0.755425
$836$	1.84333	0.0637527
$837$	10.4460	0.361067
$838$	−8.19191	−0.282985
$839$	−28.4327	−0.981605	−0.490803	−	0.871271i	$-0.663296\pi$
$839$	−28.4327	−0.981605	−0.490803	+	0.871271i	$0.663296\pi$
$840$	−3.20587	−0.110613
$841$	21.3452	0.736042
$842$	−24.7799	−0.853971
$843$	−26.7108	−0.919969
$844$	6.69030	0.230290
$845$	0	0
$846$	20.9286	0.719539
$847$	15.6194	0.536690
$848$	−1.48663	−0.0510510
$849$	−31.4287	−1.07863
$850$	1.69981	0.0583029
$851$	−108.751	−3.72794
$852$	−9.34350	−0.320103
$853$	7.19717	0.246426	0.123213	−	0.992380i	$-0.460680\pi$
$853$	7.19717	0.246426	0.123213	+	0.992380i	$0.460680\pi$
$854$	1.39257	0.0476527
$855$	11.7068	0.400365
$856$	6.46441	0.220949
$857$	30.2786	1.03430	0.517148	−	0.855896i	$-0.326994\pi$
$857$	30.2786	1.03430	0.517148	+	0.855896i	$0.326994\pi$
$858$	0	0
$859$	−11.0133	−0.375769	−0.187884	−	0.982191i	$-0.560163\pi$
$859$	−11.0133	−0.375769	−0.187884	+	0.982191i	$0.560163\pi$
$860$	4.77478	0.162819
$861$	8.95153	0.305067
$862$	−1.94731	−0.0663257
$863$	22.5488	0.767571	0.383786	−	0.923422i	$-0.374620\pi$
$863$	22.5488	0.767571	0.383786	+	0.923422i	$0.374620\pi$
$864$	2.22522	0.0757035
$865$	4.20426	0.142949
$866$	23.7805	0.808095
$867$	−31.5694	−1.07215
$868$	6.72675	0.228321
$869$	2.14746	0.0728476
$870$	15.8744	0.538193
$871$	0	0
$872$	−16.6406	−0.563521
$873$	23.9983	0.812220
$874$	−54.4562	−1.84201
$875$	1.43294	0.0484421
$876$	−19.7747	−0.668125
$877$	−55.2018	−1.86403	−0.932016	−	0.362418i	$-0.881951\pi$
$877$	−55.2018	−1.86403	−0.932016	+	0.362418i	$0.881951\pi$
$878$	−33.4007	−1.12722
$879$	−64.2885	−2.16840
$880$	0.315763	0.0106444
$881$	−1.80062	−0.0606644	−0.0303322	−	0.999540i	$-0.509657\pi$
$881$	−1.80062	−0.0606644	−0.0303322	+	0.999540i	$0.509657\pi$
$882$	9.92003	0.334025
$883$	−28.0786	−0.944919	−0.472459	−	0.881352i	$-0.656634\pi$
$883$	−28.0786	−0.944919	−0.472459	+	0.881352i	$0.656634\pi$
$884$	0	0
$885$	−8.81014	−0.296150
$886$	−15.5768	−0.523312
$887$	−3.74577	−0.125771	−0.0628854	−	0.998021i	$-0.520030\pi$
$887$	−3.74577	−0.125771	−0.0628854	+	0.998021i	$0.520030\pi$
$888$	−26.0823	−0.875265
$889$	−23.9172	−0.802158
$890$	−1.48615	−0.0498159
$891$	3.47169	0.116306
$892$	1.36952	0.0458548
$893$	60.9231	2.03871
$894$	−6.01135	−0.201050
$895$	23.5902	0.788534
$896$	1.43294	0.0478711
$897$	0	0
$898$	31.2239	1.04195
$899$	−33.3087	−1.11091
$900$	2.00539	0.0668462
$901$	2.52698	0.0841858
$902$	−0.881684	−0.0293569
$903$	15.3073	0.509396
$904$	14.2361	0.473487
$905$	−16.4103	−0.545495
$906$	34.3294	1.14052
$907$	26.8418	0.891269	0.445634	−	0.895215i	$-0.352978\pi$
$907$	26.8418	0.891269	0.445634	+	0.895215i	$0.352978\pi$
$908$	−1.46657	−0.0486699
$909$	33.4447	1.10929
$910$	0	0
$911$	−22.6354	−0.749945	−0.374972	−	0.927036i	$-0.622348\pi$
$911$	−22.6354	−0.749945	−0.374972	+	0.927036i	$0.622348\pi$
$912$	−13.0605	−0.432476
$913$	3.51559	0.116349
$914$	−22.4950	−0.744067
$915$	−2.17424	−0.0718783
$916$	7.36084	0.243209
$917$	−7.48546	−0.247192
$918$	−3.78244	−0.124839
$919$	5.61518	0.185228	0.0926138	−	0.995702i	$-0.470478\pi$
$919$	5.61518	0.185228	0.0926138	+	0.995702i	$0.470478\pi$
$920$	−9.32839	−0.307548
$921$	19.5126	0.642963
$922$	−18.4663	−0.608154
$923$	0	0
$924$	1.01230	0.0333021
$925$	11.6581	0.383315
$926$	20.3900	0.670056
$927$	−15.8749	−0.521400
$928$	−7.09544	−0.232919
$929$	−10.9655	−0.359766	−0.179883	−	0.983688i	$-0.557572\pi$
$929$	−10.9655	−0.359766	−0.179883	+	0.983688i	$0.557572\pi$
$930$	−10.5026	−0.344394
$931$	28.8772	0.946413
$932$	15.5806	0.510359
$933$	52.8272	1.72949
$934$	28.0498	0.917819
$935$	−0.536736	−0.0175532
$936$	0	0
$937$	53.8795	1.76017	0.880083	−	0.474820i	$-0.157487\pi$
$937$	53.8795	1.76017	0.880083	+	0.474820i	$0.157487\pi$
$938$	18.3143	0.597984
$939$	49.9326	1.62949
$940$	10.4362	0.340390
$941$	−7.42918	−0.242184	−0.121092	−	0.992641i	$-0.538640\pi$
$941$	−7.42918	−0.242184	−0.121092	+	0.992641i	$0.538640\pi$
$942$	12.5073	0.407511
$943$	26.0470	0.848207
$944$	3.93789	0.128167
$945$	−3.18860	−0.103725
$946$	−1.50770	−0.0490196
$947$	−15.5223	−0.504407	−0.252204	−	0.967674i	$-0.581155\pi$
$947$	−15.5223	−0.504407	−0.252204	+	0.967674i	$0.581155\pi$
$948$	−15.2154	−0.494172
$949$	0	0
$950$	5.83768	0.189399
$951$	−50.0377	−1.62258
$952$	−2.43572	−0.0789420
$953$	−13.4032	−0.434171	−0.217085	−	0.976153i	$-0.569655\pi$
$953$	−13.4032	−0.434171	−0.217085	+	0.976153i	$0.569655\pi$
$954$	2.98126	0.0965219
$955$	21.8818	0.708079
$956$	9.78398	0.316436
$957$	−5.01256	−0.162033
$958$	−8.09122	−0.261416
$959$	15.5368	0.501708
$960$	−2.23727	−0.0722077
$961$	−8.96282	−0.289123
$962$	0	0
$963$	−12.9637	−0.417748
$964$	11.7487	0.378401
$965$	−0.440205	−0.0141707
$966$	−29.9056	−0.962197
$967$	−27.1373	−0.872677	−0.436338	−	0.899783i	$-0.643725\pi$
$967$	−27.1373	−0.872677	−0.436338	+	0.899783i	$0.643725\pi$
$968$	10.9003	0.350349
$969$	22.2003	0.713176
$970$	11.9669	0.384235
$971$	−54.9714	−1.76411	−0.882057	−	0.471142i	$-0.843842\pi$
$971$	−54.9714	−1.76411	−0.882057	+	0.471142i	$0.843842\pi$
$972$	−17.9222	−0.574855
$973$	−18.2708	−0.585734
$974$	−14.3897	−0.461074
$975$	0	0
$976$	0.971828	0.0311075
$977$	−58.0043	−1.85572	−0.927860	−	0.372928i	$-0.878354\pi$
$977$	−58.0043	−1.85572	−0.927860	+	0.372928i	$0.878354\pi$
$978$	9.70141	0.310217
$979$	0.469272	0.0149980
$980$	4.94669	0.158016
$981$	33.3708	1.06545
$982$	10.2104	0.325827
$983$	26.4277	0.842914	0.421457	−	0.906848i	$-0.361519\pi$
$983$	26.4277	0.842914	0.421457	+	0.906848i	$0.361519\pi$
$984$	6.24698	0.199146
$985$	13.7762	0.438947
$986$	12.0609	0.384096
$987$	33.4570	1.06495
$988$	0	0
$989$	44.5410	1.41632
$990$	−0.633227	−0.0201253
$991$	−18.2519	−0.579790	−0.289895	−	0.957059i	$-0.593620\pi$
$991$	−18.2519	−0.579790	−0.289895	+	0.957059i	$0.593620\pi$
$992$	4.69438	0.149047
$993$	−35.0352	−1.11181
$994$	−5.98436	−0.189812
$995$	−2.53270	−0.0802921
$996$	−24.9089	−0.789270
$997$	−9.13634	−0.289351	−0.144675	−	0.989479i	$-0.546214\pi$
$997$	−9.13634	−0.289351	−0.144675	+	0.989479i	$0.546214\pi$
$998$	3.13991	0.0993922
$999$	−25.9418	−0.820762

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1690.2.a.v.1.5	✓	6
5.4	even	2		8450.2.a.cq.1.2		6
13.2	odd	12		1690.2.l.n.1161.5		24
13.3	even	3		1690.2.e.v.191.2		12
13.4	even	6		1690.2.e.u.991.2		12
13.5	odd	4		1690.2.d.l.1351.11		12
13.6	odd	12		1690.2.l.n.361.11		24
13.7	odd	12		1690.2.l.n.361.5		24
13.8	odd	4		1690.2.d.l.1351.5		12
13.9	even	3		1690.2.e.v.991.2		12
13.10	even	6		1690.2.e.u.191.2		12
13.11	odd	12		1690.2.l.n.1161.11		24
13.12	even	2		1690.2.a.w.1.5	yes	6
65.64	even	2		8450.2.a.cp.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1690.2.a.v.1.5	✓	6	1.1	even	1	trivial
1690.2.a.w.1.5	yes	6	13.12	even	2
1690.2.d.l.1351.5		12	13.8	odd	4
1690.2.d.l.1351.11		12	13.5	odd	4
1690.2.e.u.191.2		12	13.10	even	6
1690.2.e.u.991.2		12	13.4	even	6
1690.2.e.v.191.2		12	13.3	even	3
1690.2.e.v.991.2		12	13.9	even	3
1690.2.l.n.361.5		24	13.7	odd	12
1690.2.l.n.361.11		24	13.6	odd	12
1690.2.l.n.1161.5		24	13.2	odd	12
1690.2.l.n.1161.11		24	13.11	odd	12
8450.2.a.cp.1.2		6	65.64	even	2
8450.2.a.cq.1.2		6	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1690 = 2 \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1690.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +2.23727 q^{3} +1.00000 q^{4} -1.00000 q^{5} -2.23727 q^{6} -1.43294 q^{7} -1.00000 q^{8} +2.00539 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +2.23727 q^{3} +1.00000 q^{4} -1.00000 q^{5} -2.23727 q^{6} -1.43294 q^{7} -1.00000 q^{8} +2.00539 q^{9} +1.00000 q^{10} -0.315763 q^{11} +2.23727 q^{12} +1.43294 q^{14} -2.23727 q^{15} +1.00000 q^{16} -1.69981 q^{17} -2.00539 q^{18} -5.83768 q^{19} -1.00000 q^{20} -3.20587 q^{21} +0.315763 q^{22} -9.32839 q^{23} -2.23727 q^{24} +1.00000 q^{25} -2.22522 q^{27} -1.43294 q^{28} +7.09544 q^{29} +2.23727 q^{30} -4.69438 q^{31} -1.00000 q^{32} -0.706448 q^{33} +1.69981 q^{34} +1.43294 q^{35} +2.00539 q^{36} +11.6581 q^{37} +5.83768 q^{38} +1.00000 q^{40} -2.79223 q^{41} +3.20587 q^{42} -4.77478 q^{43} -0.315763 q^{44} -2.00539 q^{45} +9.32839 q^{46} -10.4362 q^{47} +2.23727 q^{48} -4.94669 q^{49} -1.00000 q^{50} -3.80293 q^{51} -1.48663 q^{53} +2.22522 q^{54} +0.315763 q^{55} +1.43294 q^{56} -13.0605 q^{57} -7.09544 q^{58} +3.93789 q^{59} -2.23727 q^{60} +0.971828 q^{61} +4.69438 q^{62} -2.87359 q^{63} +1.00000 q^{64} +0.706448 q^{66} +12.7810 q^{67} -1.69981 q^{68} -20.8701 q^{69} -1.43294 q^{70} -4.17629 q^{71} -2.00539 q^{72} -8.83875 q^{73} -11.6581 q^{74} +2.23727 q^{75} -5.83768 q^{76} +0.452469 q^{77} -6.80085 q^{79} -1.00000 q^{80} -10.9946 q^{81} +2.79223 q^{82} -11.1336 q^{83} -3.20587 q^{84} +1.69981 q^{85} +4.77478 q^{86} +15.8744 q^{87} +0.315763 q^{88} -1.48615 q^{89} +2.00539 q^{90} -9.32839 q^{92} -10.5026 q^{93} +10.4362 q^{94} +5.83768 q^{95} -2.23727 q^{96} +11.9669 q^{97} +4.94669 q^{98} -0.633227 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{2} - 2 q^{3} + 6 q^{4} - 6 q^{5} + 2 q^{6} - 3 q^{7} - 6 q^{8} + 16 q^{9}+O(q^{10}) \) 6 * q - 6 * q^2 - 2 * q^3 + 6 * q^4 - 6 * q^5 + 2 * q^6 - 3 * q^7 - 6 * q^8 + 16 * q^9	\( 6 q - 6 q^{2} - 2 q^{3} + 6 q^{4} - 6 q^{5} + 2 q^{6} - 3 q^{7} - 6 q^{8} + 16 q^{9} + 6 q^{10} - 15 q^{11} - 2 q^{12} + 3 q^{14} + 2 q^{15} + 6 q^{16} - 3 q^{17} - 16 q^{18} - q^{19} - 6 q^{20} + 2 q^{21} + 15 q^{22} - 3 q^{23} + 2 q^{24} + 6 q^{25} - 20 q^{27} - 3 q^{28} + 7 q^{29} - 2 q^{30} - 6 q^{32} - 4 q^{33} + 3 q^{34} + 3 q^{35} + 16 q^{36} - 6 q^{37} + q^{38} + 6 q^{40} - 2 q^{41} - 2 q^{42} - 22 q^{43} - 15 q^{44} - 16 q^{45} + 3 q^{46} - 7 q^{47} - 2 q^{48} + 31 q^{49} - 6 q^{50} - 22 q^{51} - 16 q^{53} + 20 q^{54} + 15 q^{55} + 3 q^{56} - 2 q^{57} - 7 q^{58} - 15 q^{59} + 2 q^{60} + 33 q^{61} - 25 q^{63} + 6 q^{64} + 4 q^{66} + 8 q^{67} - 3 q^{68} - 6 q^{69} - 3 q^{70} - 40 q^{71} - 16 q^{72} - 21 q^{73} + 6 q^{74} - 2 q^{75} - q^{76} - 34 q^{77} + 20 q^{79} - 6 q^{80} - 2 q^{81} + 2 q^{82} - 22 q^{83} + 2 q^{84} + 3 q^{85} + 22 q^{86} - 39 q^{87} + 15 q^{88} - 20 q^{89} + 16 q^{90} - 3 q^{92} - 48 q^{93} + 7 q^{94} + q^{95} + 2 q^{96} + 7 q^{97} - 31 q^{98} - 55 q^{99}+O(q^{100}) \) 6 * q - 6 * q^2 - 2 * q^3 + 6 * q^4 - 6 * q^5 + 2 * q^6 - 3 * q^7 - 6 * q^8 + 16 * q^9 + 6 * q^10 - 15 * q^11 - 2 * q^12 + 3 * q^14 + 2 * q^15 + 6 * q^16 - 3 * q^17 - 16 * q^18 - q^19 - 6 * q^20 + 2 * q^21 + 15 * q^22 - 3 * q^23 + 2 * q^24 + 6 * q^25 - 20 * q^27 - 3 * q^28 + 7 * q^29 - 2 * q^30 - 6 * q^32 - 4 * q^33 + 3 * q^34 + 3 * q^35 + 16 * q^36 - 6 * q^37 + q^38 + 6 * q^40 - 2 * q^41 - 2 * q^42 - 22 * q^43 - 15 * q^44 - 16 * q^45 + 3 * q^46 - 7 * q^47 - 2 * q^48 + 31 * q^49 - 6 * q^50 - 22 * q^51 - 16 * q^53 + 20 * q^54 + 15 * q^55 + 3 * q^56 - 2 * q^57 - 7 * q^58 - 15 * q^59 + 2 * q^60 + 33 * q^61 - 25 * q^63 + 6 * q^64 + 4 * q^66 + 8 * q^67 - 3 * q^68 - 6 * q^69 - 3 * q^70 - 40 * q^71 - 16 * q^72 - 21 * q^73 + 6 * q^74 - 2 * q^75 - q^76 - 34 * q^77 + 20 * q^79 - 6 * q^80 - 2 * q^81 + 2 * q^82 - 22 * q^83 + 2 * q^84 + 3 * q^85 + 22 * q^86 - 39 * q^87 + 15 * q^88 - 20 * q^89 + 16 * q^90 - 3 * q^92 - 48 * q^93 + 7 * q^94 + q^95 + 2 * q^96 + 7 * q^97 - 31 * q^98 - 55 * q^99

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