LMFDB - Embedded newform 445.2.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [445,2,Mod(1,445)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(445, 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("445.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 445 = 5 \cdot 89 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	445.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:	$3.55334288995$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 8x^{5} + 6x^{4} + 19x^{3} - 10x^{2} - 12x + 6 $ x^7 - x^6 - 8x^5 + 6x^4 + 19x^3 - 10x^2 - 12*x + 6
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.498937$ of defining polynomial
Character	$\chi$	$=$	445.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.75106 q^{2} -0.459905 q^{3} +5.56834 q^{4} +1.00000 q^{5} +1.26523 q^{6} +0.587818 q^{7} -9.81674 q^{8} -2.78849 q^{9} +O(q^{10})$	\(q-2.75106 q^{2} -0.459905 q^{3} +5.56834 q^{4} +1.00000 q^{5} +1.26523 q^{6} +0.587818 q^{7} -9.81674 q^{8} -2.78849 q^{9} -2.75106 q^{10} -0.892165 q^{11} -2.56091 q^{12} -1.31436 q^{13} -1.61712 q^{14} -0.459905 q^{15} +15.8698 q^{16} -6.89306 q^{17} +7.67130 q^{18} +2.78585 q^{19} +5.56834 q^{20} -0.270341 q^{21} +2.45440 q^{22} -0.636773 q^{23} +4.51477 q^{24} +1.00000 q^{25} +3.61587 q^{26} +2.66216 q^{27} +3.27317 q^{28} +5.55621 q^{29} +1.26523 q^{30} -9.03241 q^{31} -24.0252 q^{32} +0.410311 q^{33} +18.9632 q^{34} +0.587818 q^{35} -15.5273 q^{36} +0.632709 q^{37} -7.66404 q^{38} +0.604479 q^{39} -9.81674 q^{40} -7.36941 q^{41} +0.743724 q^{42} -7.41164 q^{43} -4.96788 q^{44} -2.78849 q^{45} +1.75180 q^{46} -6.16627 q^{47} -7.29859 q^{48} -6.65447 q^{49} -2.75106 q^{50} +3.17016 q^{51} -7.31878 q^{52} +3.18988 q^{53} -7.32376 q^{54} -0.892165 q^{55} -5.77045 q^{56} -1.28123 q^{57} -15.2855 q^{58} +5.68022 q^{59} -2.56091 q^{60} -4.98727 q^{61} +24.8487 q^{62} -1.63912 q^{63} +34.3554 q^{64} -1.31436 q^{65} -1.12879 q^{66} +7.78542 q^{67} -38.3829 q^{68} +0.292855 q^{69} -1.61712 q^{70} -14.9564 q^{71} +27.3738 q^{72} +10.9297 q^{73} -1.74062 q^{74} -0.459905 q^{75} +15.5126 q^{76} -0.524431 q^{77} -1.66296 q^{78} -11.7250 q^{79} +15.8698 q^{80} +7.14112 q^{81} +20.2737 q^{82} -14.3750 q^{83} -1.50535 q^{84} -6.89306 q^{85} +20.3899 q^{86} -2.55533 q^{87} +8.75815 q^{88} +1.00000 q^{89} +7.67130 q^{90} -0.772602 q^{91} -3.54577 q^{92} +4.15405 q^{93} +16.9638 q^{94} +2.78585 q^{95} +11.0493 q^{96} +5.86420 q^{97} +18.3069 q^{98} +2.48779 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 4 q^{2} - 8 q^{3} + 8 q^{4} + 7 q^{5} - 2 q^{6} - 16 q^{7} - 12 q^{8} + 11 q^{9}+O(q^{10}) $ 7 * q - 4 * q^2 - 8 * q^3 + 8 * q^4 + 7 * q^5 - 2 * q^6 - 16 * q^7 - 12 * q^8 + 11 * q^9	\( 7 q - 4 q^{2} - 8 q^{3} + 8 q^{4} + 7 q^{5} - 2 q^{6} - 16 q^{7} - 12 q^{8} + 11 q^{9} - 4 q^{10} - 10 q^{11} - 11 q^{12} - 7 q^{13} + 3 q^{14} - 8 q^{15} + 10 q^{16} - 13 q^{17} + 4 q^{18} - 7 q^{19} + 8 q^{20} + 16 q^{21} + 2 q^{22} - 13 q^{23} + 4 q^{24} + 7 q^{25} + q^{26} - 23 q^{27} - 21 q^{28} - 4 q^{29} - 2 q^{30} + q^{31} - 13 q^{32} - 6 q^{33} + 10 q^{34} - 16 q^{35} + 20 q^{36} - 5 q^{37} - 40 q^{38} - 13 q^{39} - 12 q^{40} + 5 q^{41} + 30 q^{42} - 31 q^{43} - 21 q^{44} + 11 q^{45} + 16 q^{46} - 14 q^{47} - 7 q^{48} + 19 q^{49} - 4 q^{50} - q^{51} - 13 q^{53} - 17 q^{54} - 10 q^{55} - q^{56} + 21 q^{57} + 17 q^{58} - 14 q^{59} - 11 q^{60} + 3 q^{61} + 26 q^{62} - 54 q^{63} + 14 q^{64} - 7 q^{65} + 36 q^{66} + q^{67} - 35 q^{68} + 31 q^{69} + 3 q^{70} - 8 q^{71} + 53 q^{72} + 9 q^{73} - 35 q^{74} - 8 q^{75} + 40 q^{76} + 42 q^{77} + 46 q^{78} + 9 q^{79} + 10 q^{80} + 35 q^{81} + 29 q^{82} - 42 q^{83} + 55 q^{84} - 13 q^{85} + 35 q^{86} + 6 q^{87} + 30 q^{88} + 7 q^{89} + 4 q^{90} + 31 q^{91} + 19 q^{92} + 24 q^{93} + 37 q^{94} - 7 q^{95} + 44 q^{96} - 7 q^{97} + 9 q^{98} + 5 q^{99}+O(q^{100}) \) 7 * q - 4 * q^2 - 8 * q^3 + 8 * q^4 + 7 * q^5 - 2 * q^6 - 16 * q^7 - 12 * q^8 + 11 * q^9 - 4 * q^10 - 10 * q^11 - 11 * q^12 - 7 * q^13 + 3 * q^14 - 8 * q^15 + 10 * q^16 - 13 * q^17 + 4 * q^18 - 7 * q^19 + 8 * q^20 + 16 * q^21 + 2 * q^22 - 13 * q^23 + 4 * q^24 + 7 * q^25 + q^26 - 23 * q^27 - 21 * q^28 - 4 * q^29 - 2 * q^30 + q^31 - 13 * q^32 - 6 * q^33 + 10 * q^34 - 16 * q^35 + 20 * q^36 - 5 * q^37 - 40 * q^38 - 13 * q^39 - 12 * q^40 + 5 * q^41 + 30 * q^42 - 31 * q^43 - 21 * q^44 + 11 * q^45 + 16 * q^46 - 14 * q^47 - 7 * q^48 + 19 * q^49 - 4 * q^50 - q^51 - 13 * q^53 - 17 * q^54 - 10 * q^55 - q^56 + 21 * q^57 + 17 * q^58 - 14 * q^59 - 11 * q^60 + 3 * q^61 + 26 * q^62 - 54 * q^63 + 14 * q^64 - 7 * q^65 + 36 * q^66 + q^67 - 35 * q^68 + 31 * q^69 + 3 * q^70 - 8 * q^71 + 53 * q^72 + 9 * q^73 - 35 * q^74 - 8 * q^75 + 40 * q^76 + 42 * q^77 + 46 * q^78 + 9 * q^79 + 10 * q^80 + 35 * q^81 + 29 * q^82 - 42 * q^83 + 55 * q^84 - 13 * q^85 + 35 * q^86 + 6 * q^87 + 30 * q^88 + 7 * q^89 + 4 * q^90 + 31 * q^91 + 19 * q^92 + 24 * q^93 + 37 * q^94 - 7 * q^95 + 44 * q^96 - 7 * q^97 + 9 * q^98 + 5 * 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 $

$2$	−2.75106	−1.94529	−0.972647	−	0.232287i	$-0.925379\pi$
$2$	−2.75106	−1.94529	−0.972647	+	0.232287i	$0.925379\pi$
$3$	−0.459905	−0.265526	−0.132763	−	0.991148i	$-0.542385\pi$
$3$	−0.459905	−0.265526	−0.132763	+	0.991148i	$0.542385\pi$
$4$	5.56834	2.78417
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	1.26523	0.516527
$7$	0.587818	0.222174	0.111087	−	0.993811i	$-0.464567\pi$
$7$	0.587818	0.222174	0.111087	+	0.993811i	$0.464567\pi$
$8$	−9.81674	−3.47074
$9$	−2.78849	−0.929496
$10$	−2.75106	−0.869962
$11$	−0.892165	−0.268998	−0.134499	−	0.990914i	$-0.542942\pi$
$11$	−0.892165	−0.268998	−0.134499	+	0.990914i	$0.542942\pi$
$12$	−2.56091	−0.739271
$13$	−1.31436	−0.364537	−0.182268	−	0.983249i	$-0.558344\pi$
$13$	−1.31436	−0.364537	−0.182268	+	0.983249i	$0.558344\pi$
$14$	−1.61712	−0.432195
$15$	−0.459905	−0.118747
$16$	15.8698	3.96744
$17$	−6.89306	−1.67181	−0.835906	−	0.548872i	$-0.815057\pi$
$17$	−6.89306	−1.67181	−0.835906	+	0.548872i	$0.815057\pi$
$18$	7.67130	1.80814
$19$	2.78585	0.639118	0.319559	−	0.947566i	$-0.396465\pi$
$19$	2.78585	0.639118	0.319559	+	0.947566i	$0.396465\pi$
$20$	5.56834	1.24512
$21$	−0.270341	−0.0589932
$22$	2.45440	0.523280
$23$	−0.636773	−0.132776	−0.0663881	−	0.997794i	$-0.521148\pi$
$23$	−0.636773	−0.132776	−0.0663881	+	0.997794i	$0.521148\pi$
$24$	4.51477	0.921573
$25$	1.00000	0.200000
$26$	3.61587	0.709131
$27$	2.66216	0.512332
$28$	3.27317	0.618571
$29$	5.55621	1.03176	0.515881	−	0.856660i	$-0.327465\pi$
$29$	5.55621	1.03176	0.515881	+	0.856660i	$0.327465\pi$
$30$	1.26523	0.230998
$31$	−9.03241	−1.62227	−0.811134	−	0.584860i	$-0.801149\pi$
$31$	−9.03241	−1.62227	−0.811134	+	0.584860i	$0.801149\pi$
$32$	−24.0252	−4.24710
$33$	0.410311	0.0714260
$34$	18.9632	3.25217
$35$	0.587818	0.0993594
$36$	−15.5273	−2.58788
$37$	0.632709	0.104017	0.0520084	−	0.998647i	$-0.483438\pi$
$37$	0.632709	0.104017	0.0520084	+	0.998647i	$0.483438\pi$
$38$	−7.66404	−1.24327
$39$	0.604479	0.0967941
$40$	−9.81674	−1.55216
$41$	−7.36941	−1.15091	−0.575454	−	0.817834i	$-0.695175\pi$
$41$	−7.36941	−1.15091	−0.575454	+	0.817834i	$0.695175\pi$
$42$	0.743724	0.114759
$43$	−7.41164	−1.13026	−0.565132	−	0.825000i	$-0.691175\pi$
$43$	−7.41164	−1.13026	−0.565132	+	0.825000i	$0.691175\pi$
$44$	−4.96788	−0.748936
$45$	−2.78849	−0.415683
$46$	1.75180	0.258289
$47$	−6.16627	−0.899442	−0.449721	−	0.893169i	$-0.648477\pi$
$47$	−6.16627	−0.899442	−0.449721	+	0.893169i	$0.648477\pi$
$48$	−7.29859	−1.05346
$49$	−6.65447	−0.950639
$50$	−2.75106	−0.389059
$51$	3.17016	0.443911
$52$	−7.31878	−1.01493
$53$	3.18988	0.438163	0.219082	−	0.975707i	$-0.429694\pi$
$53$	3.18988	0.438163	0.219082	+	0.975707i	$0.429694\pi$
$54$	−7.32376	−0.996637
$55$	−0.892165	−0.120299
$56$	−5.77045	−0.771109
$57$	−1.28123	−0.169703
$58$	−15.2855	−2.00708
$59$	5.68022	0.739502	0.369751	−	0.929131i	$-0.379443\pi$
$59$	5.68022	0.739502	0.369751	+	0.929131i	$0.379443\pi$
$60$	−2.56091	−0.330612
$61$	−4.98727	−0.638555	−0.319277	−	0.947661i	$-0.603440\pi$
$61$	−4.98727	−0.638555	−0.319277	+	0.947661i	$0.603440\pi$
$62$	24.8487	3.15579
$63$	−1.63912	−0.206510
$64$	34.3554	4.29443
$65$	−1.31436	−0.163026
$66$	−1.12879	−0.138945
$67$	7.78542	0.951140	0.475570	−	0.879678i	$-0.342242\pi$
$67$	7.78542	0.951140	0.475570	+	0.879678i	$0.342242\pi$
$68$	−38.3829	−4.65461
$69$	0.292855	0.0352556
$70$	−1.61712	−0.193283
$71$	−14.9564	−1.77499	−0.887496	−	0.460815i	$-0.847557\pi$
$71$	−14.9564	−1.77499	−0.887496	+	0.460815i	$0.847557\pi$
$72$	27.3738	3.22604
$73$	10.9297	1.27922	0.639611	−	0.768698i	$-0.279095\pi$
$73$	10.9297	1.27922	0.639611	+	0.768698i	$0.279095\pi$
$74$	−1.74062	−0.202343
$75$	−0.459905	−0.0531053
$76$	15.5126	1.77941
$77$	−0.524431	−0.0597644
$78$	−1.66296	−0.188293
$79$	−11.7250	−1.31917	−0.659584	−	0.751631i	$-0.729267\pi$
$79$	−11.7250	−1.31917	−0.659584	+	0.751631i	$0.729267\pi$
$80$	15.8698	1.77429
$81$	7.14112	0.793458
$82$	20.2737	2.23886
$83$	−14.3750	−1.57786	−0.788930	−	0.614483i	$-0.789365\pi$
$83$	−14.3750	−1.57786	−0.788930	+	0.614483i	$0.789365\pi$
$84$	−1.50535	−0.164247
$85$	−6.89306	−0.747657
$86$	20.3899	2.19870
$87$	−2.55533	−0.273960
$88$	8.75815	0.933622
$89$	1.00000	0.106000
$89$	1.00000	0.106000
$90$	7.67130	0.808626
$91$	−0.772602	−0.0809907
$92$	−3.54577	−0.369672
$93$	4.15405	0.430755
$94$	16.9638	1.74968
$95$	2.78585	0.285822
$96$	11.0493	1.12772
$97$	5.86420	0.595420	0.297710	−	0.954656i	$-0.403777\pi$
$97$	5.86420	0.595420	0.297710	+	0.954656i	$0.403777\pi$
$98$	18.3069	1.84927
$99$	2.48779	0.250032
$100$	5.56834	0.556834
$101$	−10.6416	−1.05888	−0.529439	−	0.848348i	$-0.677597\pi$
$101$	−10.6416	−1.05888	−0.529439	+	0.848348i	$0.677597\pi$
$102$	−8.72129	−0.863537
$103$	1.82241	0.179568	0.0897839	−	0.995961i	$-0.471382\pi$
$103$	1.82241	0.179568	0.0897839	+	0.995961i	$0.471382\pi$
$104$	12.9027	1.26521
$105$	−0.270341	−0.0263825
$106$	−8.77555	−0.852357
$107$	−3.76328	−0.363810	−0.181905	−	0.983316i	$-0.558226\pi$
$107$	−3.76328	−0.363810	−0.181905	+	0.983316i	$0.558226\pi$
$108$	14.8238	1.42642
$109$	5.53929	0.530568	0.265284	−	0.964170i	$-0.414534\pi$
$109$	5.53929	0.530568	0.265284	+	0.964170i	$0.414534\pi$
$110$	2.45440	0.234018
$111$	−0.290986	−0.0276192
$112$	9.32853	0.881464
$113$	14.1421	1.33037	0.665186	−	0.746677i	$-0.268352\pi$
$113$	14.1421	1.33037	0.665186	+	0.746677i	$0.268352\pi$
$114$	3.52473	0.330122
$115$	−0.636773	−0.0593794
$116$	30.9389	2.87260
$117$	3.66506	0.338835
$118$	−15.6266	−1.43855
$119$	−4.05186	−0.371434
$120$	4.51477	0.412140
$121$	−10.2040	−0.927640
$122$	13.7203	1.24218
$123$	3.38923	0.305597
$124$	−50.2955	−4.51667
$125$	1.00000	0.0894427
$126$	4.50933	0.401723
$127$	−11.9731	−1.06244	−0.531219	−	0.847235i	$-0.678266\pi$
$127$	−11.9731	−1.06244	−0.531219	+	0.847235i	$0.678266\pi$
$128$	−46.4634	−4.10682
$129$	3.40865	0.300115
$130$	3.61587	0.317133
$131$	−21.0827	−1.84201	−0.921003	−	0.389556i	$-0.872629\pi$
$131$	−21.0827	−1.84201	−0.921003	+	0.389556i	$0.872629\pi$
$132$	2.28476	0.198862
$133$	1.63757	0.141995
$134$	−21.4182	−1.85025
$135$	2.66216	0.229122
$136$	67.6674	5.80243
$137$	21.7300	1.85652	0.928260	−	0.371931i	$-0.121304\pi$
$137$	21.7300	1.85652	0.928260	+	0.371931i	$0.121304\pi$
$138$	−0.805663	−0.0685826
$139$	−7.18857	−0.609726	−0.304863	−	0.952396i	$-0.598611\pi$
$139$	−7.18857	−0.609726	−0.304863	+	0.952396i	$0.598611\pi$
$140$	3.27317	0.276634
$141$	2.83590	0.238826
$142$	41.1459	3.45288
$143$	1.17262	0.0980596
$144$	−44.2526	−3.68772
$145$	5.55621	0.461418
$146$	−30.0682	−2.48847
$147$	3.06043	0.252420
$148$	3.52314	0.289601
$149$	17.9452	1.47013	0.735063	−	0.677999i	$-0.237153\pi$
$149$	17.9452	1.47013	0.735063	+	0.677999i	$0.237153\pi$
$150$	1.26523	0.103305
$151$	8.60851	0.700551	0.350275	−	0.936647i	$-0.386088\pi$
$151$	8.60851	0.700551	0.350275	+	0.936647i	$0.386088\pi$
$152$	−27.3479	−2.21821
$153$	19.2212	1.55394
$154$	1.44274	0.116259
$155$	−9.03241	−0.725500
$156$	3.36595	0.269491
$157$	−1.31247	−0.104746	−0.0523731	−	0.998628i	$-0.516679\pi$
$157$	−1.31247	−0.104746	−0.0523731	+	0.998628i	$0.516679\pi$
$158$	32.2563	2.56617
$159$	−1.46704	−0.116344
$160$	−24.0252	−1.89936
$161$	−0.374306	−0.0294995
$162$	−19.6457	−1.54351
$163$	−4.32734	−0.338943	−0.169472	−	0.985535i	$-0.554206\pi$
$163$	−4.32734	−0.338943	−0.169472	+	0.985535i	$0.554206\pi$
$164$	−41.0354	−3.20433
$165$	0.410311	0.0319427
$166$	39.5465	3.06940
$167$	10.1225	0.783306	0.391653	−	0.920113i	$-0.371903\pi$
$167$	10.1225	0.783306	0.391653	+	0.920113i	$0.371903\pi$
$168$	2.65386	0.204750
$169$	−11.2725	−0.867113
$170$	18.9632	1.45441
$171$	−7.76830	−0.594057
$172$	−41.2705	−3.14685
$173$	−4.46799	−0.339695	−0.169847	−	0.985470i	$-0.554327\pi$
$173$	−4.46799	−0.339695	−0.169847	+	0.985470i	$0.554327\pi$
$174$	7.02987	0.532933
$175$	0.587818	0.0444349
$176$	−14.1584	−1.06723
$177$	−2.61236	−0.196357
$178$	−2.75106	−0.206201
$179$	17.1331	1.28059	0.640293	−	0.768131i	$-0.278813\pi$
$179$	17.1331	1.28059	0.640293	+	0.768131i	$0.278813\pi$
$180$	−15.5273	−1.15733
$181$	0.766969	0.0570083	0.0285042	−	0.999594i	$-0.490926\pi$
$181$	0.766969	0.0570083	0.0285042	+	0.999594i	$0.490926\pi$
$182$	2.12548	0.157551
$183$	2.29367	0.169553
$184$	6.25103	0.460832
$185$	0.632709	0.0465177
$186$	−11.4281	−0.837946
$187$	6.14975	0.449714
$188$	−34.3359	−2.50420
$189$	1.56486	0.113827
$190$	−7.66404	−0.556008
$191$	22.8474	1.65318	0.826590	−	0.562805i	$-0.190278\pi$
$191$	22.8474	1.65318	0.826590	+	0.562805i	$0.190278\pi$
$192$	−15.8002	−1.14028
$193$	−13.9655	−1.00526	−0.502630	−	0.864501i	$-0.667634\pi$
$193$	−13.9655	−1.00526	−0.502630	+	0.864501i	$0.667634\pi$
$194$	−16.1328	−1.15827
$195$	0.604479	0.0432876
$196$	−37.0544	−2.64674
$197$	−14.4721	−1.03110	−0.515549	−	0.856860i	$-0.672412\pi$
$197$	−14.4721	−1.03110	−0.515549	+	0.856860i	$0.672412\pi$
$198$	−6.84407	−0.486387
$199$	9.60667	0.680999	0.340499	−	0.940245i	$-0.389404\pi$
$199$	9.60667	0.680999	0.340499	+	0.940245i	$0.389404\pi$
$200$	−9.81674	−0.694148
$201$	−3.58056	−0.252553
$202$	29.2757	2.05983
$203$	3.26604	0.229231
$204$	17.6525	1.23592
$205$	−7.36941	−0.514702
$206$	−5.01358	−0.349312
$207$	1.77563	0.123415
$208$	−20.8585	−1.44628
$209$	−2.48544	−0.171921
$210$	0.743724	0.0513218
$211$	2.98766	0.205679	0.102839	−	0.994698i	$-0.467207\pi$
$211$	2.98766	0.205679	0.102839	+	0.994698i	$0.467207\pi$
$212$	17.7623	1.21992
$213$	6.87851	0.471307
$214$	10.3530	0.707717
$215$	−7.41164	−0.505469
$216$	−26.1337	−1.77817
$217$	−5.30941	−0.360426
$218$	−15.2389	−1.03211
$219$	−5.02662	−0.339667
$220$	−4.96788	−0.334934
$221$	9.05993	0.609437
$222$	0.800522	0.0537275
$223$	−23.4737	−1.57192	−0.785958	−	0.618280i	$-0.787830\pi$
$223$	−23.4737	−1.57192	−0.785958	+	0.618280i	$0.787830\pi$
$224$	−14.1225	−0.943597
$225$	−2.78849	−0.185899
$226$	−38.9057	−2.58797
$227$	11.3599	0.753982	0.376991	−	0.926217i	$-0.376959\pi$
$227$	11.3599	0.753982	0.376991	+	0.926217i	$0.376959\pi$
$228$	−7.13431	−0.472481
$229$	5.54933	0.366710	0.183355	−	0.983047i	$-0.441304\pi$
$229$	5.54933	0.366710	0.183355	+	0.983047i	$0.441304\pi$
$230$	1.75180	0.115510
$231$	0.241188	0.0158690
$232$	−54.5439	−3.58098
$233$	5.52225	0.361775	0.180887	−	0.983504i	$-0.442103\pi$
$233$	5.52225	0.361775	0.180887	+	0.983504i	$0.442103\pi$
$234$	−10.0828	−0.659134
$235$	−6.16627	−0.402243
$236$	31.6294	2.05890
$237$	5.39240	0.350274
$238$	11.1469	0.722548
$239$	12.7494	0.824693	0.412347	−	0.911027i	$-0.364709\pi$
$239$	12.7494	0.824693	0.412347	+	0.911027i	$0.364709\pi$
$240$	−7.29859	−0.471122
$241$	18.6081	1.19865	0.599325	−	0.800506i	$-0.295436\pi$
$241$	18.6081	1.19865	0.599325	+	0.800506i	$0.295436\pi$
$242$	28.0720	1.80453
$243$	−11.2707	−0.723016
$244$	−27.7708	−1.77785
$245$	−6.65447	−0.425138
$246$	−9.32399	−0.594476
$247$	−3.66160	−0.232982
$248$	88.6688	5.63047
$249$	6.61114	0.418964
$250$	−2.75106	−0.173992
$251$	10.2929	0.649685	0.324842	−	0.945768i	$-0.394689\pi$
$251$	10.2929	0.649685	0.324842	+	0.945768i	$0.394689\pi$
$252$	−9.12720	−0.574959
$253$	0.568106	0.0357165
$254$	32.9386	2.06675
$255$	3.17016	0.198523
$256$	59.1128	3.69455
$257$	9.53876	0.595011	0.297506	−	0.954720i	$-0.403845\pi$
$257$	9.53876	0.595011	0.297506	+	0.954720i	$0.403845\pi$
$258$	−9.37741	−0.583812
$259$	0.371918	0.0231099
$260$	−7.31878	−0.453892
$261$	−15.4934	−0.959019
$262$	57.9999	3.58324
$263$	17.1059	1.05480	0.527399	−	0.849618i	$-0.323167\pi$
$263$	17.1059	1.05480	0.527399	+	0.849618i	$0.323167\pi$
$264$	−4.02792	−0.247901
$265$	3.18988	0.195953
$266$	−4.50506	−0.276223
$267$	−0.459905	−0.0281457
$268$	43.3519	2.64814
$269$	31.8968	1.94478	0.972391	−	0.233356i	$-0.0749707\pi$
$269$	31.8968	1.94478	0.972391	+	0.233356i	$0.0749707\pi$
$270$	−7.32376	−0.445710
$271$	−29.7027	−1.80431	−0.902154	−	0.431414i	$-0.858015\pi$
$271$	−29.7027	−1.80431	−0.902154	+	0.431414i	$0.858015\pi$
$272$	−109.391	−6.63282
$273$	0.355324	0.0215052
$274$	−59.7806	−3.61148
$275$	−0.892165	−0.0537996
$276$	1.63072	0.0981577
$277$	−11.5680	−0.695052	−0.347526	−	0.937670i	$-0.612978\pi$
$277$	−11.5680	−0.695052	−0.347526	+	0.937670i	$0.612978\pi$
$278$	19.7762	1.18610
$279$	25.1867	1.50789
$280$	−5.77045	−0.344851
$281$	15.3722	0.917031	0.458515	−	0.888686i	$-0.348381\pi$
$281$	15.3722	0.917031	0.458515	+	0.888686i	$0.348381\pi$
$282$	−7.80173	−0.464587
$283$	2.75053	0.163502	0.0817509	−	0.996653i	$-0.473949\pi$
$283$	2.75053	0.163502	0.0817509	+	0.996653i	$0.473949\pi$
$284$	−83.2821	−4.94188
$285$	−1.28123	−0.0758933
$286$	−3.22596	−0.190755
$287$	−4.33187	−0.255702
$288$	66.9941	3.94766
$289$	30.5143	1.79496
$290$	−15.2855	−0.897594
$291$	−2.69698	−0.158100
$292$	60.8602	3.56158
$293$	30.7950	1.79906	0.899531	−	0.436856i	$-0.143908\pi$
$293$	30.7950	1.79906	0.899531	+	0.436856i	$0.143908\pi$
$294$	−8.41942	−0.491031
$295$	5.68022	0.330715
$296$	−6.21114	−0.361015
$297$	−2.37508	−0.137816
$298$	−49.3683	−2.85983
$299$	0.836946	0.0484018
$300$	−2.56091	−0.147854
$301$	−4.35669	−0.251116
$302$	−23.6826	−1.36278
$303$	4.89412	0.281160
$304$	44.2108	2.53566
$305$	−4.98727	−0.285570
$306$	−52.8787	−3.02288
$307$	10.6937	0.610320	0.305160	−	0.952301i	$-0.401290\pi$
$307$	10.6937	0.610320	0.305160	+	0.952301i	$0.401290\pi$
$308$	−2.92021	−0.166394
$309$	−0.838138	−0.0476800
$310$	24.8487	1.41131
$311$	6.00458	0.340488	0.170244	−	0.985402i	$-0.445544\pi$
$311$	6.00458	0.340488	0.170244	+	0.985402i	$0.445544\pi$
$312$	−5.93401	−0.335947
$313$	4.68825	0.264995	0.132498	−	0.991183i	$-0.457700\pi$
$313$	4.68825	0.264995	0.132498	+	0.991183i	$0.457700\pi$
$314$	3.61068	0.203762
$315$	−1.63912	−0.0923541
$316$	−65.2889	−3.67279
$317$	−31.9932	−1.79692	−0.898459	−	0.439057i	$-0.855313\pi$
$317$	−31.9932	−1.79692	−0.898459	+	0.439057i	$0.855313\pi$
$318$	4.03592	0.226323
$319$	−4.95706	−0.277542
$320$	34.3554	1.92053
$321$	1.73075	0.0966011
$322$	1.02974	0.0573852
$323$	−19.2030	−1.06848
$324$	39.7642	2.20912
$325$	−1.31436	−0.0729073
$326$	11.9048	0.659345
$327$	−2.54755	−0.140880
$328$	72.3436	3.99451
$329$	−3.62464	−0.199833
$330$	−1.12879	−0.0621380
$331$	−5.24309	−0.288187	−0.144093	−	0.989564i	$-0.546027\pi$
$331$	−5.24309	−0.288187	−0.144093	+	0.989564i	$0.546027\pi$
$332$	−80.0449	−4.39303
$333$	−1.76430	−0.0966832
$334$	−27.8477	−1.52376
$335$	7.78542	0.425363
$336$	−4.29024	−0.234052
$337$	2.66495	0.145169	0.0725844	−	0.997362i	$-0.476875\pi$
$337$	2.66495	0.145169	0.0725844	+	0.997362i	$0.476875\pi$
$338$	31.0113	1.68679
$339$	−6.50401	−0.353249
$340$	−38.3829	−2.08161
$341$	8.05840	0.436387
$342$	21.3711	1.15562
$343$	−8.02634	−0.433382
$344$	72.7581	3.92285
$345$	0.292855	0.0157668
$346$	12.2917	0.660807
$347$	−11.0892	−0.595301	−0.297650	−	0.954675i	$-0.596203\pi$
$347$	−11.0892	−0.595301	−0.297650	+	0.954675i	$0.596203\pi$
$348$	−14.2290	−0.762752
$349$	−5.81916	−0.311493	−0.155746	−	0.987797i	$-0.549778\pi$
$349$	−5.81916	−0.311493	−0.155746	+	0.987797i	$0.549778\pi$
$350$	−1.61712	−0.0864389
$351$	−3.49902	−0.186764
$352$	21.4345	1.14246
$353$	−6.35008	−0.337980	−0.168990	−	0.985618i	$-0.554051\pi$
$353$	−6.35008	−0.337980	−0.168990	+	0.985618i	$0.554051\pi$
$354$	7.18678	0.381973
$355$	−14.9564	−0.793801
$356$	5.56834	0.295122
$357$	1.86347	0.0986255
$358$	−47.1342	−2.49112
$359$	−33.9604	−1.79236	−0.896181	−	0.443688i	$-0.853670\pi$
$359$	−33.9604	−1.79236	−0.896181	+	0.443688i	$0.853670\pi$
$360$	27.3738	1.44273
$361$	−11.2390	−0.591529
$362$	−2.10998	−0.110898
$363$	4.69289	0.246313
$364$	−4.30211	−0.225492
$365$	10.9297	0.572086
$366$	−6.31004	−0.329831
$367$	−13.4491	−0.702038	−0.351019	−	0.936368i	$-0.614165\pi$
$367$	−13.4491	−0.702038	−0.351019	+	0.936368i	$0.614165\pi$
$368$	−10.1054	−0.526782
$369$	20.5495	1.06976
$370$	−1.74062	−0.0904907
$371$	1.87507	0.0973487
$372$	23.1312	1.19930
$373$	−10.4418	−0.540658	−0.270329	−	0.962768i	$-0.587133\pi$
$373$	−10.4418	−0.540658	−0.270329	+	0.962768i	$0.587133\pi$
$374$	−16.9183	−0.874826
$375$	−0.459905	−0.0237494
$376$	60.5326	3.12173
$377$	−7.30284	−0.376115
$378$	−4.30504	−0.221427
$379$	29.1034	1.49494	0.747469	−	0.664296i	$-0.231269\pi$
$379$	29.1034	1.49494	0.747469	+	0.664296i	$0.231269\pi$
$380$	15.5126	0.795778
$381$	5.50647	0.282105
$382$	−62.8546	−3.21592
$383$	26.7367	1.36618	0.683092	−	0.730333i	$-0.260635\pi$
$383$	26.7367	1.36618	0.683092	+	0.730333i	$0.260635\pi$
$384$	21.3688	1.09047
$385$	−0.524431	−0.0267275
$386$	38.4200	1.95553
$387$	20.6673	1.05058
$388$	32.6539	1.65775
$389$	−34.9180	−1.77041	−0.885205	−	0.465201i	$-0.845982\pi$
$389$	−34.9180	−1.77041	−0.885205	+	0.465201i	$0.845982\pi$
$390$	−1.66296	−0.0842072
$391$	4.38931	0.221977
$392$	65.3252	3.29942
$393$	9.69605	0.489101
$394$	39.8138	2.00579
$395$	−11.7250	−0.589950
$396$	13.8529	0.696133
$397$	35.4392	1.77864	0.889322	−	0.457282i	$-0.151177\pi$
$397$	35.4392	1.77864	0.889322	+	0.457282i	$0.151177\pi$
$398$	−26.4286	−1.32474
$399$	−0.753128	−0.0377036
$400$	15.8698	0.793488
$401$	−22.9962	−1.14837	−0.574187	−	0.818724i	$-0.694682\pi$
$401$	−22.9962	−1.14837	−0.574187	+	0.818724i	$0.694682\pi$
$402$	9.85033	0.491290
$403$	11.8718	0.591376
$404$	−59.2560	−2.94810
$405$	7.14112	0.354845
$406$	−8.98508	−0.445922
$407$	−0.564481	−0.0279803
$408$	−31.1206	−1.54070
$409$	24.8868	1.23057	0.615286	−	0.788304i	$-0.289040\pi$
$409$	24.8868	1.23057	0.615286	+	0.788304i	$0.289040\pi$
$410$	20.2737	1.00125
$411$	−9.99375	−0.492955
$412$	10.1478	0.499948
$413$	3.33894	0.164298
$414$	−4.88487	−0.240078
$415$	−14.3750	−0.705641
$416$	31.5777	1.54822
$417$	3.30606	0.161898
$418$	6.83759	0.334438
$419$	−38.2037	−1.86637	−0.933185	−	0.359396i	$-0.882983\pi$
$419$	−38.2037	−1.86637	−0.933185	+	0.359396i	$0.882983\pi$
$420$	−1.50535	−0.0734535
$421$	9.53347	0.464633	0.232316	−	0.972640i	$-0.425370\pi$
$421$	9.53347	0.464633	0.232316	+	0.972640i	$0.425370\pi$
$422$	−8.21923	−0.400106
$423$	17.1946	0.836028
$424$	−31.3142	−1.52075
$425$	−6.89306	−0.334363
$426$	−18.9232	−0.916832
$427$	−2.93161	−0.141870
$428$	−20.9552	−1.01291
$429$	−0.539295	−0.0260374
$430$	20.3899	0.983287
$431$	15.3802	0.740838	0.370419	−	0.928865i	$-0.379214\pi$
$431$	15.3802	0.740838	0.370419	+	0.928865i	$0.379214\pi$
$432$	42.2478	2.03265
$433$	−17.4466	−0.838432	−0.419216	−	0.907887i	$-0.637695\pi$
$433$	−17.4466	−0.838432	−0.419216	+	0.907887i	$0.637695\pi$
$434$	14.6065	0.701135
$435$	−2.55533	−0.122519
$436$	30.8447	1.47719
$437$	−1.77395	−0.0848596
$438$	13.8285	0.660753
$439$	−15.5530	−0.742302	−0.371151	−	0.928573i	$-0.621037\pi$
$439$	−15.5530	−0.742302	−0.371151	+	0.928573i	$0.621037\pi$
$440$	8.75815	0.417528
$441$	18.5559	0.883614
$442$	−24.9244	−1.18553
$443$	−23.9793	−1.13929	−0.569646	−	0.821890i	$-0.692920\pi$
$443$	−23.9793	−1.13929	−0.569646	+	0.821890i	$0.692920\pi$
$444$	−1.62031	−0.0768966
$445$	1.00000	0.0474045
$446$	64.5776	3.05784
$447$	−8.25308	−0.390357
$448$	20.1947	0.954111
$449$	26.7348	1.26170	0.630848	−	0.775907i	$-0.282707\pi$
$449$	26.7348	1.26170	0.630848	+	0.775907i	$0.282707\pi$
$450$	7.67130	0.361629
$451$	6.57473	0.309592
$452$	78.7478	3.70399
$453$	−3.95910	−0.186015
$454$	−31.2518	−1.46672
$455$	−0.772602	−0.0362201
$456$	12.5775	0.588994
$457$	−23.7958	−1.11312	−0.556561	−	0.830807i	$-0.687879\pi$
$457$	−23.7958	−1.11312	−0.556561	+	0.830807i	$0.687879\pi$
$458$	−15.2665	−0.713359
$459$	−18.3504	−0.856523
$460$	−3.54577	−0.165322
$461$	−14.3367	−0.667728	−0.333864	−	0.942621i	$-0.608353\pi$
$461$	−14.3367	−0.667728	−0.333864	+	0.942621i	$0.608353\pi$
$462$	−0.663524	−0.0308699
$463$	5.29222	0.245950	0.122975	−	0.992410i	$-0.460756\pi$
$463$	5.29222	0.245950	0.122975	+	0.992410i	$0.460756\pi$
$464$	88.1758	4.09346
$465$	4.15405	0.192640
$466$	−15.1921	−0.703759
$467$	−23.7078	−1.09707	−0.548534	−	0.836128i	$-0.684814\pi$
$467$	−23.7078	−1.09707	−0.548534	+	0.836128i	$0.684814\pi$
$468$	20.4083	0.943375
$469$	4.57641	0.211319
$470$	16.9638	0.782481
$471$	0.603610	0.0278129
$472$	−55.7612	−2.56662
$473$	6.61240	0.304039
$474$	−14.8348	−0.681386
$475$	2.78585	0.127824
$476$	−22.5622	−1.03414
$477$	−8.89493	−0.407271
$478$	−35.0745	−1.60427
$479$	−5.23009	−0.238969	−0.119484	−	0.992836i	$-0.538124\pi$
$479$	−5.23009	−0.238969	−0.119484	+	0.992836i	$0.538124\pi$
$480$	11.0493	0.504331
$481$	−0.831605	−0.0379179
$482$	−51.1919	−2.33173
$483$	0.172145	0.00783289
$484$	−56.8196	−2.58271
$485$	5.86420	0.266280
$486$	31.0064	1.40648
$487$	−7.29308	−0.330481	−0.165240	−	0.986253i	$-0.552840\pi$
$487$	−7.29308	−0.330481	−0.165240	+	0.986253i	$0.552840\pi$
$488$	48.9587	2.21626
$489$	1.99017	0.0899984
$490$	18.3069	0.827020
$491$	−17.7901	−0.802857	−0.401428	−	0.915890i	$-0.631486\pi$
$491$	−17.7901	−0.802857	−0.401428	+	0.915890i	$0.631486\pi$
$492$	18.8724	0.850834
$493$	−38.2993	−1.72491
$494$	10.0733	0.453218
$495$	2.48779	0.111818
$496$	−143.342	−6.43625
$497$	−8.79161	−0.394358
$498$	−18.1876	−0.815008
$499$	27.6810	1.23917	0.619586	−	0.784929i	$-0.287300\pi$
$499$	27.6810	1.23917	0.619586	+	0.784929i	$0.287300\pi$
$500$	5.56834	0.249024
$501$	−4.65541	−0.207988
$502$	−28.3165	−1.26383
$503$	9.64823	0.430193	0.215097	−	0.976593i	$-0.430993\pi$
$503$	9.64823	0.430193	0.215097	+	0.976593i	$0.430993\pi$
$504$	16.0908	0.716743
$505$	−10.6416	−0.473544
$506$	−1.56290	−0.0694792
$507$	5.18427	0.230241
$508$	−66.6701	−2.95801
$509$	−25.1703	−1.11565	−0.557826	−	0.829958i	$-0.688364\pi$
$509$	−25.1703	−1.11565	−0.557826	+	0.829958i	$0.688364\pi$
$510$	−8.72129	−0.386185
$511$	6.42466	0.284210
$512$	−69.6963	−3.08017
$513$	7.41636	0.327440
$514$	−26.2417	−1.15747
$515$	1.82241	0.0803052
$516$	18.9805	0.835572
$517$	5.50133	0.241948
$518$	−1.02317	−0.0449555
$519$	2.05485	0.0901980
$520$	12.9027	0.565820
$521$	−25.9914	−1.13870	−0.569351	−	0.822094i	$-0.692805\pi$
$521$	−25.9914	−1.13870	−0.569351	+	0.822094i	$0.692805\pi$
$522$	42.6234	1.86557
$523$	−11.7696	−0.514650	−0.257325	−	0.966325i	$-0.582841\pi$
$523$	−11.7696	−0.514650	−0.257325	+	0.966325i	$0.582841\pi$
$524$	−117.396	−5.12846
$525$	−0.270341	−0.0117986
$526$	−47.0595	−2.05189
$527$	62.2609	2.71213
$528$	6.51155	0.283379
$529$	−22.5945	−0.982370
$530$	−8.77555	−0.381186
$531$	−15.8392	−0.687364
$532$	9.11856	0.395340
$533$	9.68603	0.419548
$534$	1.26523	0.0547518
$535$	−3.76328	−0.162701
$536$	−76.4274	−3.30116
$537$	−7.87959	−0.340030
$538$	−87.7501	−3.78318
$539$	5.93688	0.255720
$540$	14.8238	0.637915
$541$	18.3468	0.788790	0.394395	−	0.918941i	$-0.370954\pi$
$541$	18.3468	0.788790	0.394395	+	0.918941i	$0.370954\pi$
$542$	81.7139	3.50991
$543$	−0.352733	−0.0151372
$544$	165.607	7.10036
$545$	5.53929	0.237277
$546$	−0.977517	−0.0418339
$547$	−24.1996	−1.03470	−0.517351	−	0.855774i	$-0.673082\pi$
$547$	−24.1996	−1.03470	−0.517351	+	0.855774i	$0.673082\pi$
$548$	121.000	5.16887
$549$	13.9069	0.593534
$550$	2.45440	0.104656
$551$	15.4788	0.659417
$552$	−2.87488	−0.122363
$553$	−6.89218	−0.293085
$554$	31.8242	1.35208
$555$	−0.290986	−0.0123517
$556$	−40.0284	−1.69758
$557$	28.5443	1.20946	0.604730	−	0.796431i	$-0.293281\pi$
$557$	28.5443	1.20946	0.604730	+	0.796431i	$0.293281\pi$
$558$	−69.2903	−2.93329
$559$	9.74152	0.412023
$560$	9.32853	0.394202
$561$	−2.82830	−0.119411
$562$	−42.2900	−1.78390
$563$	7.53424	0.317531	0.158765	−	0.987316i	$-0.449249\pi$
$563$	7.53424	0.317531	0.158765	+	0.987316i	$0.449249\pi$
$564$	15.7913	0.664932
$565$	14.1421	0.594961
$566$	−7.56687	−0.318059
$567$	4.19768	0.176286
$568$	146.823	6.16054
$569$	−13.2148	−0.553994	−0.276997	−	0.960871i	$-0.589339\pi$
$569$	−13.2148	−0.553994	−0.276997	+	0.960871i	$0.589339\pi$
$570$	3.52473	0.147635
$571$	1.66699	0.0697612	0.0348806	−	0.999391i	$-0.488895\pi$
$571$	1.66699	0.0697612	0.0348806	+	0.999391i	$0.488895\pi$
$572$	6.52956	0.273015
$573$	−10.5076	−0.438963
$574$	11.9173	0.497417
$575$	−0.636773	−0.0265553
$576$	−95.7996	−3.99165
$577$	9.12732	0.379975	0.189988	−	0.981786i	$-0.439155\pi$
$577$	9.12732	0.379975	0.189988	+	0.981786i	$0.439155\pi$
$578$	−83.9467	−3.49172
$579$	6.42282	0.266923
$580$	30.9389	1.28467
$581$	−8.44988	−0.350560
$582$	7.41956	0.307550
$583$	−2.84590	−0.117865
$584$	−107.294	−4.43985
$585$	3.66506	0.151532
$586$	−84.7189	−3.49971
$587$	2.10109	0.0867213	0.0433607	−	0.999059i	$-0.486194\pi$
$587$	2.10109	0.0867213	0.0433607	+	0.999059i	$0.486194\pi$
$588$	17.0415	0.702780
$589$	−25.1629	−1.03682
$590$	−15.6266	−0.643339
$591$	6.65581	0.273784
$592$	10.0409	0.412680
$593$	10.7003	0.439407	0.219704	−	0.975567i	$-0.429491\pi$
$593$	10.7003	0.439407	0.219704	+	0.975567i	$0.429491\pi$
$594$	6.53400	0.268093
$595$	−4.05186	−0.166110
$596$	99.9249	4.09308
$597$	−4.41816	−0.180823
$598$	−2.30249	−0.0941558
$599$	20.4277	0.834654	0.417327	−	0.908756i	$-0.362967\pi$
$599$	20.4277	0.834654	0.417327	+	0.908756i	$0.362967\pi$
$600$	4.51477	0.184315
$601$	−31.1566	−1.27090	−0.635452	−	0.772141i	$-0.719186\pi$
$601$	−31.1566	−1.27090	−0.635452	+	0.772141i	$0.719186\pi$
$602$	11.9855	0.488494
$603$	−21.7095	−0.884081
$604$	47.9352	1.95045
$605$	−10.2040	−0.414853
$606$	−13.4640	−0.546939
$607$	−6.90856	−0.280410	−0.140205	−	0.990123i	$-0.544776\pi$
$607$	−6.90856	−0.280410	−0.140205	+	0.990123i	$0.544776\pi$
$608$	−66.9307	−2.71440
$609$	−1.50207	−0.0608669
$610$	13.7203	0.555519
$611$	8.10467	0.327880
$612$	107.030	4.32644
$613$	30.4773	1.23097	0.615484	−	0.788150i	$-0.288961\pi$
$613$	30.4773	1.23097	0.615484	+	0.788150i	$0.288961\pi$
$614$	−29.4189	−1.18725
$615$	3.38923	0.136667
$616$	5.14820	0.207427
$617$	−38.7431	−1.55974	−0.779869	−	0.625943i	$-0.784714\pi$
$617$	−38.7431	−1.55974	−0.779869	+	0.625943i	$0.784714\pi$
$618$	2.30577	0.0927517
$619$	−10.1914	−0.409626	−0.204813	−	0.978801i	$-0.565659\pi$
$619$	−10.1914	−0.409626	−0.204813	+	0.978801i	$0.565659\pi$
$620$	−50.2955	−2.01992
$621$	−1.69519	−0.0680256
$622$	−16.5190	−0.662350
$623$	0.587818	0.0235504
$624$	9.59294	0.384025
$625$	1.00000	0.0400000
$626$	−12.8977	−0.515494
$627$	1.14307	0.0456496
$628$	−7.30826	−0.291631
$629$	−4.36130	−0.173897
$630$	4.50933	0.179656
$631$	15.2025	0.605201	0.302600	−	0.953117i	$-0.402145\pi$
$631$	15.2025	0.605201	0.302600	+	0.953117i	$0.402145\pi$
$632$	115.101	4.57849
$633$	−1.37404	−0.0546131
$634$	88.0154	3.49554
$635$	−11.9731	−0.475136
$636$	−8.16899	−0.323922
$637$	8.74634	0.346543
$638$	13.6372	0.539901
$639$	41.7056	1.64985
$640$	−46.4634	−1.83663
$641$	19.5368	0.771657	0.385829	−	0.922570i	$-0.373916\pi$
$641$	19.5368	0.771657	0.385829	+	0.922570i	$0.373916\pi$
$642$	−4.76140	−0.187918
$643$	−3.76227	−0.148369	−0.0741847	−	0.997245i	$-0.523635\pi$
$643$	−3.76227	−0.148369	−0.0741847	+	0.997245i	$0.523635\pi$
$644$	−2.08427	−0.0821316
$645$	3.40865	0.134216
$646$	52.8287	2.07852
$647$	−44.2880	−1.74114	−0.870570	−	0.492044i	$-0.836250\pi$
$647$	−44.2880	−1.74114	−0.870570	+	0.492044i	$0.836250\pi$
$648$	−70.1025	−2.75389
$649$	−5.06770	−0.198924
$650$	3.61587	0.141826
$651$	2.44183	0.0957027
$652$	−24.0961	−0.943677
$653$	5.24704	0.205333	0.102666	−	0.994716i	$-0.467263\pi$
$653$	5.24704	0.205333	0.102666	+	0.994716i	$0.467263\pi$
$654$	7.00847	0.274053
$655$	−21.0827	−0.823770
$656$	−116.951	−4.56616
$657$	−30.4773	−1.18903
$658$	9.97162	0.388734
$659$	−8.67677	−0.337999	−0.169000	−	0.985616i	$-0.554054\pi$
$659$	−8.67677	−0.337999	−0.169000	+	0.985616i	$0.554054\pi$
$660$	2.28476	0.0889340
$661$	25.1920	0.979856	0.489928	−	0.871763i	$-0.337023\pi$
$661$	25.1920	0.979856	0.489928	+	0.871763i	$0.337023\pi$
$662$	14.4241	0.560608
$663$	−4.16671	−0.161822
$664$	141.116	5.47634
$665$	1.63757	0.0635023
$666$	4.85370	0.188077
$667$	−3.53804	−0.136994
$668$	56.3658	2.18086
$669$	10.7957	0.417385
$670$	−21.4182	−0.827456
$671$	4.44947	0.171770
$672$	6.49500	0.250550
$673$	23.2928	0.897870	0.448935	−	0.893564i	$-0.351803\pi$
$673$	23.2928	0.897870	0.448935	+	0.893564i	$0.351803\pi$
$674$	−7.33143	−0.282396
$675$	2.66216	0.102466
$676$	−62.7690	−2.41419
$677$	20.2570	0.778541	0.389270	−	0.921124i	$-0.372727\pi$
$677$	20.2570	0.778541	0.389270	+	0.921124i	$0.372727\pi$
$678$	17.8929	0.687174
$679$	3.44708	0.132287
$680$	67.6674	2.59492
$681$	−5.22447	−0.200202
$682$	−22.1692	−0.848901
$683$	−48.9807	−1.87419	−0.937096	−	0.349071i	$-0.886497\pi$
$683$	−48.9807	−1.87419	−0.937096	+	0.349071i	$0.886497\pi$
$684$	−43.2566	−1.65396
$685$	21.7300	0.830261
$686$	22.0810	0.843055
$687$	−2.55217	−0.0973712
$688$	−117.621	−4.48426
$689$	−4.19263	−0.159727
$690$	−0.805663	−0.0306711
$691$	−13.3790	−0.508959	−0.254480	−	0.967078i	$-0.581904\pi$
$691$	−13.3790	−0.508959	−0.254480	+	0.967078i	$0.581904\pi$
$692$	−24.8793	−0.945769
$693$	1.46237	0.0555508
$694$	30.5072	1.15804
$695$	−7.18857	−0.272678
$696$	25.0850	0.950845
$697$	50.7978	1.92410
$698$	16.0089	0.605945
$699$	−2.53971	−0.0960608
$700$	3.27317	0.123714
$701$	−3.46359	−0.130818	−0.0654089	−	0.997859i	$-0.520835\pi$
$701$	−3.46359	−0.130818	−0.0654089	+	0.997859i	$0.520835\pi$
$702$	9.62602	0.363311
$703$	1.76263	0.0664790
$704$	−30.6507	−1.15519
$705$	2.83590	0.106806
$706$	17.4695	0.657472
$707$	−6.25532	−0.235255
$708$	−14.5465	−0.546693
$709$	11.6352	0.436970	0.218485	−	0.975840i	$-0.429888\pi$
$709$	11.6352	0.436970	0.218485	+	0.975840i	$0.429888\pi$
$710$	41.1459	1.54418
$711$	32.6951	1.22616
$712$	−9.81674	−0.367898
$713$	5.75159	0.215399
$714$	−5.12653	−0.191856
$715$	1.17262	0.0438536
$716$	95.4028	3.56537
$717$	−5.86354	−0.218978
$718$	93.4272	3.48667
$719$	−5.56903	−0.207690	−0.103845	−	0.994594i	$-0.533115\pi$
$719$	−5.56903	−0.207690	−0.103845	+	0.994594i	$0.533115\pi$
$720$	−44.2526	−1.64920
$721$	1.07125	0.0398954
$722$	30.9193	1.15070
$723$	−8.55795	−0.318273
$724$	4.27075	0.158721
$725$	5.55621	0.206352
$726$	−12.9104	−0.479151
$727$	−18.8775	−0.700126	−0.350063	−	0.936726i	$-0.613840\pi$
$727$	−18.8775	−0.700126	−0.350063	+	0.936726i	$0.613840\pi$
$728$	7.58443	0.281098
$729$	−16.2399	−0.601478
$730$	−30.0682	−1.11288
$731$	51.0889	1.88959
$732$	12.7720	0.472065
$733$	17.6145	0.650607	0.325304	−	0.945610i	$-0.394534\pi$
$733$	17.6145	0.650607	0.325304	+	0.945610i	$0.394534\pi$
$734$	36.9994	1.36567
$735$	3.06043	0.112886
$736$	15.2986	0.563914
$737$	−6.94588	−0.255855
$738$	−56.5330	−2.08101
$739$	16.1072	0.592515	0.296257	−	0.955108i	$-0.404261\pi$
$739$	16.1072	0.592515	0.296257	+	0.955108i	$0.404261\pi$
$740$	3.52314	0.129513
$741$	1.68399	0.0618628
$742$	−5.15843	−0.189372
$743$	−29.8955	−1.09676	−0.548380	−	0.836229i	$-0.684755\pi$
$743$	−29.8955	−1.09676	−0.548380	+	0.836229i	$0.684755\pi$
$744$	−40.7792	−1.49504
$745$	17.9452	0.657460
$746$	28.7262	1.05174
$747$	40.0845	1.46661
$748$	34.2439	1.25208
$749$	−2.21212	−0.0808292
$750$	1.26523	0.0461996
$751$	9.25061	0.337559	0.168780	−	0.985654i	$-0.446017\pi$
$751$	9.25061	0.337559	0.168780	+	0.985654i	$0.446017\pi$
$752$	−97.8572	−3.56849
$753$	−4.73378	−0.172509
$754$	20.0906	0.731655
$755$	8.60851	0.313296
$756$	8.71370	0.316914
$757$	−32.8131	−1.19261	−0.596307	−	0.802756i	$-0.703366\pi$
$757$	−32.8131	−1.19261	−0.596307	+	0.802756i	$0.703366\pi$
$758$	−80.0651	−2.90810
$759$	−0.261275	−0.00948368
$760$	−27.3479	−0.992014
$761$	24.7217	0.896160	0.448080	−	0.893994i	$-0.352108\pi$
$761$	24.7217	0.896160	0.448080	+	0.893994i	$0.352108\pi$
$762$	−15.1487	−0.548778
$763$	3.25610	0.117879
$764$	127.222	4.60274
$765$	19.2212	0.694944
$766$	−73.5544	−2.65763
$767$	−7.46583	−0.269576
$768$	−27.1863	−0.981001
$769$	22.3736	0.806813	0.403407	−	0.915021i	$-0.367826\pi$
$769$	22.3736	0.806813	0.403407	+	0.915021i	$0.367826\pi$
$770$	1.44274	0.0519928
$771$	−4.38692	−0.157991
$772$	−77.7649	−2.79882
$773$	49.9309	1.79589	0.897945	−	0.440107i	$-0.145059\pi$
$773$	49.9309	1.79589	0.897945	+	0.440107i	$0.145059\pi$
$774$	−56.8569	−2.04368
$775$	−9.03241	−0.324454
$776$	−57.5673	−2.06655
$777$	−0.171047	−0.00613628
$778$	96.0615	3.44397
$779$	−20.5301	−0.735566
$780$	3.36595	0.120520
$781$	13.3435	0.477469
$782$	−12.0753	−0.431811
$783$	14.7915	0.528605
$784$	−105.605	−3.77160
$785$	−1.31247	−0.0468439
$786$	−26.6745	−0.951446
$787$	−35.1401	−1.25261	−0.626304	−	0.779579i	$-0.715433\pi$
$787$	−35.1401	−1.25261	−0.626304	+	0.779579i	$0.715433\pi$
$788$	−80.5858	−2.87075
$789$	−7.86711	−0.280077
$790$	32.2563	1.14763
$791$	8.31295	0.295575
$792$	−24.4220	−0.867797
$793$	6.55505	0.232777
$794$	−97.4955	−3.45999
$795$	−1.46704	−0.0520306
$796$	53.4933	1.89602
$797$	−23.9334	−0.847766	−0.423883	−	0.905717i	$-0.639333\pi$
$797$	−23.9334	−0.847766	−0.423883	+	0.905717i	$0.639333\pi$
$798$	2.07190	0.0733445
$799$	42.5044	1.50370
$800$	−24.0252	−0.849421
$801$	−2.78849	−0.0985263
$802$	63.2639	2.23393
$803$	−9.75108	−0.344108
$804$	−19.9378	−0.703151
$805$	−0.374306	−0.0131926
$806$	−32.6600	−1.15040
$807$	−14.6695	−0.516391
$808$	104.466	3.67509
$809$	−42.2870	−1.48673	−0.743366	−	0.668885i	$-0.766772\pi$
$809$	−42.2870	−1.48673	−0.743366	+	0.668885i	$0.766772\pi$
$810$	−19.6457	−0.690278
$811$	13.8234	0.485406	0.242703	−	0.970101i	$-0.421966\pi$
$811$	13.8234	0.485406	0.242703	+	0.970101i	$0.421966\pi$
$812$	18.1864	0.638219
$813$	13.6604	0.479092
$814$	1.55292	0.0544299
$815$	−4.32734	−0.151580
$816$	50.3096	1.76119
$817$	−20.6477	−0.722372
$818$	−68.4651	−2.39383
$819$	2.15439	0.0752805
$820$	−41.0354	−1.43302
$821$	32.0266	1.11774	0.558868	−	0.829257i	$-0.311236\pi$
$821$	32.0266	1.11774	0.558868	+	0.829257i	$0.311236\pi$
$822$	27.4934	0.958944
$823$	38.3450	1.33662	0.668311	−	0.743882i	$-0.267018\pi$
$823$	38.3450	1.33662	0.668311	+	0.743882i	$0.267018\pi$
$824$	−17.8902	−0.623233
$825$	0.410311	0.0142852
$826$	−9.18562	−0.319609
$827$	−20.0855	−0.698440	−0.349220	−	0.937041i	$-0.613553\pi$
$827$	−20.0855	−0.698440	−0.349220	+	0.937041i	$0.613553\pi$
$828$	9.88733	0.343608
$829$	−43.3908	−1.50702	−0.753512	−	0.657434i	$-0.771642\pi$
$829$	−43.3908	−1.50702	−0.753512	+	0.657434i	$0.771642\pi$
$830$	39.5465	1.37268
$831$	5.32018	0.184555
$832$	−45.1552	−1.56548
$833$	45.8697	1.58929
$834$	−9.09518	−0.314940
$835$	10.1225	0.350305
$836$	−13.8398	−0.478658
$837$	−24.0457	−0.831140
$838$	105.101	3.63064
$839$	22.3152	0.770406	0.385203	−	0.922832i	$-0.374131\pi$
$839$	22.3152	0.770406	0.385203	+	0.922832i	$0.374131\pi$
$840$	2.65386	0.0915670
$841$	1.87148	0.0645336
$842$	−26.2272	−0.903848
$843$	−7.06978	−0.243496
$844$	16.6363	0.572645
$845$	−11.2725	−0.387785
$846$	−47.3033	−1.62632
$847$	−5.99812	−0.206098
$848$	50.6226	1.73839
$849$	−1.26498	−0.0434141
$850$	18.9632	0.650434
$851$	−0.402892	−0.0138110
$852$	38.3019	1.31220
$853$	−43.5152	−1.48993	−0.744965	−	0.667103i	$-0.767534\pi$
$853$	−43.5152	−1.48993	−0.744965	+	0.667103i	$0.767534\pi$
$854$	8.06504	0.275980
$855$	−7.76830	−0.265670
$856$	36.9431	1.26269
$857$	19.0470	0.650633	0.325316	−	0.945605i	$-0.394529\pi$
$857$	19.0470	0.650633	0.325316	+	0.945605i	$0.394529\pi$
$858$	1.48363	0.0506504
$859$	−0.896256	−0.0305799	−0.0152899	−	0.999883i	$-0.504867\pi$
$859$	−0.896256	−0.0305799	−0.0152899	+	0.999883i	$0.504867\pi$
$860$	−41.2705	−1.40731
$861$	1.99225	0.0678958
$862$	−42.3119	−1.44115
$863$	−22.6236	−0.770117	−0.385058	−	0.922892i	$-0.625819\pi$
$863$	−22.6236	−0.770117	−0.385058	+	0.922892i	$0.625819\pi$
$864$	−63.9589	−2.17593
$865$	−4.46799	−0.151916
$866$	47.9968	1.63100
$867$	−14.0337	−0.476609
$868$	−29.5646	−1.00349
$869$	10.4606	0.354853
$870$	7.02987	0.238335
$871$	−10.2328	−0.346725
$872$	−54.3778	−1.84146
$873$	−16.3523	−0.553440
$874$	4.88025	0.165077
$875$	0.587818	0.0198719
$876$	−27.9899	−0.945693
$877$	−6.52027	−0.220174	−0.110087	−	0.993922i	$-0.535113\pi$
$877$	−6.52027	−0.220174	−0.110087	+	0.993922i	$0.535113\pi$
$878$	42.7871	1.44400
$879$	−14.1628	−0.477699
$880$	−14.1584	−0.477281
$881$	−22.7030	−0.764884	−0.382442	−	0.923979i	$-0.624917\pi$
$881$	−22.7030	−0.764884	−0.382442	+	0.923979i	$0.624917\pi$
$882$	−51.0484	−1.71889
$883$	−45.7804	−1.54063	−0.770317	−	0.637662i	$-0.779902\pi$
$883$	−45.7804	−1.54063	−0.770317	+	0.637662i	$0.779902\pi$
$884$	50.4488	1.69678
$885$	−2.61236	−0.0878137
$886$	65.9686	2.21626
$887$	−45.0961	−1.51418	−0.757089	−	0.653311i	$-0.773379\pi$
$887$	−45.0961	−1.51418	−0.757089	+	0.653311i	$0.773379\pi$
$888$	2.85654	0.0958591
$889$	−7.03798	−0.236046
$890$	−2.75106	−0.0922158
$891$	−6.37106	−0.213438
$892$	−130.710	−4.37648
$893$	−17.1783	−0.574849
$894$	22.7047	0.759360
$895$	17.1331	0.572696
$896$	−27.3120	−0.912430
$897$	−0.384916	−0.0128520
$898$	−73.5492	−2.45437
$899$	−50.1860	−1.67380
$900$	−15.5273	−0.517575
$901$	−21.9880	−0.732527
$902$	−18.0875	−0.602248
$903$	2.00367	0.0666778
$904$	−138.829	−4.61738
$905$	0.766969	0.0254949
$906$	10.8917	0.361854
$907$	43.5083	1.44467	0.722335	−	0.691543i	$-0.243069\pi$
$907$	43.5083	1.44467	0.722335	+	0.691543i	$0.243069\pi$
$908$	63.2557	2.09922
$909$	29.6739	0.984222
$910$	2.12548	0.0704588
$911$	14.1257	0.468006	0.234003	−	0.972236i	$-0.424817\pi$
$911$	14.1257	0.468006	0.234003	+	0.972236i	$0.424817\pi$
$912$	−20.3328	−0.673285
$913$	12.8249	0.424441
$914$	65.4638	2.16535
$915$	2.29367	0.0758265
$916$	30.9006	1.02098
$917$	−12.3928	−0.409246
$918$	50.4831	1.66619
$919$	7.32776	0.241721	0.120860	−	0.992670i	$-0.461435\pi$
$919$	7.32776	0.241721	0.120860	+	0.992670i	$0.461435\pi$
$920$	6.25103	0.206090
$921$	−4.91807	−0.162056
$922$	39.4412	1.29893
$923$	19.6580	0.647050
$924$	1.34302	0.0441821
$925$	0.632709	0.0208034
$926$	−14.5592	−0.478446
$927$	−5.08178	−0.166908
$928$	−133.489	−4.38200
$929$	35.9672	1.18004	0.590022	−	0.807387i	$-0.299119\pi$
$929$	35.9672	1.18004	0.590022	+	0.807387i	$0.299119\pi$
$930$	−11.4281	−0.374741
$931$	−18.5383	−0.607570
$932$	30.7498	1.00724
$933$	−2.76154	−0.0904086
$934$	65.2217	2.13412
$935$	6.14975	0.201118
$936$	−35.9790	−1.17601
$937$	34.6839	1.13307	0.566536	−	0.824037i	$-0.308283\pi$
$937$	34.6839	1.13307	0.566536	+	0.824037i	$0.308283\pi$
$938$	−12.5900	−0.411078
$939$	−2.15615	−0.0703633
$940$	−34.3359	−1.11991
$941$	−22.1865	−0.723260	−0.361630	−	0.932322i	$-0.617780\pi$
$941$	−22.1865	−0.723260	−0.361630	+	0.932322i	$0.617780\pi$
$942$	−1.66057	−0.0541043
$943$	4.69264	0.152813
$944$	90.1438	2.93393
$945$	1.56486	0.0509050
$946$	−18.1911	−0.591445
$947$	30.6653	0.996488	0.498244	−	0.867037i	$-0.333978\pi$
$947$	30.6653	0.996488	0.498244	+	0.867037i	$0.333978\pi$
$948$	30.0267	0.975223
$949$	−14.3655	−0.466323
$950$	−7.66404	−0.248654
$951$	14.7139	0.477129
$952$	39.7761	1.28915
$953$	16.1294	0.522483	0.261241	−	0.965273i	$-0.415868\pi$
$953$	16.1294	0.522483	0.261241	+	0.965273i	$0.415868\pi$
$954$	24.4705	0.792262
$955$	22.8474	0.739324
$956$	70.9933	2.29609
$957$	2.27978	0.0736947
$958$	14.3883	0.464865
$959$	12.7733	0.412471
$960$	−15.8002	−0.509950
$961$	50.5844	1.63175
$962$	2.28780	0.0737615
$963$	10.4938	0.338159
$964$	103.616	3.33725
$965$	−13.9655	−0.449566
$966$	−0.473583	−0.0152373
$967$	37.6822	1.21178	0.605888	−	0.795550i	$-0.292818\pi$
$967$	37.6822	1.21178	0.605888	+	0.795550i	$0.292818\pi$
$968$	100.170	3.21960
$969$	8.83157	0.283711
$970$	−16.1328	−0.517993
$971$	−8.73227	−0.280232	−0.140116	−	0.990135i	$-0.544748\pi$
$971$	−8.73227	−0.280232	−0.140116	+	0.990135i	$0.544748\pi$
$972$	−62.7592	−2.01300
$973$	−4.22557	−0.135465
$974$	20.0637	0.642883
$975$	0.604479	0.0193588
$976$	−79.1468	−2.53343
$977$	27.1633	0.869030	0.434515	−	0.900665i	$-0.356920\pi$
$977$	27.1633	0.869030	0.434515	+	0.900665i	$0.356920\pi$
$978$	−5.47507	−0.175073
$979$	−0.892165	−0.0285137
$980$	−37.0544	−1.18366
$981$	−15.4462	−0.493161
$982$	48.9417	1.56179
$983$	−36.1763	−1.15385	−0.576923	−	0.816798i	$-0.695747\pi$
$983$	−36.1763	−1.15385	−0.576923	+	0.816798i	$0.695747\pi$
$984$	−33.2712	−1.06065
$985$	−14.4721	−0.461121
$986$	105.364	3.35547
$987$	1.66699	0.0530609
$988$	−20.3890	−0.648661
$989$	4.71953	0.150072
$990$	−6.84407	−0.217519
$991$	39.8979	1.26740	0.633699	−	0.773580i	$-0.281536\pi$
$991$	39.8979	1.26740	0.633699	+	0.773580i	$0.281536\pi$
$992$	217.006	6.88994
$993$	2.41133	0.0765211
$994$	24.1863	0.767142
$995$	9.60667	0.304552
$996$	36.8131	1.16647
$997$	−21.5114	−0.681272	−0.340636	−	0.940195i	$-0.610642\pi$
$997$	−21.5114	−0.681272	−0.340636	+	0.940195i	$0.610642\pi$
$998$	−76.1522	−2.41056
$999$	1.68437	0.0532911

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	445.2.a.f.1.1	✓	7
3.2	odd	2		4005.2.a.o.1.7		7
4.3	odd	2		7120.2.a.bj.1.3		7
5.4	even	2		2225.2.a.k.1.7		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
445.2.a.f.1.1	✓	7	1.1	even	1	trivial
2225.2.a.k.1.7		7	5.4	even	2
4005.2.a.o.1.7		7	3.2	odd	2
7120.2.a.bj.1.3		7	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 445 = 5 \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	445.a (trivial)

\(f(q)\)	\(=\)	\(q-2.75106 q^{2} -0.459905 q^{3} +5.56834 q^{4} +1.00000 q^{5} +1.26523 q^{6} +0.587818 q^{7} -9.81674 q^{8} -2.78849 q^{9} +O(q^{10})\)	\(q-2.75106 q^{2} -0.459905 q^{3} +5.56834 q^{4} +1.00000 q^{5} +1.26523 q^{6} +0.587818 q^{7} -9.81674 q^{8} -2.78849 q^{9} -2.75106 q^{10} -0.892165 q^{11} -2.56091 q^{12} -1.31436 q^{13} -1.61712 q^{14} -0.459905 q^{15} +15.8698 q^{16} -6.89306 q^{17} +7.67130 q^{18} +2.78585 q^{19} +5.56834 q^{20} -0.270341 q^{21} +2.45440 q^{22} -0.636773 q^{23} +4.51477 q^{24} +1.00000 q^{25} +3.61587 q^{26} +2.66216 q^{27} +3.27317 q^{28} +5.55621 q^{29} +1.26523 q^{30} -9.03241 q^{31} -24.0252 q^{32} +0.410311 q^{33} +18.9632 q^{34} +0.587818 q^{35} -15.5273 q^{36} +0.632709 q^{37} -7.66404 q^{38} +0.604479 q^{39} -9.81674 q^{40} -7.36941 q^{41} +0.743724 q^{42} -7.41164 q^{43} -4.96788 q^{44} -2.78849 q^{45} +1.75180 q^{46} -6.16627 q^{47} -7.29859 q^{48} -6.65447 q^{49} -2.75106 q^{50} +3.17016 q^{51} -7.31878 q^{52} +3.18988 q^{53} -7.32376 q^{54} -0.892165 q^{55} -5.77045 q^{56} -1.28123 q^{57} -15.2855 q^{58} +5.68022 q^{59} -2.56091 q^{60} -4.98727 q^{61} +24.8487 q^{62} -1.63912 q^{63} +34.3554 q^{64} -1.31436 q^{65} -1.12879 q^{66} +7.78542 q^{67} -38.3829 q^{68} +0.292855 q^{69} -1.61712 q^{70} -14.9564 q^{71} +27.3738 q^{72} +10.9297 q^{73} -1.74062 q^{74} -0.459905 q^{75} +15.5126 q^{76} -0.524431 q^{77} -1.66296 q^{78} -11.7250 q^{79} +15.8698 q^{80} +7.14112 q^{81} +20.2737 q^{82} -14.3750 q^{83} -1.50535 q^{84} -6.89306 q^{85} +20.3899 q^{86} -2.55533 q^{87} +8.75815 q^{88} +1.00000 q^{89} +7.67130 q^{90} -0.772602 q^{91} -3.54577 q^{92} +4.15405 q^{93} +16.9638 q^{94} +2.78585 q^{95} +11.0493 q^{96} +5.86420 q^{97} +18.3069 q^{98} +2.48779 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 4 q^{2} - 8 q^{3} + 8 q^{4} + 7 q^{5} - 2 q^{6} - 16 q^{7} - 12 q^{8} + 11 q^{9}+O(q^{10}) \) 7 * q - 4 * q^2 - 8 * q^3 + 8 * q^4 + 7 * q^5 - 2 * q^6 - 16 * q^7 - 12 * q^8 + 11 * q^9	\( 7 q - 4 q^{2} - 8 q^{3} + 8 q^{4} + 7 q^{5} - 2 q^{6} - 16 q^{7} - 12 q^{8} + 11 q^{9} - 4 q^{10} - 10 q^{11} - 11 q^{12} - 7 q^{13} + 3 q^{14} - 8 q^{15} + 10 q^{16} - 13 q^{17} + 4 q^{18} - 7 q^{19} + 8 q^{20} + 16 q^{21} + 2 q^{22} - 13 q^{23} + 4 q^{24} + 7 q^{25} + q^{26} - 23 q^{27} - 21 q^{28} - 4 q^{29} - 2 q^{30} + q^{31} - 13 q^{32} - 6 q^{33} + 10 q^{34} - 16 q^{35} + 20 q^{36} - 5 q^{37} - 40 q^{38} - 13 q^{39} - 12 q^{40} + 5 q^{41} + 30 q^{42} - 31 q^{43} - 21 q^{44} + 11 q^{45} + 16 q^{46} - 14 q^{47} - 7 q^{48} + 19 q^{49} - 4 q^{50} - q^{51} - 13 q^{53} - 17 q^{54} - 10 q^{55} - q^{56} + 21 q^{57} + 17 q^{58} - 14 q^{59} - 11 q^{60} + 3 q^{61} + 26 q^{62} - 54 q^{63} + 14 q^{64} - 7 q^{65} + 36 q^{66} + q^{67} - 35 q^{68} + 31 q^{69} + 3 q^{70} - 8 q^{71} + 53 q^{72} + 9 q^{73} - 35 q^{74} - 8 q^{75} + 40 q^{76} + 42 q^{77} + 46 q^{78} + 9 q^{79} + 10 q^{80} + 35 q^{81} + 29 q^{82} - 42 q^{83} + 55 q^{84} - 13 q^{85} + 35 q^{86} + 6 q^{87} + 30 q^{88} + 7 q^{89} + 4 q^{90} + 31 q^{91} + 19 q^{92} + 24 q^{93} + 37 q^{94} - 7 q^{95} + 44 q^{96} - 7 q^{97} + 9 q^{98} + 5 q^{99}+O(q^{100}) \) 7 * q - 4 * q^2 - 8 * q^3 + 8 * q^4 + 7 * q^5 - 2 * q^6 - 16 * q^7 - 12 * q^8 + 11 * q^9 - 4 * q^10 - 10 * q^11 - 11 * q^12 - 7 * q^13 + 3 * q^14 - 8 * q^15 + 10 * q^16 - 13 * q^17 + 4 * q^18 - 7 * q^19 + 8 * q^20 + 16 * q^21 + 2 * q^22 - 13 * q^23 + 4 * q^24 + 7 * q^25 + q^26 - 23 * q^27 - 21 * q^28 - 4 * q^29 - 2 * q^30 + q^31 - 13 * q^32 - 6 * q^33 + 10 * q^34 - 16 * q^35 + 20 * q^36 - 5 * q^37 - 40 * q^38 - 13 * q^39 - 12 * q^40 + 5 * q^41 + 30 * q^42 - 31 * q^43 - 21 * q^44 + 11 * q^45 + 16 * q^46 - 14 * q^47 - 7 * q^48 + 19 * q^49 - 4 * q^50 - q^51 - 13 * q^53 - 17 * q^54 - 10 * q^55 - q^56 + 21 * q^57 + 17 * q^58 - 14 * q^59 - 11 * q^60 + 3 * q^61 + 26 * q^62 - 54 * q^63 + 14 * q^64 - 7 * q^65 + 36 * q^66 + q^67 - 35 * q^68 + 31 * q^69 + 3 * q^70 - 8 * q^71 + 53 * q^72 + 9 * q^73 - 35 * q^74 - 8 * q^75 + 40 * q^76 + 42 * q^77 + 46 * q^78 + 9 * q^79 + 10 * q^80 + 35 * q^81 + 29 * q^82 - 42 * q^83 + 55 * q^84 - 13 * q^85 + 35 * q^86 + 6 * q^87 + 30 * q^88 + 7 * q^89 + 4 * q^90 + 31 * q^91 + 19 * q^92 + 24 * q^93 + 37 * q^94 - 7 * q^95 + 44 * q^96 - 7 * q^97 + 9 * q^98 + 5 * q^99

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