LMFDB - Embedded newform 445.2.a.f.1.6

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			1.6
Root			$-1.96388$ 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)

$f(q)$	$=$	$q+0.856822 q^{2} -0.931146 q^{3} -1.26586 q^{4} +1.00000 q^{5} -0.797826 q^{6} -0.580377 q^{7} -2.79826 q^{8} -2.13297 q^{9} +O(q^{10})$	\(q+0.856822 q^{2} -0.931146 q^{3} -1.26586 q^{4} +1.00000 q^{5} -0.797826 q^{6} -0.580377 q^{7} -2.79826 q^{8} -2.13297 q^{9} +0.856822 q^{10} -4.74359 q^{11} +1.17870 q^{12} +6.25246 q^{13} -0.497280 q^{14} -0.931146 q^{15} +0.134107 q^{16} -5.87935 q^{17} -1.82757 q^{18} -5.41445 q^{19} -1.26586 q^{20} +0.540416 q^{21} -4.06441 q^{22} -8.56239 q^{23} +2.60558 q^{24} +1.00000 q^{25} +5.35724 q^{26} +4.77954 q^{27} +0.734675 q^{28} -3.96542 q^{29} -0.797826 q^{30} +5.68893 q^{31} +5.71142 q^{32} +4.41697 q^{33} -5.03755 q^{34} -0.580377 q^{35} +2.70003 q^{36} +0.495300 q^{37} -4.63922 q^{38} -5.82195 q^{39} -2.79826 q^{40} +11.8503 q^{41} +0.463040 q^{42} -4.78767 q^{43} +6.00471 q^{44} -2.13297 q^{45} -7.33644 q^{46} -3.78707 q^{47} -0.124873 q^{48} -6.66316 q^{49} +0.856822 q^{50} +5.47453 q^{51} -7.91471 q^{52} -7.91925 q^{53} +4.09521 q^{54} -4.74359 q^{55} +1.62404 q^{56} +5.04164 q^{57} -3.39766 q^{58} +10.6416 q^{59} +1.17870 q^{60} +9.77193 q^{61} +4.87439 q^{62} +1.23793 q^{63} +4.62545 q^{64} +6.25246 q^{65} +3.78456 q^{66} +1.35061 q^{67} +7.44242 q^{68} +7.97283 q^{69} -0.497280 q^{70} -10.7514 q^{71} +5.96859 q^{72} +1.31208 q^{73} +0.424384 q^{74} -0.931146 q^{75} +6.85392 q^{76} +2.75307 q^{77} -4.98837 q^{78} +0.492431 q^{79} +0.134107 q^{80} +1.94845 q^{81} +10.1536 q^{82} +1.62413 q^{83} -0.684089 q^{84} -5.87935 q^{85} -4.10218 q^{86} +3.69239 q^{87} +13.2738 q^{88} +1.00000 q^{89} -1.82757 q^{90} -3.62878 q^{91} +10.8388 q^{92} -5.29722 q^{93} -3.24485 q^{94} -5.41445 q^{95} -5.31816 q^{96} -9.61533 q^{97} -5.70914 q^{98} +10.1179 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.856822	0.605864	0.302932	−	0.953012i	$-0.402034\pi$
$2$	0.856822	0.605864	0.302932	+	0.953012i	$0.402034\pi$
$3$	−0.931146	−0.537597	−0.268799	−	0.963196i	$-0.586627\pi$
$3$	−0.931146	−0.537597	−0.268799	+	0.963196i	$0.586627\pi$
$4$	−1.26586	−0.632928
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−0.797826	−0.325711
$7$	−0.580377	−0.219362	−0.109681	−	0.993967i	$-0.534983\pi$
$7$	−0.580377	−0.219362	−0.109681	+	0.993967i	$0.534983\pi$
$8$	−2.79826	−0.989333
$9$	−2.13297	−0.710989
$10$	0.856822	0.270951
$11$	−4.74359	−1.43025	−0.715123	−	0.698998i	$-0.753629\pi$
$11$	−4.74359	−1.43025	−0.715123	+	0.698998i	$0.753629\pi$
$12$	1.17870	0.340261
$13$	6.25246	1.73412	0.867060	−	0.498204i	$-0.166007\pi$
$13$	6.25246	1.73412	0.867060	+	0.498204i	$0.166007\pi$
$14$	−0.497280	−0.132904
$15$	−0.931146	−0.240421
$16$	0.134107	0.0335267
$17$	−5.87935	−1.42595	−0.712976	−	0.701189i	$-0.752653\pi$
$17$	−5.87935	−1.42595	−0.712976	+	0.701189i	$0.752653\pi$
$18$	−1.82757	−0.430763
$19$	−5.41445	−1.24216	−0.621080	−	0.783747i	$-0.713306\pi$
$19$	−5.41445	−1.24216	−0.621080	+	0.783747i	$0.713306\pi$
$20$	−1.26586	−0.283054
$21$	0.540416	0.117928
$22$	−4.06441	−0.866535
$23$	−8.56239	−1.78538	−0.892691	−	0.450669i	$-0.851185\pi$
$23$	−8.56239	−1.78538	−0.892691	+	0.450669i	$0.851185\pi$
$24$	2.60558	0.531863
$25$	1.00000	0.200000
$26$	5.35724	1.05064
$27$	4.77954	0.919823
$28$	0.734675	0.138840
$29$	−3.96542	−0.736361	−0.368180	−	0.929754i	$-0.620019\pi$
$29$	−3.96542	−0.736361	−0.368180	+	0.929754i	$0.620019\pi$
$30$	−0.797826	−0.145662
$31$	5.68893	1.02176	0.510881	−	0.859652i	$-0.329319\pi$
$31$	5.68893	1.02176	0.510881	+	0.859652i	$0.329319\pi$
$32$	5.71142	1.00965
$33$	4.41697	0.768897
$34$	−5.03755	−0.863933
$35$	−0.580377	−0.0981017
$36$	2.70003	0.450005
$37$	0.495300	0.0814269	0.0407134	−	0.999171i	$-0.487037\pi$
$37$	0.495300	0.0814269	0.0407134	+	0.999171i	$0.487037\pi$
$38$	−4.63922	−0.752581
$39$	−5.82195	−0.932258
$40$	−2.79826	−0.442443
$41$	11.8503	1.85071	0.925356	−	0.379098i	$-0.123766\pi$
$41$	11.8503	1.85071	0.925356	+	0.379098i	$0.123766\pi$
$42$	0.463040	0.0714486
$43$	−4.78767	−0.730113	−0.365056	−	0.930985i	$-0.618950\pi$
$43$	−4.78767	−0.730113	−0.365056	+	0.930985i	$0.618950\pi$
$44$	6.00471	0.905244
$45$	−2.13297	−0.317964
$46$	−7.33644	−1.08170
$47$	−3.78707	−0.552401	−0.276201	−	0.961100i	$-0.589075\pi$
$47$	−3.78707	−0.552401	−0.276201	+	0.961100i	$0.589075\pi$
$48$	−0.124873	−0.0180239
$49$	−6.66316	−0.951880
$50$	0.856822	0.121173
$51$	5.47453	0.766588
$52$	−7.91471	−1.09757
$53$	−7.91925	−1.08779	−0.543897	−	0.839152i	$-0.683052\pi$
$53$	−7.91925	−1.08779	−0.543897	+	0.839152i	$0.683052\pi$
$54$	4.09521	0.557288
$55$	−4.74359	−0.639626
$56$	1.62404	0.217022
$57$	5.04164	0.667782
$58$	−3.39766	−0.446135
$59$	10.6416	1.38542	0.692711	−	0.721216i	$-0.256416\pi$
$59$	10.6416	1.38542	0.692711	+	0.721216i	$0.256416\pi$
$60$	1.17870	0.152169
$61$	9.77193	1.25117	0.625584	−	0.780157i	$-0.284861\pi$
$61$	9.77193	1.25117	0.625584	+	0.780157i	$0.284861\pi$
$62$	4.87439	0.619049
$63$	1.23793	0.155964
$64$	4.62545	0.578182
$65$	6.25246	0.775522
$66$	3.78456	0.465847
$67$	1.35061	0.165004	0.0825019	−	0.996591i	$-0.473709\pi$
$67$	1.35061	0.165004	0.0825019	+	0.996591i	$0.473709\pi$
$68$	7.44242	0.902525
$69$	7.97283	0.959816
$70$	−0.497280	−0.0594363
$71$	−10.7514	−1.27595	−0.637976	−	0.770056i	$-0.720228\pi$
$71$	−10.7514	−1.27595	−0.637976	+	0.770056i	$0.720228\pi$
$72$	5.96859	0.703405
$73$	1.31208	0.153567	0.0767834	−	0.997048i	$-0.475535\pi$
$73$	1.31208	0.153567	0.0767834	+	0.997048i	$0.475535\pi$
$74$	0.424384	0.0493336
$75$	−0.931146	−0.107519
$76$	6.85392	0.786198
$77$	2.75307	0.313742
$78$	−4.98837	−0.564822
$79$	0.492431	0.0554028	0.0277014	−	0.999616i	$-0.491181\pi$
$79$	0.492431	0.0554028	0.0277014	+	0.999616i	$0.491181\pi$
$80$	0.134107	0.0149936
$81$	1.94845	0.216495
$82$	10.1536	1.12128
$83$	1.62413	0.178272	0.0891359	−	0.996019i	$-0.471589\pi$
$83$	1.62413	0.178272	0.0891359	+	0.996019i	$0.471589\pi$
$84$	−0.684089	−0.0746402
$85$	−5.87935	−0.637705
$86$	−4.10218	−0.442349
$87$	3.69239	0.395866
$88$	13.2738	1.41499
$89$	1.00000	0.106000
$89$	1.00000	0.106000
$90$	−1.82757	−0.192643
$91$	−3.62878	−0.380400
$92$	10.8388	1.13002
$93$	−5.29722	−0.549296
$94$	−3.24485	−0.334680
$95$	−5.41445	−0.555511
$96$	−5.31816	−0.542783
$97$	−9.61533	−0.976289	−0.488144	−	0.872763i	$-0.662326\pi$
$97$	−9.61533	−0.976289	−0.488144	+	0.872763i	$0.662326\pi$
$98$	−5.70914	−0.576710
$99$	10.1179	1.01689
$100$	−1.26586	−0.126586
$101$	4.72415	0.470070	0.235035	−	0.971987i	$-0.424479\pi$
$101$	4.72415	0.470070	0.235035	+	0.971987i	$0.424479\pi$
$102$	4.69070	0.464448
$103$	5.17345	0.509755	0.254877	−	0.966973i	$-0.417965\pi$
$103$	5.17345	0.509755	0.254877	+	0.966973i	$0.417965\pi$
$104$	−17.4960	−1.71562
$105$	0.540416	0.0527392
$106$	−6.78539	−0.659055
$107$	5.68158	0.549259	0.274630	−	0.961550i	$-0.411445\pi$
$107$	5.68158	0.549259	0.274630	+	0.961550i	$0.411445\pi$
$108$	−6.05021	−0.582182
$109$	−6.48821	−0.621458	−0.310729	−	0.950499i	$-0.600573\pi$
$109$	−6.48821	−0.621458	−0.310729	+	0.950499i	$0.600573\pi$
$110$	−4.06441	−0.387526
$111$	−0.461197	−0.0437749
$112$	−0.0778326	−0.00735449
$113$	9.49005	0.892749	0.446375	−	0.894846i	$-0.352715\pi$
$113$	9.49005	0.892749	0.446375	+	0.894846i	$0.352715\pi$
$114$	4.31979	0.404585
$115$	−8.56239	−0.798447
$116$	5.01966	0.466064
$117$	−13.3363	−1.23294
$118$	9.11797	0.839377
$119$	3.41224	0.312800
$120$	2.60558	0.237856
$121$	11.5017	1.04561
$122$	8.37280	0.758038
$123$	−11.0344	−0.994938
$124$	−7.20137	−0.646702
$125$	1.00000	0.0894427
$126$	1.06068	0.0944931
$127$	−8.22863	−0.730173	−0.365087	−	0.930974i	$-0.618961\pi$
$127$	−8.22863	−0.730173	−0.365087	+	0.930974i	$0.618961\pi$
$128$	−7.45965	−0.659346
$129$	4.45802	0.392507
$130$	5.35724	0.469861
$131$	−4.92961	−0.430702	−0.215351	−	0.976537i	$-0.569090\pi$
$131$	−4.92961	−0.430702	−0.215351	+	0.976537i	$0.569090\pi$
$132$	−5.59126	−0.486657
$133$	3.14242	0.272483
$134$	1.15723	0.0999699
$135$	4.77954	0.411357
$136$	16.4519	1.41074
$137$	−11.9905	−1.02442	−0.512211	−	0.858860i	$-0.671173\pi$
$137$	−11.9905	−1.02442	−0.512211	+	0.858860i	$0.671173\pi$
$138$	6.83130	0.581519
$139$	18.0960	1.53488	0.767440	−	0.641120i	$-0.221530\pi$
$139$	18.0960	1.53488	0.767440	+	0.641120i	$0.221530\pi$
$140$	0.734675	0.0620913
$141$	3.52632	0.296969
$142$	−9.21200	−0.773054
$143$	−29.6591	−2.48022
$144$	−0.286046	−0.0238371
$145$	−3.96542	−0.329311
$146$	1.12421	0.0930406
$147$	6.20438	0.511728
$148$	−0.626979	−0.0515374
$149$	−19.9604	−1.63522	−0.817609	−	0.575773i	$-0.804701\pi$
$149$	−19.9604	−1.63522	−0.817609	+	0.575773i	$0.804701\pi$
$150$	−0.797826	−0.0651422
$151$	−9.01450	−0.733589	−0.366795	−	0.930302i	$-0.619545\pi$
$151$	−9.01450	−0.733589	−0.366795	+	0.930302i	$0.619545\pi$
$152$	15.1510	1.22891
$153$	12.5405	1.01384
$154$	2.35889	0.190085
$155$	5.68893	0.456946
$156$	7.36975	0.590052
$157$	−9.23610	−0.737121	−0.368561	−	0.929604i	$-0.620149\pi$
$157$	−9.23610	−0.737121	−0.368561	+	0.929604i	$0.620149\pi$
$158$	0.421925	0.0335666
$159$	7.37398	0.584795
$160$	5.71142	0.451527
$161$	4.96942	0.391645
$162$	1.66948	0.131167
$163$	−19.2838	−1.51043	−0.755213	−	0.655479i	$-0.772467\pi$
$163$	−19.2838	−1.51043	−0.755213	+	0.655479i	$0.772467\pi$
$164$	−15.0008	−1.17137
$165$	4.41697	0.343861
$166$	1.39159	0.108009
$167$	−24.8161	−1.92033	−0.960163	−	0.279439i	$-0.909852\pi$
$167$	−24.8161	−1.92033	−0.960163	+	0.279439i	$0.909852\pi$
$168$	−1.51222	−0.116670
$169$	26.0932	2.00717
$170$	−5.03755	−0.386363
$171$	11.5488	0.883163
$172$	6.06050	0.462109
$173$	−17.0185	−1.29389	−0.646945	−	0.762537i	$-0.723954\pi$
$173$	−17.0185	−1.29389	−0.646945	+	0.762537i	$0.723954\pi$
$174$	3.16372	0.239841
$175$	−0.580377	−0.0438724
$176$	−0.636148	−0.0479515
$177$	−9.90890	−0.744799
$178$	0.856822	0.0642215
$179$	−6.10678	−0.456442	−0.228221	−	0.973609i	$-0.573291\pi$
$179$	−6.10678	−0.456442	−0.228221	+	0.973609i	$0.573291\pi$
$180$	2.70003	0.201248
$181$	−1.02757	−0.0763785	−0.0381892	−	0.999271i	$-0.512159\pi$
$181$	−1.02757	−0.0763785	−0.0381892	+	0.999271i	$0.512159\pi$
$182$	−3.10922	−0.230471
$183$	−9.09909	−0.672624
$184$	23.9598	1.76634
$185$	0.495300	0.0364152
$186$	−4.53877	−0.332799
$187$	27.8892	2.03946
$188$	4.79389	0.349630
$189$	−2.77394	−0.201774
$190$	−4.63922	−0.336564
$191$	−11.3023	−0.817808	−0.408904	−	0.912577i	$-0.634089\pi$
$191$	−11.3023	−0.817808	−0.408904	+	0.912577i	$0.634089\pi$
$192$	−4.30697	−0.310829
$193$	−19.0759	−1.37311	−0.686555	−	0.727077i	$-0.740878\pi$
$193$	−19.0759	−1.37311	−0.686555	+	0.727077i	$0.740878\pi$
$194$	−8.23862	−0.591499
$195$	−5.82195	−0.416918
$196$	8.43461	0.602472
$197$	20.1996	1.43916	0.719580	−	0.694409i	$-0.244334\pi$
$197$	20.1996	1.43916	0.719580	+	0.694409i	$0.244334\pi$
$198$	8.66926	0.616097
$199$	−11.2701	−0.798915	−0.399457	−	0.916752i	$-0.630801\pi$
$199$	−11.2701	−0.798915	−0.399457	+	0.916752i	$0.630801\pi$
$200$	−2.79826	−0.197867
$201$	−1.25762	−0.0887055
$202$	4.04775	0.284799
$203$	2.30144	0.161530
$204$	−6.92997	−0.485195
$205$	11.8503	0.827664
$206$	4.43272	0.308842
$207$	18.2633	1.26939
$208$	0.838497	0.0581393
$209$	25.6839	1.77660
$210$	0.463040	0.0319528
$211$	6.09599	0.419665	0.209833	−	0.977737i	$-0.432708\pi$
$211$	6.09599	0.419665	0.209833	+	0.977737i	$0.432708\pi$
$212$	10.0246	0.688495
$213$	10.0111	0.685948
$214$	4.86810	0.332777
$215$	−4.78767	−0.326516
$216$	−13.3744	−0.910011
$217$	−3.30172	−0.224136
$218$	−5.55924	−0.376519
$219$	−1.22173	−0.0825571
$220$	6.00471	0.404837
$221$	−36.7604	−2.47277
$222$	−0.395163	−0.0265216
$223$	7.65809	0.512824	0.256412	−	0.966568i	$-0.417460\pi$
$223$	7.65809	0.512824	0.256412	+	0.966568i	$0.417460\pi$
$224$	−3.31478	−0.221478
$225$	−2.13297	−0.142198
$226$	8.13128	0.540885
$227$	−29.5274	−1.95980	−0.979900	−	0.199490i	$-0.936072\pi$
$227$	−29.5274	−1.95980	−0.979900	+	0.199490i	$0.936072\pi$
$228$	−6.38200	−0.422658
$229$	7.14724	0.472303	0.236151	−	0.971716i	$-0.424114\pi$
$229$	7.14724	0.472303	0.236151	+	0.971716i	$0.424114\pi$
$230$	−7.33644	−0.483751
$231$	−2.56351	−0.168667
$232$	11.0963	0.728506
$233$	−10.9605	−0.718045	−0.359023	−	0.933329i	$-0.616890\pi$
$233$	−10.9605	−0.718045	−0.359023	+	0.933329i	$0.616890\pi$
$234$	−11.4268	−0.746994
$235$	−3.78707	−0.247041
$236$	−13.4708	−0.876873
$237$	−0.458525	−0.0297844
$238$	2.92368	0.189514
$239$	19.7441	1.27714	0.638571	−	0.769563i	$-0.279526\pi$
$239$	19.7441	1.27714	0.638571	+	0.769563i	$0.279526\pi$
$240$	−0.124873	−0.00806052
$241$	−19.2285	−1.23862	−0.619309	−	0.785148i	$-0.712587\pi$
$241$	−19.2285	−1.23862	−0.619309	+	0.785148i	$0.712587\pi$
$242$	9.85487	0.633495
$243$	−16.1529	−1.03621
$244$	−12.3699	−0.791899
$245$	−6.66316	−0.425694
$246$	−9.45451	−0.602798
$247$	−33.8536	−2.15405
$248$	−15.9191	−1.01086
$249$	−1.51231	−0.0958385
$250$	0.856822	0.0541902
$251$	−2.44743	−0.154481	−0.0772403	−	0.997013i	$-0.524611\pi$
$251$	−2.44743	−0.154481	−0.0772403	+	0.997013i	$0.524611\pi$
$252$	−1.56704	−0.0987141
$253$	40.6165	2.55354
$254$	−7.05047	−0.442386
$255$	5.47453	0.342828
$256$	−15.6425	−0.977656
$257$	20.5627	1.28267	0.641333	−	0.767263i	$-0.278382\pi$
$257$	20.5627	1.28267	0.641333	+	0.767263i	$0.278382\pi$
$258$	3.81973	0.237806
$259$	−0.287461	−0.0178620
$260$	−7.91471	−0.490850
$261$	8.45812	0.523545
$262$	−4.22379	−0.260947
$263$	10.4549	0.644679	0.322340	−	0.946624i	$-0.395531\pi$
$263$	10.4549	0.644679	0.322340	+	0.946624i	$0.395531\pi$
$264$	−12.3598	−0.760695
$265$	−7.91925	−0.486476
$266$	2.69250	0.165088
$267$	−0.931146	−0.0569852
$268$	−1.70968	−0.104436
$269$	26.2193	1.59862	0.799308	−	0.600921i	$-0.205199\pi$
$269$	26.2193	1.59862	0.799308	+	0.600921i	$0.205199\pi$
$270$	4.09521	0.249227
$271$	20.6659	1.25536	0.627681	−	0.778470i	$-0.284004\pi$
$271$	20.6659	1.25536	0.627681	+	0.778470i	$0.284004\pi$
$272$	−0.788461	−0.0478075
$273$	3.37893	0.204502
$274$	−10.2738	−0.620660
$275$	−4.74359	−0.286049
$276$	−10.0925	−0.607495
$277$	0.433915	0.0260714	0.0130357	−	0.999915i	$-0.495850\pi$
$277$	0.433915	0.0260714	0.0130357	+	0.999915i	$0.495850\pi$
$278$	15.5050	0.929930
$279$	−12.1343	−0.726461
$280$	1.62404	0.0970552
$281$	16.7226	0.997585	0.498793	−	0.866721i	$-0.333777\pi$
$281$	16.7226	0.997585	0.498793	+	0.866721i	$0.333777\pi$
$282$	3.02142	0.179923
$283$	−10.4735	−0.622587	−0.311293	−	0.950314i	$-0.600762\pi$
$283$	−10.4735	−0.622587	−0.311293	+	0.950314i	$0.600762\pi$
$284$	13.6097	0.807586
$285$	5.04164	0.298641
$286$	−25.4126	−1.50268
$287$	−6.87767	−0.405976
$288$	−12.1823	−0.717847
$289$	17.5668	1.03334
$290$	−3.39766	−0.199518
$291$	8.95327	0.524850
$292$	−1.66090	−0.0971968
$293$	7.94600	0.464210	0.232105	−	0.972691i	$-0.425439\pi$
$293$	7.94600	0.464210	0.232105	+	0.972691i	$0.425439\pi$
$294$	5.31604	0.310038
$295$	10.6416	0.619579
$296$	−1.38598	−0.0805583
$297$	−22.6722	−1.31557
$298$	−17.1025	−0.990721
$299$	−53.5360	−3.09607
$300$	1.17870	0.0680521
$301$	2.77865	0.160159
$302$	−7.72382	−0.444456
$303$	−4.39887	−0.252708
$304$	−0.726115	−0.0416456
$305$	9.77193	0.559539
$306$	10.7449	0.614247
$307$	−2.11312	−0.120602	−0.0603011	−	0.998180i	$-0.519206\pi$
$307$	−2.11312	−0.120602	−0.0603011	+	0.998180i	$0.519206\pi$
$308$	−3.48500	−0.198576
$309$	−4.81723	−0.274043
$310$	4.87439	0.276847
$311$	−8.66507	−0.491351	−0.245675	−	0.969352i	$-0.579010\pi$
$311$	−8.66507	−0.491351	−0.245675	+	0.969352i	$0.579010\pi$
$312$	16.2913	0.922313
$313$	16.0660	0.908105	0.454052	−	0.890975i	$-0.349978\pi$
$313$	16.0660	0.908105	0.454052	+	0.890975i	$0.349978\pi$
$314$	−7.91369	−0.446595
$315$	1.23793	0.0697492
$316$	−0.623347	−0.0350660
$317$	−1.70619	−0.0958291	−0.0479146	−	0.998851i	$-0.515258\pi$
$317$	−1.70619	−0.0958291	−0.0479146	+	0.998851i	$0.515258\pi$
$318$	6.31818	0.354306
$319$	18.8104	1.05318
$320$	4.62545	0.258571
$321$	−5.29038	−0.295280
$322$	4.25790	0.237284
$323$	31.8335	1.77126
$324$	−2.46646	−0.137026
$325$	6.25246	0.346824
$326$	−16.5228	−0.915113
$327$	6.04147	0.334094
$328$	−33.1603	−1.83097
$329$	2.19793	0.121176
$330$	3.78456	0.208333
$331$	−7.90149	−0.434305	−0.217153	−	0.976138i	$-0.569677\pi$
$331$	−7.90149	−0.434305	−0.217153	+	0.976138i	$0.569677\pi$
$332$	−2.05592	−0.112833
$333$	−1.05646	−0.0578936
$334$	−21.2630	−1.16346
$335$	1.35061	0.0737919
$336$	0.0724735	0.00395375
$337$	7.99703	0.435626	0.217813	−	0.975991i	$-0.430108\pi$
$337$	7.99703	0.435626	0.217813	+	0.975991i	$0.430108\pi$
$338$	22.3572	1.21607
$339$	−8.83662	−0.479939
$340$	7.44242	0.403622
$341$	−26.9859	−1.46137
$342$	9.89530	0.535077
$343$	7.92979	0.428168
$344$	13.3971	0.722325
$345$	7.97283	0.429243
$346$	−14.5818	−0.783922
$347$	6.24790	0.335405	0.167702	−	0.985838i	$-0.446365\pi$
$347$	6.24790	0.335405	0.167702	+	0.985838i	$0.446365\pi$
$348$	−4.67403	−0.250555
$349$	7.07360	0.378641	0.189320	−	0.981915i	$-0.439372\pi$
$349$	7.07360	0.378641	0.189320	+	0.981915i	$0.439372\pi$
$350$	−0.497280	−0.0265807
$351$	29.8839	1.59508
$352$	−27.0926	−1.44404
$353$	17.2887	0.920184	0.460092	−	0.887871i	$-0.347816\pi$
$353$	17.2887	0.920184	0.460092	+	0.887871i	$0.347816\pi$
$354$	−8.49016	−0.451247
$355$	−10.7514	−0.570623
$356$	−1.26586	−0.0670903
$357$	−3.17729	−0.168160
$358$	−5.23242	−0.276542
$359$	12.3384	0.651193	0.325597	−	0.945509i	$-0.394435\pi$
$359$	12.3384	0.651193	0.325597	+	0.945509i	$0.394435\pi$
$360$	5.96859	0.314572
$361$	10.3163	0.542962
$362$	−0.880442	−0.0462750
$363$	−10.7097	−0.562115
$364$	4.59352	0.240766
$365$	1.31208	0.0686771
$366$	−7.79630	−0.407519
$367$	30.6605	1.60047	0.800233	−	0.599689i	$-0.204709\pi$
$367$	30.6605	1.60047	0.800233	+	0.599689i	$0.204709\pi$
$368$	−1.14828	−0.0598580
$369$	−25.2764	−1.31584
$370$	0.424384	0.0220627
$371$	4.59616	0.238621
$372$	6.70552	0.347665
$373$	−5.45234	−0.282312	−0.141156	−	0.989987i	$-0.545082\pi$
$373$	−5.45234	−0.282312	−0.141156	+	0.989987i	$0.545082\pi$
$374$	23.8961	1.23564
$375$	−0.931146	−0.0480842
$376$	10.5972	0.546509
$377$	−24.7936	−1.27694
$378$	−2.37677	−0.122248
$379$	−35.5232	−1.82471	−0.912353	−	0.409405i	$-0.865736\pi$
$379$	−35.5232	−1.82471	−0.912353	+	0.409405i	$0.865736\pi$
$380$	6.85392	0.351599
$381$	7.66206	0.392539
$382$	−9.68408	−0.495481
$383$	−12.8946	−0.658882	−0.329441	−	0.944176i	$-0.606860\pi$
$383$	−12.8946	−0.658882	−0.329441	+	0.944176i	$0.606860\pi$
$384$	6.94602	0.354463
$385$	2.75307	0.140310
$386$	−16.3446	−0.831919
$387$	10.2119	0.519102
$388$	12.1716	0.617921
$389$	−28.2948	−1.43460	−0.717300	−	0.696764i	$-0.754622\pi$
$389$	−28.2948	−1.43460	−0.717300	+	0.696764i	$0.754622\pi$
$390$	−4.98837	−0.252596
$391$	50.3413	2.54587
$392$	18.6452	0.941727
$393$	4.59018	0.231544
$394$	17.3074	0.871936
$395$	0.492431	0.0247769
$396$	−12.8078	−0.643618
$397$	−32.7862	−1.64549	−0.822745	−	0.568411i	$-0.807558\pi$
$397$	−32.7862	−1.64549	−0.822745	+	0.568411i	$0.807558\pi$
$398$	−9.65645	−0.484034
$399$	−2.92606	−0.146486
$400$	0.134107	0.00670534
$401$	19.8756	0.992538	0.496269	−	0.868169i	$-0.334703\pi$
$401$	19.8756	0.992538	0.496269	+	0.868169i	$0.334703\pi$
$402$	−1.07755	−0.0537435
$403$	35.5698	1.77186
$404$	−5.98009	−0.297521
$405$	1.94845	0.0968194
$406$	1.97193	0.0978650
$407$	−2.34950	−0.116461
$408$	−15.3191	−0.758411
$409$	24.6891	1.22080	0.610398	−	0.792095i	$-0.291009\pi$
$409$	24.6891	1.22080	0.610398	+	0.792095i	$0.291009\pi$
$410$	10.1536	0.501452
$411$	11.1649	0.550726
$412$	−6.54884	−0.322638
$413$	−6.17616	−0.303909
$414$	15.6484	0.769076
$415$	1.62413	0.0797256
$416$	35.7104	1.75085
$417$	−16.8500	−0.825148
$418$	22.0066	1.07638
$419$	−23.6692	−1.15632	−0.578159	−	0.815924i	$-0.696229\pi$
$419$	−23.6692	−1.15632	−0.578159	+	0.815924i	$0.696229\pi$
$420$	−0.684089	−0.0333801
$421$	15.9257	0.776169	0.388085	−	0.921624i	$-0.373137\pi$
$421$	15.9257	0.776169	0.388085	+	0.921624i	$0.373137\pi$
$422$	5.22317	0.254260
$423$	8.07770	0.392751
$424$	22.1601	1.07619
$425$	−5.87935	−0.285190
$426$	8.57772	0.415592
$427$	−5.67141	−0.274459
$428$	−7.19207	−0.347642
$429$	27.6169	1.33336
$430$	−4.10218	−0.197825
$431$	7.78225	0.374858	0.187429	−	0.982278i	$-0.439985\pi$
$431$	7.78225	0.374858	0.187429	+	0.982278i	$0.439985\pi$
$432$	0.640969	0.0308386
$433$	−3.59930	−0.172971	−0.0864856	−	0.996253i	$-0.527564\pi$
$433$	−3.59930	−0.172971	−0.0864856	+	0.996253i	$0.527564\pi$
$434$	−2.82899	−0.135796
$435$	3.69239	0.177036
$436$	8.21314	0.393338
$437$	46.3606	2.21773
$438$	−1.04681	−0.0500184
$439$	30.2840	1.44538	0.722689	−	0.691173i	$-0.242906\pi$
$439$	30.2840	1.44538	0.722689	+	0.691173i	$0.242906\pi$
$440$	13.2738	0.632803
$441$	14.2123	0.676777
$442$	−31.4971	−1.49816
$443$	15.1094	0.717870	0.358935	−	0.933362i	$-0.383140\pi$
$443$	15.1094	0.717870	0.358935	+	0.933362i	$0.383140\pi$
$444$	0.583809	0.0277064
$445$	1.00000	0.0474045
$446$	6.56162	0.310702
$447$	18.5860	0.879089
$448$	−2.68451	−0.126831
$449$	18.3101	0.864106	0.432053	−	0.901848i	$-0.357789\pi$
$449$	18.3101	0.864106	0.432053	+	0.901848i	$0.357789\pi$
$450$	−1.82757	−0.0861526
$451$	−56.2132	−2.64698
$452$	−12.0130	−0.565046
$453$	8.39381	0.394376
$454$	−25.2997	−1.18737
$455$	−3.62878	−0.170120
$456$	−14.1078	−0.660659
$457$	−21.0509	−0.984720	−0.492360	−	0.870392i	$-0.663866\pi$
$457$	−21.0509	−0.984720	−0.492360	+	0.870392i	$0.663866\pi$
$458$	6.12391	0.286151
$459$	−28.1006	−1.31162
$460$	10.8388	0.505360
$461$	−38.8056	−1.80735	−0.903677	−	0.428214i	$-0.859143\pi$
$461$	−38.8056	−1.80735	−0.903677	+	0.428214i	$0.859143\pi$
$462$	−2.19647	−0.102189
$463$	−12.1126	−0.562921	−0.281460	−	0.959573i	$-0.590819\pi$
$463$	−12.1126	−0.562921	−0.281460	+	0.959573i	$0.590819\pi$
$464$	−0.531791	−0.0246878
$465$	−5.29722	−0.245653
$466$	−9.39118	−0.435038
$467$	2.25635	0.104411	0.0522056	−	0.998636i	$-0.483375\pi$
$467$	2.25635	0.104411	0.0522056	+	0.998636i	$0.483375\pi$
$468$	16.8818	0.780363
$469$	−0.783866	−0.0361955
$470$	−3.24485	−0.149674
$471$	8.60016	0.396274
$472$	−29.7780	−1.37064
$473$	22.7107	1.04424
$474$	−0.392874	−0.0180453
$475$	−5.41445	−0.248432
$476$	−4.31941	−0.197980
$477$	16.8915	0.773409
$478$	16.9172	0.773775
$479$	0.237254	0.0108404	0.00542022	−	0.999985i	$-0.498275\pi$
$479$	0.237254	0.0108404	0.00542022	+	0.999985i	$0.498275\pi$
$480$	−5.31816	−0.242740
$481$	3.09684	0.141204
$482$	−16.4754	−0.750434
$483$	−4.62725	−0.210547
$484$	−14.5595	−0.661793
$485$	−9.61533	−0.436610
$486$	−13.8402	−0.627803
$487$	1.69314	0.0767237	0.0383619	−	0.999264i	$-0.487786\pi$
$487$	1.69314	0.0767237	0.0383619	+	0.999264i	$0.487786\pi$
$488$	−27.3444	−1.23782
$489$	17.9561	0.812001
$490$	−5.70914	−0.257913
$491$	−18.2064	−0.821641	−0.410821	−	0.911716i	$-0.634758\pi$
$491$	−18.2064	−0.821641	−0.410821	+	0.911716i	$0.634758\pi$
$492$	13.9680	0.629725
$493$	23.3141	1.05002
$494$	−29.0065	−1.30506
$495$	10.1179	0.454767
$496$	0.762924	0.0342563
$497$	6.23985	0.279896
$498$	−1.29578	−0.0580651
$499$	20.0254	0.896462	0.448231	−	0.893918i	$-0.352054\pi$
$499$	20.0254	0.896462	0.448231	+	0.893918i	$0.352054\pi$
$500$	−1.26586	−0.0566108
$501$	23.1074	1.03236
$502$	−2.09701	−0.0935943
$503$	33.4990	1.49365	0.746824	−	0.665021i	$-0.231578\pi$
$503$	33.4990	1.49365	0.746824	+	0.665021i	$0.231578\pi$
$504$	−3.46404	−0.154300
$505$	4.72415	0.210222
$506$	34.8011	1.54710
$507$	−24.2966	−1.07905
$508$	10.4163	0.462147
$509$	−15.2860	−0.677541	−0.338770	−	0.940869i	$-0.610011\pi$
$509$	−15.2860	−0.677541	−0.338770	+	0.940869i	$0.610011\pi$
$510$	4.69070	0.207708
$511$	−0.761499	−0.0336867
$512$	1.51647	0.0670192
$513$	−25.8786	−1.14257
$514$	17.6186	0.777122
$515$	5.17345	0.227969
$516$	−5.64321	−0.248429
$517$	17.9643	0.790070
$518$	−0.246303	−0.0108219
$519$	15.8467	0.695592
$520$	−17.4960	−0.767249
$521$	−23.7850	−1.04204	−0.521021	−	0.853544i	$-0.674449\pi$
$521$	−23.7850	−1.04204	−0.521021	+	0.853544i	$0.674449\pi$
$522$	7.24710	0.317197
$523$	26.9989	1.18058	0.590290	−	0.807192i	$-0.299014\pi$
$523$	26.9989	1.18058	0.590290	+	0.807192i	$0.299014\pi$
$524$	6.24018	0.272603
$525$	0.540416	0.0235857
$526$	8.95802	0.390588
$527$	−33.4472	−1.45698
$528$	0.592347	0.0257786
$529$	50.3145	2.18759
$530$	−6.78539	−0.294738
$531$	−22.6982	−0.985020
$532$	−3.97786	−0.172462
$533$	74.0938	3.20936
$534$	−0.797826	−0.0345253
$535$	5.68158	0.245636
$536$	−3.77936	−0.163244
$537$	5.68630	0.245382
$538$	22.4652	0.968545
$539$	31.6073	1.36142
$540$	−6.05021	−0.260360
$541$	−21.0584	−0.905370	−0.452685	−	0.891671i	$-0.649534\pi$
$541$	−21.0584	−0.905370	−0.452685	+	0.891671i	$0.649534\pi$
$542$	17.7070	0.760580
$543$	0.956815	0.0410609
$544$	−33.5794	−1.43971
$545$	−6.48821	−0.277924
$546$	2.89514	0.123900
$547$	12.6410	0.540491	0.270245	−	0.962792i	$-0.412895\pi$
$547$	12.6410	0.540491	0.270245	+	0.962792i	$0.412895\pi$
$548$	15.1783	0.648385
$549$	−20.8432	−0.889567
$550$	−4.06441	−0.173307
$551$	21.4706	0.914678
$552$	−22.3100	−0.949578
$553$	−0.285796	−0.0121533
$554$	0.371788	0.0157957
$555$	−0.461197	−0.0195767
$556$	−22.9069	−0.971470
$557$	7.81870	0.331289	0.165644	−	0.986186i	$-0.447030\pi$
$557$	7.81870	0.331289	0.165644	+	0.986186i	$0.447030\pi$
$558$	−10.3969	−0.440137
$559$	−29.9347	−1.26610
$560$	−0.0778326	−0.00328903
$561$	−25.9689	−1.09641
$562$	14.3283	0.604401
$563$	−20.7792	−0.875737	−0.437869	−	0.899039i	$-0.644266\pi$
$563$	−20.7792	−0.875737	−0.437869	+	0.899039i	$0.644266\pi$
$564$	−4.46381	−0.187960
$565$	9.49005	0.399250
$566$	−8.97394	−0.377203
$567$	−1.13084	−0.0474908
$568$	30.0851	1.26234
$569$	−19.4719	−0.816307	−0.408153	−	0.912913i	$-0.633827\pi$
$569$	−19.4719	−0.816307	−0.408153	+	0.912913i	$0.633827\pi$
$570$	4.31979	0.180936
$571$	−9.07799	−0.379902	−0.189951	−	0.981794i	$-0.560833\pi$
$571$	−9.07799	−0.379902	−0.189951	+	0.981794i	$0.560833\pi$
$572$	37.5442	1.56980
$573$	10.5241	0.439651
$574$	−5.89294	−0.245966
$575$	−8.56239	−0.357076
$576$	−9.86594	−0.411081
$577$	9.99714	0.416186	0.208093	−	0.978109i	$-0.433274\pi$
$577$	9.99714	0.416186	0.208093	+	0.978109i	$0.433274\pi$
$578$	15.0516	0.626063
$579$	17.7624	0.738181
$580$	5.01966	0.208430
$581$	−0.942611	−0.0391061
$582$	7.67136	0.317988
$583$	37.5657	1.55581
$584$	−3.67152	−0.151929
$585$	−13.3363	−0.551388
$586$	6.80830	0.281248
$587$	−29.0697	−1.19984	−0.599918	−	0.800062i	$-0.704800\pi$
$587$	−29.0697	−1.19984	−0.599918	+	0.800062i	$0.704800\pi$
$588$	−7.85385	−0.323887
$589$	−30.8024	−1.26919
$590$	9.11797	0.375381
$591$	−18.8088	−0.773689
$592$	0.0664232	0.00272998
$593$	−7.26916	−0.298509	−0.149254	−	0.988799i	$-0.547687\pi$
$593$	−7.26916	−0.298509	−0.149254	+	0.988799i	$0.547687\pi$
$594$	−19.4260	−0.797059
$595$	3.41224	0.139888
$596$	25.2670	1.03498
$597$	10.4941	0.429494
$598$	−45.8708	−1.87580
$599$	15.9436	0.651438	0.325719	−	0.945467i	$-0.394394\pi$
$599$	15.9436	0.651438	0.325719	+	0.945467i	$0.394394\pi$
$600$	2.60558	0.106373
$601$	−7.56369	−0.308530	−0.154265	−	0.988030i	$-0.549301\pi$
$601$	−7.56369	−0.308530	−0.154265	+	0.988030i	$0.549301\pi$
$602$	2.38081	0.0970346
$603$	−2.88081	−0.117316
$604$	11.4111	0.464310
$605$	11.5017	0.467609
$606$	−3.76905	−0.153107
$607$	−1.88969	−0.0767000	−0.0383500	−	0.999264i	$-0.512210\pi$
$607$	−1.88969	−0.0767000	−0.0383500	+	0.999264i	$0.512210\pi$
$608$	−30.9242	−1.25414
$609$	−2.14298	−0.0868379
$610$	8.37280	0.339005
$611$	−23.6785	−0.957930
$612$	−15.8744	−0.641686
$613$	−12.4920	−0.504548	−0.252274	−	0.967656i	$-0.581178\pi$
$613$	−12.4920	−0.504548	−0.252274	+	0.967656i	$0.581178\pi$
$614$	−1.81057	−0.0730686
$615$	−11.0344	−0.444950
$616$	−7.70381	−0.310395
$617$	−10.0989	−0.406567	−0.203284	−	0.979120i	$-0.565161\pi$
$617$	−10.0989	−0.406567	−0.203284	+	0.979120i	$0.565161\pi$
$618$	−4.12751	−0.166033
$619$	−30.5158	−1.22653	−0.613266	−	0.789876i	$-0.710145\pi$
$619$	−30.5158	−1.22653	−0.613266	+	0.789876i	$0.710145\pi$
$620$	−7.20137	−0.289214
$621$	−40.9243	−1.64224
$622$	−7.42442	−0.297692
$623$	−0.580377	−0.0232523
$624$	−0.780763	−0.0312555
$625$	1.00000	0.0400000
$626$	13.7657	0.550188
$627$	−23.9155	−0.955093
$628$	11.6916	0.466545
$629$	−2.91204	−0.116111
$630$	1.06068	0.0422586
$631$	−15.5337	−0.618386	−0.309193	−	0.950999i	$-0.600059\pi$
$631$	−15.5337	−0.618386	−0.309193	+	0.950999i	$0.600059\pi$
$632$	−1.37795	−0.0548118
$633$	−5.67625	−0.225611
$634$	−1.46190	−0.0580595
$635$	−8.22863	−0.326543
$636$	−9.33440	−0.370133
$637$	−41.6611	−1.65067
$638$	16.1171	0.638083
$639$	22.9323	0.907188
$640$	−7.45965	−0.294869
$641$	−17.5661	−0.693818	−0.346909	−	0.937899i	$-0.612769\pi$
$641$	−17.5661	−0.693818	−0.346909	+	0.937899i	$0.612769\pi$
$642$	−4.53291	−0.178900
$643$	−40.7523	−1.60712	−0.803558	−	0.595227i	$-0.797062\pi$
$643$	−40.7523	−1.60712	−0.803558	+	0.595227i	$0.797062\pi$
$644$	−6.29057	−0.247883
$645$	4.45802	0.175534
$646$	27.2756	1.07314
$647$	−20.1397	−0.791775	−0.395887	−	0.918299i	$-0.629563\pi$
$647$	−20.1397	−0.791775	−0.395887	+	0.918299i	$0.629563\pi$
$648$	−5.45227	−0.214186
$649$	−50.4795	−1.98149
$650$	5.35724	0.210128
$651$	3.07439	0.120495
$652$	24.4106	0.955992
$653$	−31.5360	−1.23410	−0.617050	−	0.786924i	$-0.711672\pi$
$653$	−31.5360	−1.23410	−0.617050	+	0.786924i	$0.711672\pi$
$654$	5.17646	0.202416
$655$	−4.92961	−0.192616
$656$	1.58921	0.0620483
$657$	−2.79861	−0.109184
$658$	1.88324	0.0734162
$659$	7.91412	0.308290	0.154145	−	0.988048i	$-0.450738\pi$
$659$	7.91412	0.308290	0.154145	+	0.988048i	$0.450738\pi$
$660$	−5.59126	−0.217639
$661$	−39.7578	−1.54640	−0.773200	−	0.634163i	$-0.781345\pi$
$661$	−39.7578	−1.54640	−0.773200	+	0.634163i	$0.781345\pi$
$662$	−6.77017	−0.263130
$663$	34.2293	1.32935
$664$	−4.54474	−0.176370
$665$	3.14242	0.121858
$666$	−0.905197	−0.0350757
$667$	33.9535	1.31469
$668$	31.4136	1.21543
$669$	−7.13080	−0.275693
$670$	1.15723	0.0447079
$671$	−46.3540	−1.78948
$672$	3.08654	0.119066
$673$	21.6297	0.833763	0.416882	−	0.908961i	$-0.363123\pi$
$673$	21.6297	0.833763	0.416882	+	0.908961i	$0.363123\pi$
$674$	6.85202	0.263930
$675$	4.77954	0.183965
$676$	−33.0303	−1.27039
$677$	−16.7735	−0.644658	−0.322329	−	0.946628i	$-0.604466\pi$
$677$	−16.7735	−0.644658	−0.322329	+	0.946628i	$0.604466\pi$
$678$	−7.57141	−0.290778
$679$	5.58052	0.214161
$680$	16.4519	0.630903
$681$	27.4943	1.05358
$682$	−23.1221	−0.885392
$683$	24.7665	0.947663	0.473832	−	0.880615i	$-0.342871\pi$
$683$	24.7665	0.947663	0.473832	+	0.880615i	$0.342871\pi$
$684$	−14.6192	−0.558979
$685$	−11.9905	−0.458135
$686$	6.79442	0.259412
$687$	−6.65512	−0.253909
$688$	−0.642059	−0.0244783
$689$	−49.5148	−1.88636
$690$	6.83130	0.260063
$691$	47.4194	1.80392	0.901959	−	0.431822i	$-0.142129\pi$
$691$	47.4194	1.80392	0.901959	+	0.431822i	$0.142129\pi$
$692$	21.5429	0.818940
$693$	−5.87222	−0.223067
$694$	5.35333	0.203210
$695$	18.0960	0.686420
$696$	−10.3322	−0.391643
$697$	−69.6723	−2.63903
$698$	6.06081	0.229405
$699$	10.2058	0.386019
$700$	0.734675	0.0277681
$701$	34.9075	1.31844	0.659218	−	0.751952i	$-0.270887\pi$
$701$	34.9075	1.31844	0.659218	+	0.751952i	$0.270887\pi$
$702$	25.6051	0.966404
$703$	−2.68178	−0.101145
$704$	−21.9413	−0.826942
$705$	3.52632	0.132809
$706$	14.8133	0.557507
$707$	−2.74179	−0.103116
$708$	12.5433	0.471404
$709$	48.6918	1.82866	0.914330	−	0.404970i	$-0.132718\pi$
$709$	48.6918	1.82866	0.914330	+	0.404970i	$0.132718\pi$
$710$	−9.21200	−0.345720
$711$	−1.05034	−0.0393908
$712$	−2.79826	−0.104869
$713$	−48.7108	−1.82423
$714$	−2.72237	−0.101882
$715$	−29.6591	−1.10919
$716$	7.73030	0.288895
$717$	−18.3847	−0.686588
$718$	10.5718	0.394535
$719$	35.5414	1.32547	0.662736	−	0.748853i	$-0.269395\pi$
$719$	35.5414	1.32547	0.662736	+	0.748853i	$0.269395\pi$
$720$	−0.286046	−0.0106603
$721$	−3.00255	−0.111821
$722$	8.83921	0.328961
$723$	17.9045	0.665877
$724$	1.30075	0.0483421
$725$	−3.96542	−0.147272
$726$	−9.17632	−0.340565
$727$	2.22393	0.0824812	0.0412406	−	0.999149i	$-0.486869\pi$
$727$	2.22393	0.0824812	0.0412406	+	0.999149i	$0.486869\pi$
$728$	10.1543	0.376342
$729$	9.19536	0.340569
$730$	1.12421	0.0416090
$731$	28.1484	1.04111
$732$	11.5181	0.425723
$733$	−9.92710	−0.366666	−0.183333	−	0.983051i	$-0.558689\pi$
$733$	−9.92710	−0.366666	−0.183333	+	0.983051i	$0.558689\pi$
$734$	26.2706	0.969665
$735$	6.20438	0.228852
$736$	−48.9034	−1.80260
$737$	−6.40676	−0.235996
$738$	−21.6574	−0.797219
$739$	−10.1642	−0.373896	−0.186948	−	0.982370i	$-0.559860\pi$
$739$	−10.1642	−0.373896	−0.186948	+	0.982370i	$0.559860\pi$
$740$	−0.626979	−0.0230482
$741$	31.5227	1.15801
$742$	3.93809	0.144572
$743$	−43.3395	−1.58997	−0.794985	−	0.606629i	$-0.792522\pi$
$743$	−43.3395	−1.58997	−0.794985	+	0.606629i	$0.792522\pi$
$744$	14.8230	0.543437
$745$	−19.9604	−0.731292
$746$	−4.67168	−0.171042
$747$	−3.46423	−0.126749
$748$	−35.3038	−1.29083
$749$	−3.29746	−0.120487
$750$	−0.797826	−0.0291325
$751$	−3.92124	−0.143088	−0.0715440	−	0.997437i	$-0.522793\pi$
$751$	−3.92124	−0.143088	−0.0715440	+	0.997437i	$0.522793\pi$
$752$	−0.507872	−0.0185202
$753$	2.27892	0.0830484
$754$	−21.2437	−0.773651
$755$	−9.01450	−0.328071
$756$	3.51141	0.127709
$757$	7.19956	0.261672	0.130836	−	0.991404i	$-0.458234\pi$
$757$	7.19956	0.261672	0.130836	+	0.991404i	$0.458234\pi$
$758$	−30.4371	−1.10552
$759$	−37.8199	−1.37277
$760$	15.1510	0.549585
$761$	40.8629	1.48128	0.740639	−	0.671903i	$-0.234523\pi$
$761$	40.8629	1.48128	0.740639	+	0.671903i	$0.234523\pi$
$762$	6.56501	0.237825
$763$	3.76561	0.136324
$764$	14.3071	0.517614
$765$	12.5405	0.453401
$766$	−11.0484	−0.399193
$767$	66.5363	2.40249
$768$	14.5654	0.525585
$769$	51.3277	1.85092	0.925462	−	0.378840i	$-0.123677\pi$
$769$	51.3277	1.85092	0.925462	+	0.378840i	$0.123677\pi$
$770$	2.35889	0.0850086
$771$	−19.1469	−0.689558
$772$	24.1473	0.869081
$773$	−2.66946	−0.0960137	−0.0480069	−	0.998847i	$-0.515287\pi$
$773$	−2.66946	−0.0960137	−0.0480069	+	0.998847i	$0.515287\pi$
$774$	8.74981	0.314506
$775$	5.68893	0.204352
$776$	26.9062	0.965875
$777$	0.267668	0.00960254
$778$	−24.2436	−0.869173
$779$	−64.1631	−2.29888
$780$	7.36975	0.263879
$781$	51.0001	1.82493
$782$	43.1335	1.54245
$783$	−18.9529	−0.677322
$784$	−0.893576	−0.0319134
$785$	−9.23610	−0.329651
$786$	3.93297	0.140284
$787$	−16.3284	−0.582043	−0.291022	−	0.956716i	$-0.593995\pi$
$787$	−16.3284	−0.582043	−0.291022	+	0.956716i	$0.593995\pi$
$788$	−25.5698	−0.910886
$789$	−9.73507	−0.346578
$790$	0.421925	0.0150114
$791$	−5.50781	−0.195835
$792$	−28.3126	−1.00604
$793$	61.0986	2.16967
$794$	−28.0919	−0.996944
$795$	7.37398	0.261528
$796$	14.2663	0.505656
$797$	34.8557	1.23465	0.617325	−	0.786708i	$-0.288216\pi$
$797$	34.8557	1.23465	0.617325	+	0.786708i	$0.288216\pi$
$798$	−2.50711	−0.0887506
$799$	22.2655	0.787698
$800$	5.71142	0.201929
$801$	−2.13297	−0.0753647
$802$	17.0298	0.601344
$803$	−6.22395	−0.219638
$804$	1.59196	0.0561443
$805$	4.96942	0.175149
$806$	30.4769	1.07350
$807$	−24.4140	−0.859412
$808$	−13.2194	−0.465056
$809$	−41.4204	−1.45626	−0.728132	−	0.685437i	$-0.759611\pi$
$809$	−41.4204	−1.45626	−0.728132	+	0.685437i	$0.759611\pi$
$810$	1.66948	0.0586594
$811$	7.34996	0.258092	0.129046	−	0.991639i	$-0.458808\pi$
$811$	7.34996	0.258092	0.129046	+	0.991639i	$0.458808\pi$
$812$	−2.91330	−0.102237
$813$	−19.2429	−0.674880
$814$	−2.01310	−0.0705593
$815$	−19.2838	−0.675483
$816$	0.734172	0.0257012
$817$	25.9226	0.906917
$818$	21.1541	0.739637
$819$	7.74008	0.270460
$820$	−15.0008	−0.523852
$821$	−38.8242	−1.35498	−0.677488	−	0.735534i	$-0.736931\pi$
$821$	−38.8242	−1.35498	−0.677488	+	0.735534i	$0.736931\pi$
$822$	9.56636	0.333665
$823$	−21.4901	−0.749099	−0.374550	−	0.927207i	$-0.622203\pi$
$823$	−21.4901	−0.749099	−0.374550	+	0.927207i	$0.622203\pi$
$824$	−14.4766	−0.504317
$825$	4.41697	0.153779
$826$	−5.29187	−0.184128
$827$	19.6349	0.682771	0.341386	−	0.939923i	$-0.389104\pi$
$827$	19.6349	0.682771	0.341386	+	0.939923i	$0.389104\pi$
$828$	−23.1187	−0.803431
$829$	−20.8272	−0.723358	−0.361679	−	0.932303i	$-0.617796\pi$
$829$	−20.8272	−0.723358	−0.361679	+	0.932303i	$0.617796\pi$
$830$	1.39159	0.0483029
$831$	−0.404038	−0.0140159
$832$	28.9204	1.00264
$833$	39.1751	1.35734
$834$	−14.4374	−0.499928
$835$	−24.8161	−0.858796
$836$	−32.5122	−1.12446
$837$	27.1905	0.939840
$838$	−20.2803	−0.700572
$839$	22.3217	0.770630	0.385315	−	0.922785i	$-0.374093\pi$
$839$	22.3217	0.770630	0.385315	+	0.922785i	$0.374093\pi$
$840$	−1.51222	−0.0521766
$841$	−13.2754	−0.457773
$842$	13.6455	0.470253
$843$	−15.5712	−0.536299
$844$	−7.71665	−0.265618
$845$	26.0932	0.897634
$846$	6.92115	0.237954
$847$	−6.67530	−0.229366
$848$	−1.06203	−0.0364701
$849$	9.75238	0.334701
$850$	−5.03755	−0.172787
$851$	−4.24096	−0.145378
$852$	−12.6726	−0.434156
$853$	−4.79245	−0.164090	−0.0820452	−	0.996629i	$-0.526145\pi$
$853$	−4.79245	−0.164090	−0.0820452	+	0.996629i	$0.526145\pi$
$854$	−4.85938	−0.166285
$855$	11.5488	0.394962
$856$	−15.8985	−0.543400
$857$	49.1822	1.68003	0.840016	−	0.542561i	$-0.182545\pi$
$857$	49.1822	1.68003	0.840016	+	0.542561i	$0.182545\pi$
$858$	23.6628	0.807834
$859$	45.6426	1.55731	0.778653	−	0.627454i	$-0.215903\pi$
$859$	45.6426	1.55731	0.778653	+	0.627454i	$0.215903\pi$
$860$	6.06050	0.206661
$861$	6.40412	0.218252
$862$	6.66800	0.227113
$863$	16.9728	0.577761	0.288880	−	0.957365i	$-0.406717\pi$
$863$	16.9728	0.577761	0.288880	+	0.957365i	$0.406717\pi$
$864$	27.2980	0.928695
$865$	−17.0185	−0.578645
$866$	−3.08396	−0.104797
$867$	−16.3572	−0.555520
$868$	4.17951	0.141862
$869$	−2.33589	−0.0792397
$870$	3.16372	0.107260
$871$	8.44465	0.286136
$872$	18.1557	0.614829
$873$	20.5092	0.694131
$874$	39.7228	1.34364
$875$	−0.580377	−0.0196203
$876$	1.54654	0.0522527
$877$	−6.47951	−0.218797	−0.109399	−	0.993998i	$-0.534893\pi$
$877$	−6.47951	−0.218797	−0.109399	+	0.993998i	$0.534893\pi$
$878$	25.9480	0.875703
$879$	−7.39888	−0.249558
$880$	−0.636148	−0.0214446
$881$	−13.0221	−0.438724	−0.219362	−	0.975644i	$-0.570398\pi$
$881$	−13.0221	−0.438724	−0.219362	+	0.975644i	$0.570398\pi$
$882$	12.1774	0.410035
$883$	24.1962	0.814268	0.407134	−	0.913369i	$-0.366528\pi$
$883$	24.1962	0.814268	0.407134	+	0.913369i	$0.366528\pi$
$884$	46.5334	1.56509
$885$	−9.90890	−0.333084
$886$	12.9461	0.434932
$887$	−18.6517	−0.626264	−0.313132	−	0.949710i	$-0.601378\pi$
$887$	−18.6517	−0.626264	−0.313132	+	0.949710i	$0.601378\pi$
$888$	1.29055	0.0433079
$889$	4.77571	0.160172
$890$	0.856822	0.0287207
$891$	−9.24267	−0.309641
$892$	−9.69404	−0.324581
$893$	20.5049	0.686171
$894$	15.9249	0.532609
$895$	−6.10678	−0.204127
$896$	4.32941	0.144636
$897$	49.8498	1.66444
$898$	15.6885	0.523531
$899$	−22.5590	−0.752385
$900$	2.70003	0.0900010
$901$	46.5601	1.55114
$902$	−48.1647	−1.60371
$903$	−2.58733	−0.0861010
$904$	−26.5556	−0.883226
$905$	−1.02757	−0.0341575
$906$	7.19200	0.238938
$907$	−19.1208	−0.634896	−0.317448	−	0.948276i	$-0.602826\pi$
$907$	−19.1208	−0.634896	−0.317448	+	0.948276i	$0.602826\pi$
$908$	37.3774	1.24041
$909$	−10.0765	−0.334215
$910$	−3.10922	−0.103070
$911$	6.04981	0.200439	0.100220	−	0.994965i	$-0.468045\pi$
$911$	6.04981	0.200439	0.100220	+	0.994965i	$0.468045\pi$
$912$	0.676119	0.0223885
$913$	−7.70423	−0.254973
$914$	−18.0369	−0.596607
$915$	−9.09909	−0.300807
$916$	−9.04738	−0.298934
$917$	2.86103	0.0944796
$918$	−24.0772	−0.794666
$919$	19.9158	0.656962	0.328481	−	0.944511i	$-0.393463\pi$
$919$	19.9158	0.656962	0.328481	+	0.944511i	$0.393463\pi$
$920$	23.9598	0.789930
$921$	1.96762	0.0648354
$922$	−33.2494	−1.09501
$923$	−67.2224	−2.21265
$924$	3.24504	0.106754
$925$	0.495300	0.0162854
$926$	−10.3783	−0.341054
$927$	−11.0348	−0.362430
$928$	−22.6482	−0.743464
$929$	0.333759	0.0109503	0.00547514	−	0.999985i	$-0.498257\pi$
$929$	0.333759	0.0109503	0.00547514	+	0.999985i	$0.498257\pi$
$930$	−4.53877	−0.148832
$931$	36.0774	1.18239
$932$	13.8744	0.454471
$933$	8.06844	0.264149
$934$	1.93329	0.0632590
$935$	27.8892	0.912075
$936$	37.3184	1.21979
$937$	−14.6611	−0.478957	−0.239479	−	0.970902i	$-0.576977\pi$
$937$	−14.6611	−0.478957	−0.239479	+	0.970902i	$0.576977\pi$
$938$	−0.671633	−0.0219296
$939$	−14.9598	−0.488194
$940$	4.79389	0.156360
$941$	−4.63624	−0.151137	−0.0755686	−	0.997141i	$-0.524077\pi$
$941$	−4.63624	−0.151137	−0.0755686	+	0.997141i	$0.524077\pi$
$942$	7.36880	0.240088
$943$	−101.467	−3.30423
$944$	1.42711	0.0464486
$945$	−2.77394	−0.0902362
$946$	19.4591	0.632669
$947$	10.9543	0.355965	0.177983	−	0.984034i	$-0.443043\pi$
$947$	10.9543	0.355965	0.177983	+	0.984034i	$0.443043\pi$
$948$	0.580427	0.0188514
$949$	8.20369	0.266303
$950$	−4.63922	−0.150516
$951$	1.58871	0.0515175
$952$	−9.54833	−0.309463
$953$	−26.4968	−0.858315	−0.429158	−	0.903230i	$-0.641190\pi$
$953$	−26.4968	−0.858315	−0.429158	+	0.903230i	$0.641190\pi$
$954$	14.4730	0.468581
$955$	−11.3023	−0.365735
$956$	−24.9932	−0.808339
$957$	−17.5152	−0.566185
$958$	0.203285	0.00656783
$959$	6.95904	0.224719
$960$	−4.30697	−0.139007
$961$	1.36388	0.0439962
$962$	2.65344	0.0855504
$963$	−12.1186	−0.390517
$964$	24.3405	0.783956
$965$	−19.0759	−0.614074
$966$	−3.96473	−0.127563
$967$	−59.1812	−1.90314	−0.951569	−	0.307435i	$-0.900529\pi$
$967$	−59.1812	−1.90314	−0.951569	+	0.307435i	$0.900529\pi$
$968$	−32.1846	−1.03445
$969$	−29.6416	−0.952225
$970$	−8.23862	−0.264526
$971$	−5.45699	−0.175123	−0.0875616	−	0.996159i	$-0.527907\pi$
$971$	−5.45699	−0.175123	−0.0875616	+	0.996159i	$0.527907\pi$
$972$	20.4473	0.655847
$973$	−10.5025	−0.336695
$974$	1.45072	0.0464842
$975$	−5.82195	−0.186452
$976$	1.31048	0.0419475
$977$	17.0837	0.546555	0.273278	−	0.961935i	$-0.411892\pi$
$977$	17.0837	0.546555	0.273278	+	0.961935i	$0.411892\pi$
$978$	15.3851	0.491962
$979$	−4.74359	−0.151606
$980$	8.43461	0.269434
$981$	13.8391	0.441850
$982$	−15.5996	−0.497803
$983$	42.1851	1.34549	0.672747	−	0.739872i	$-0.265114\pi$
$983$	42.1851	1.34549	0.672747	+	0.739872i	$0.265114\pi$
$984$	30.8771	0.984325
$985$	20.1996	0.643612
$986$	19.9760	0.636167
$987$	−2.04659	−0.0651438
$988$	42.8538	1.36336
$989$	40.9939	1.30353
$990$	8.66926	0.275527
$991$	−11.7224	−0.372375	−0.186188	−	0.982514i	$-0.559613\pi$
$991$	−11.7224	−0.372375	−0.186188	+	0.982514i	$0.559613\pi$
$992$	32.4918	1.03162
$993$	7.35744	0.233481
$994$	5.34644	0.169579
$995$	−11.2701	−0.357286
$996$	1.91436	0.0606589
$997$	0.629575	0.0199388	0.00996942	−	0.999950i	$-0.496827\pi$
$997$	0.629575	0.0199388	0.00996942	+	0.999950i	$0.496827\pi$
$998$	17.1582	0.543134
$999$	2.36731	0.0748983

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	445.2.a.f.1.6	✓	7
3.2	odd	2		4005.2.a.o.1.2		7
4.3	odd	2		7120.2.a.bj.1.4		7
5.4	even	2		2225.2.a.k.1.2		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
445.2.a.f.1.6	✓	7	1.1	even	1	trivial
2225.2.a.k.1.2		7	5.4	even	2
4005.2.a.o.1.2		7	3.2	odd	2
7120.2.a.bj.1.4		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)

\(f(q)\)	\(=\)	\(q+0.856822 q^{2} -0.931146 q^{3} -1.26586 q^{4} +1.00000 q^{5} -0.797826 q^{6} -0.580377 q^{7} -2.79826 q^{8} -2.13297 q^{9} +O(q^{10})\)	\(q+0.856822 q^{2} -0.931146 q^{3} -1.26586 q^{4} +1.00000 q^{5} -0.797826 q^{6} -0.580377 q^{7} -2.79826 q^{8} -2.13297 q^{9} +0.856822 q^{10} -4.74359 q^{11} +1.17870 q^{12} +6.25246 q^{13} -0.497280 q^{14} -0.931146 q^{15} +0.134107 q^{16} -5.87935 q^{17} -1.82757 q^{18} -5.41445 q^{19} -1.26586 q^{20} +0.540416 q^{21} -4.06441 q^{22} -8.56239 q^{23} +2.60558 q^{24} +1.00000 q^{25} +5.35724 q^{26} +4.77954 q^{27} +0.734675 q^{28} -3.96542 q^{29} -0.797826 q^{30} +5.68893 q^{31} +5.71142 q^{32} +4.41697 q^{33} -5.03755 q^{34} -0.580377 q^{35} +2.70003 q^{36} +0.495300 q^{37} -4.63922 q^{38} -5.82195 q^{39} -2.79826 q^{40} +11.8503 q^{41} +0.463040 q^{42} -4.78767 q^{43} +6.00471 q^{44} -2.13297 q^{45} -7.33644 q^{46} -3.78707 q^{47} -0.124873 q^{48} -6.66316 q^{49} +0.856822 q^{50} +5.47453 q^{51} -7.91471 q^{52} -7.91925 q^{53} +4.09521 q^{54} -4.74359 q^{55} +1.62404 q^{56} +5.04164 q^{57} -3.39766 q^{58} +10.6416 q^{59} +1.17870 q^{60} +9.77193 q^{61} +4.87439 q^{62} +1.23793 q^{63} +4.62545 q^{64} +6.25246 q^{65} +3.78456 q^{66} +1.35061 q^{67} +7.44242 q^{68} +7.97283 q^{69} -0.497280 q^{70} -10.7514 q^{71} +5.96859 q^{72} +1.31208 q^{73} +0.424384 q^{74} -0.931146 q^{75} +6.85392 q^{76} +2.75307 q^{77} -4.98837 q^{78} +0.492431 q^{79} +0.134107 q^{80} +1.94845 q^{81} +10.1536 q^{82} +1.62413 q^{83} -0.684089 q^{84} -5.87935 q^{85} -4.10218 q^{86} +3.69239 q^{87} +13.2738 q^{88} +1.00000 q^{89} -1.82757 q^{90} -3.62878 q^{91} +10.8388 q^{92} -5.29722 q^{93} -3.24485 q^{94} -5.41445 q^{95} -5.31816 q^{96} -9.61533 q^{97} -5.70914 q^{98} +10.1179 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 4 q^{2} - 8 q^{3} + 8 q^{4} + 7 q^{5} - 2 q^{6} - 16 q^{7} - 12 q^{8} + 11 q^{9}+O(q^{10}) \) 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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