LMFDB - Embedded newform 445.2.a.f.1.3

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			1.3
Root			$-1.07810$ 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)

$f(q)$	$=$	$q-1.83770 q^{2} +2.10226 q^{3} +1.37716 q^{4} +1.00000 q^{5} -3.86333 q^{6} -4.72699 q^{7} +1.14460 q^{8} +1.41950 q^{9} +O(q^{10})$	\(q-1.83770 q^{2} +2.10226 q^{3} +1.37716 q^{4} +1.00000 q^{5} -3.86333 q^{6} -4.72699 q^{7} +1.14460 q^{8} +1.41950 q^{9} -1.83770 q^{10} -4.62787 q^{11} +2.89514 q^{12} -4.47978 q^{13} +8.68681 q^{14} +2.10226 q^{15} -4.85775 q^{16} -5.66969 q^{17} -2.60862 q^{18} +1.34308 q^{19} +1.37716 q^{20} -9.93737 q^{21} +8.50466 q^{22} +4.44187 q^{23} +2.40625 q^{24} +1.00000 q^{25} +8.23250 q^{26} -3.32263 q^{27} -6.50981 q^{28} -4.27246 q^{29} -3.86333 q^{30} +9.40035 q^{31} +6.63791 q^{32} -9.72899 q^{33} +10.4192 q^{34} -4.72699 q^{35} +1.95487 q^{36} +6.89093 q^{37} -2.46817 q^{38} -9.41765 q^{39} +1.14460 q^{40} +5.69631 q^{41} +18.2619 q^{42} -5.04294 q^{43} -6.37330 q^{44} +1.41950 q^{45} -8.16284 q^{46} -8.76271 q^{47} -10.2123 q^{48} +15.3444 q^{49} -1.83770 q^{50} -11.9192 q^{51} -6.16935 q^{52} -8.23538 q^{53} +6.10600 q^{54} -4.62787 q^{55} -5.41053 q^{56} +2.82349 q^{57} +7.85151 q^{58} -7.52173 q^{59} +2.89514 q^{60} -6.94612 q^{61} -17.2751 q^{62} -6.70996 q^{63} -2.48300 q^{64} -4.47978 q^{65} +17.8790 q^{66} -6.38703 q^{67} -7.80805 q^{68} +9.33796 q^{69} +8.68681 q^{70} +5.84983 q^{71} +1.62476 q^{72} +11.0282 q^{73} -12.6635 q^{74} +2.10226 q^{75} +1.84962 q^{76} +21.8759 q^{77} +17.3069 q^{78} +7.89145 q^{79} -4.85775 q^{80} -11.2435 q^{81} -10.4681 q^{82} -15.3880 q^{83} -13.6853 q^{84} -5.66969 q^{85} +9.26742 q^{86} -8.98182 q^{87} -5.29707 q^{88} +1.00000 q^{89} -2.60862 q^{90} +21.1759 q^{91} +6.11715 q^{92} +19.7620 q^{93} +16.1033 q^{94} +1.34308 q^{95} +13.9546 q^{96} -0.674067 q^{97} -28.1985 q^{98} -6.56926 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.83770	−1.29945	−0.649727	−	0.760168i	$-0.725117\pi$
$2$	−1.83770	−1.29945	−0.649727	+	0.760168i	$0.725117\pi$
$3$	2.10226	1.21374	0.606870	−	0.794801i	$-0.292425\pi$
$3$	2.10226	1.21374	0.606870	+	0.794801i	$0.292425\pi$
$4$	1.37716	0.688578
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−3.86333	−1.57720
$7$	−4.72699	−1.78663	−0.893317	−	0.449426i	$-0.851628\pi$
$7$	−4.72699	−1.78663	−0.893317	+	0.449426i	$0.851628\pi$
$8$	1.14460	0.404678
$9$	1.41950	0.473166
$10$	−1.83770	−0.581133
$11$	−4.62787	−1.39536	−0.697678	−	0.716412i	$-0.745783\pi$
$11$	−4.62787	−1.39536	−0.697678	+	0.716412i	$0.745783\pi$
$12$	2.89514	0.835755
$13$	−4.47978	−1.24247	−0.621233	−	0.783626i	$-0.713368\pi$
$13$	−4.47978	−1.24247	−0.621233	+	0.783626i	$0.713368\pi$
$14$	8.68681	2.32165
$15$	2.10226	0.542801
$16$	−4.85775	−1.21444
$17$	−5.66969	−1.37510	−0.687551	−	0.726136i	$-0.741314\pi$
$17$	−5.66969	−1.37510	−0.687551	+	0.726136i	$0.741314\pi$
$18$	−2.60862	−0.614857
$19$	1.34308	0.308123	0.154061	−	0.988061i	$-0.450765\pi$
$19$	1.34308	0.308123	0.154061	+	0.988061i	$0.450765\pi$
$20$	1.37716	0.307942
$21$	−9.93737	−2.16851
$22$	8.50466	1.81320
$23$	4.44187	0.926193	0.463097	−	0.886308i	$-0.346738\pi$
$23$	4.44187	0.926193	0.463097	+	0.886308i	$0.346738\pi$
$24$	2.40625	0.491174
$25$	1.00000	0.200000
$26$	8.23250	1.61453
$27$	−3.32263	−0.639440
$28$	−6.50981	−1.23024
$29$	−4.27246	−0.793376	−0.396688	−	0.917954i	$-0.629840\pi$
$29$	−4.27246	−0.793376	−0.396688	+	0.917954i	$0.629840\pi$
$30$	−3.86333	−0.705345
$31$	9.40035	1.68835	0.844176	−	0.536066i	$-0.180090\pi$
$31$	9.40035	1.68835	0.844176	+	0.536066i	$0.180090\pi$
$32$	6.63791	1.17343
$33$	−9.72899	−1.69360
$34$	10.4192	1.78688
$35$	−4.72699	−0.799007
$36$	1.95487	0.325812
$37$	6.89093	1.13286	0.566431	−	0.824109i	$-0.308324\pi$
$37$	6.89093	1.13286	0.566431	+	0.824109i	$0.308324\pi$
$38$	−2.46817	−0.400391
$39$	−9.41765	−1.50803
$40$	1.14460	0.180978
$41$	5.69631	0.889614	0.444807	−	0.895626i	$-0.353272\pi$
$41$	5.69631	0.889614	0.444807	+	0.895626i	$0.353272\pi$
$42$	18.2619	2.81788
$43$	−5.04294	−0.769041	−0.384520	−	0.923117i	$-0.625633\pi$
$43$	−5.04294	−0.769041	−0.384520	+	0.923117i	$0.625633\pi$
$44$	−6.37330	−0.960812
$45$	1.41950	0.211606
$46$	−8.16284	−1.20354
$47$	−8.76271	−1.27817	−0.639087	−	0.769135i	$-0.720687\pi$
$47$	−8.76271	−1.27817	−0.639087	+	0.769135i	$0.720687\pi$
$48$	−10.2123	−1.47401
$49$	15.3444	2.19206
$50$	−1.83770	−0.259891
$51$	−11.9192	−1.66902
$52$	−6.16935	−0.855535
$53$	−8.23538	−1.13122	−0.565608	−	0.824674i	$-0.691358\pi$
$53$	−8.23538	−1.13122	−0.565608	+	0.824674i	$0.691358\pi$
$54$	6.10600	0.830922
$55$	−4.62787	−0.624022
$56$	−5.41053	−0.723012
$57$	2.82349	0.373981
$58$	7.85151	1.03095
$59$	−7.52173	−0.979246	−0.489623	−	0.871934i	$-0.662866\pi$
$59$	−7.52173	−0.979246	−0.489623	+	0.871934i	$0.662866\pi$
$60$	2.89514	0.373761
$61$	−6.94612	−0.889360	−0.444680	−	0.895689i	$-0.646683\pi$
$61$	−6.94612	−0.889360	−0.444680	+	0.895689i	$0.646683\pi$
$62$	−17.2751	−2.19393
$63$	−6.70996	−0.845375
$64$	−2.48300	−0.310376
$65$	−4.47978	−0.555648
$66$	17.8790	2.20075
$67$	−6.38703	−0.780300	−0.390150	−	0.920751i	$-0.627577\pi$
$67$	−6.38703	−0.780300	−0.390150	+	0.920751i	$0.627577\pi$
$68$	−7.80805	−0.946866
$69$	9.33796	1.12416
$70$	8.68681	1.03827
$71$	5.84983	0.694247	0.347123	−	0.937820i	$-0.387159\pi$
$71$	5.84983	0.694247	0.347123	+	0.937820i	$0.387159\pi$
$72$	1.62476	0.191480
$73$	11.0282	1.29076	0.645378	−	0.763863i	$-0.276700\pi$
$73$	11.0282	1.29076	0.645378	+	0.763863i	$0.276700\pi$
$74$	−12.6635	−1.47210
$75$	2.10226	0.242748
$76$	1.84962	0.212166
$77$	21.8759	2.49299
$78$	17.3069	1.95962
$79$	7.89145	0.887857	0.443929	−	0.896062i	$-0.353584\pi$
$79$	7.89145	0.887857	0.443929	+	0.896062i	$0.353584\pi$
$80$	−4.85775	−0.543113
$81$	−11.2435	−1.24928
$82$	−10.4681	−1.15601
$83$	−15.3880	−1.68905	−0.844527	−	0.535513i	$-0.820118\pi$
$83$	−15.3880	−1.68905	−0.844527	+	0.535513i	$0.820118\pi$
$84$	−13.6853	−1.49319
$85$	−5.66969	−0.614965
$86$	9.26742	0.999332
$87$	−8.98182	−0.962952
$88$	−5.29707	−0.564670
$89$	1.00000	0.106000
$89$	1.00000	0.106000
$90$	−2.60862	−0.274972
$91$	21.1759	2.21983
$92$	6.11715	0.637757
$93$	19.7620	2.04922
$94$	16.1033	1.66093
$95$	1.34308	0.137797
$96$	13.9546	1.42424
$97$	−0.674067	−0.0684412	−0.0342206	−	0.999414i	$-0.510895\pi$
$97$	−0.674067	−0.0684412	−0.0342206	+	0.999414i	$0.510895\pi$
$98$	−28.1985	−2.84848
$99$	−6.56926	−0.660235
$100$	1.37716	0.137716
$101$	13.4407	1.33740	0.668699	−	0.743533i	$-0.266851\pi$
$101$	13.4407	1.33740	0.668699	+	0.743533i	$0.266851\pi$
$102$	21.9039	2.16881
$103$	−11.4024	−1.12352	−0.561758	−	0.827302i	$-0.689875\pi$
$103$	−11.4024	−1.12352	−0.561758	+	0.827302i	$0.689875\pi$
$104$	−5.12756	−0.502799
$105$	−9.93737	−0.969788
$106$	15.1342	1.46996
$107$	−8.51924	−0.823586	−0.411793	−	0.911277i	$-0.635097\pi$
$107$	−8.51924	−0.823586	−0.411793	+	0.911277i	$0.635097\pi$
$108$	−4.57577	−0.440304
$109$	12.7180	1.21816	0.609081	−	0.793108i	$-0.291538\pi$
$109$	12.7180	1.21816	0.609081	+	0.793108i	$0.291538\pi$
$110$	8.50466	0.810887
$111$	14.4865	1.37500
$112$	22.9626	2.16976
$113$	8.25522	0.776586	0.388293	−	0.921536i	$-0.373065\pi$
$113$	8.25522	0.776586	0.388293	+	0.921536i	$0.373065\pi$
$114$	−5.18875	−0.485971
$115$	4.44187	0.414206
$116$	−5.88384	−0.546301
$117$	−6.35903	−0.587893
$118$	13.8227	1.27248
$119$	26.8006	2.45681
$120$	2.40625	0.219660
$121$	10.4172	0.947018
$122$	12.7649	1.15568
$123$	11.9751	1.07976
$124$	12.9457	1.16256
$125$	1.00000	0.0894427
$126$	12.3309	1.09853
$127$	1.77097	0.157149	0.0785743	−	0.996908i	$-0.474963\pi$
$127$	1.77097	0.157149	0.0785743	+	0.996908i	$0.474963\pi$
$128$	−8.71279	−0.770109
$129$	−10.6016	−0.933416
$130$	8.23250	0.722038
$131$	−7.28762	−0.636722	−0.318361	−	0.947969i	$-0.603132\pi$
$131$	−7.28762	−0.636722	−0.318361	+	0.947969i	$0.603132\pi$
$132$	−13.3983	−1.16618
$133$	−6.34870	−0.550503
$134$	11.7375	1.01396
$135$	−3.32263	−0.285966
$136$	−6.48955	−0.556474
$137$	0.754924	0.0644975	0.0322488	−	0.999480i	$-0.489733\pi$
$137$	0.754924	0.0644975	0.0322488	+	0.999480i	$0.489733\pi$
$138$	−17.1604	−1.46079
$139$	−6.29913	−0.534285	−0.267142	−	0.963657i	$-0.586079\pi$
$139$	−6.29913	−0.534285	−0.267142	+	0.963657i	$0.586079\pi$
$140$	−6.50981	−0.550179
$141$	−18.4215	−1.55137
$142$	−10.7502	−0.902141
$143$	20.7318	1.73368
$144$	−6.89557	−0.574631
$145$	−4.27246	−0.354808
$146$	−20.2666	−1.67728
$147$	32.2580	2.66060
$148$	9.48989	0.780064
$149$	−17.7115	−1.45098	−0.725490	−	0.688233i	$-0.758387\pi$
$149$	−17.7115	−1.45098	−0.725490	+	0.688233i	$0.758387\pi$
$150$	−3.86333	−0.315440
$151$	10.3991	0.846263	0.423132	−	0.906068i	$-0.360931\pi$
$151$	10.3991	0.846263	0.423132	+	0.906068i	$0.360931\pi$
$152$	1.53729	0.124690
$153$	−8.04812	−0.650652
$154$	−40.2014	−3.23953
$155$	9.40035	0.755054
$156$	−12.9696	−1.03840
$157$	−12.0575	−0.962292	−0.481146	−	0.876641i	$-0.659779\pi$
$157$	−12.0575	−0.962292	−0.481146	+	0.876641i	$0.659779\pi$
$158$	−14.5021	−1.15373
$159$	−17.3129	−1.37300
$160$	6.63791	0.524773
$161$	−20.9967	−1.65477
$162$	20.6623	1.62338
$163$	6.76614	0.529965	0.264982	−	0.964253i	$-0.414634\pi$
$163$	6.76614	0.529965	0.264982	+	0.964253i	$0.414634\pi$
$164$	7.84471	0.612569
$165$	−9.72899	−0.757401
$166$	28.2786	2.19485
$167$	−5.08722	−0.393661	−0.196830	−	0.980438i	$-0.563065\pi$
$167$	−5.08722	−0.393661	−0.196830	+	0.980438i	$0.563065\pi$
$168$	−11.3743	−0.877549
$169$	7.06839	0.543722
$170$	10.4192	0.799118
$171$	1.90649	0.145793
$172$	−6.94491	−0.529545
$173$	1.63396	0.124228	0.0621139	−	0.998069i	$-0.480216\pi$
$173$	1.63396	0.124228	0.0621139	+	0.998069i	$0.480216\pi$
$174$	16.5059	1.25131
$175$	−4.72699	−0.357327
$176$	22.4811	1.69457
$177$	−15.8126	−1.18855
$178$	−1.83770	−0.137742
$179$	−8.89727	−0.665014	−0.332507	−	0.943101i	$-0.607894\pi$
$179$	−8.89727	−0.665014	−0.332507	+	0.943101i	$0.607894\pi$
$180$	1.95487	0.145707
$181$	−7.14852	−0.531345	−0.265673	−	0.964063i	$-0.585594\pi$
$181$	−7.14852	−0.531345	−0.265673	+	0.964063i	$0.585594\pi$
$182$	−38.9150	−2.88457
$183$	−14.6026	−1.07945
$184$	5.08417	0.374810
$185$	6.89093	0.506631
$186$	−36.3167	−2.66287
$187$	26.2386	1.91876
$188$	−12.0676	−0.880122
$189$	15.7060	1.14245
$190$	−2.46817	−0.179060
$191$	−8.74055	−0.632444	−0.316222	−	0.948685i	$-0.602414\pi$
$191$	−8.74055	−0.632444	−0.316222	+	0.948685i	$0.602414\pi$
$192$	−5.21992	−0.376715
$193$	3.38192	0.243436	0.121718	−	0.992565i	$-0.461160\pi$
$193$	3.38192	0.243436	0.121718	+	0.992565i	$0.461160\pi$
$194$	1.23874	0.0889361
$195$	−9.41765	−0.674412
$196$	21.1317	1.50941
$197$	25.4934	1.81633	0.908164	−	0.418614i	$-0.137484\pi$
$197$	25.4934	1.81633	0.908164	+	0.418614i	$0.137484\pi$
$198$	12.0724	0.857945
$199$	−7.06669	−0.500945	−0.250472	−	0.968124i	$-0.580586\pi$
$199$	−7.06669	−0.500945	−0.250472	+	0.968124i	$0.580586\pi$
$200$	1.14460	0.0809356
$201$	−13.4272	−0.947082
$202$	−24.7000	−1.73789
$203$	20.1959	1.41747
$204$	−16.4146	−1.14925
$205$	5.69631	0.397847
$206$	20.9543	1.45995
$207$	6.30522	0.438243
$208$	21.7616	1.50890
$209$	−6.21558	−0.429941
$210$	18.2619	1.26019
$211$	−14.2330	−0.979842	−0.489921	−	0.871767i	$-0.662974\pi$
$211$	−14.2330	−0.979842	−0.489921	+	0.871767i	$0.662974\pi$
$212$	−11.3414	−0.778931
$213$	12.2979	0.842635
$214$	15.6559	1.07021
$215$	−5.04294	−0.343925
$216$	−3.80309	−0.258767
$217$	−44.4354	−3.01647
$218$	−23.3719	−1.58295
$219$	23.1842	1.56664
$220$	−6.37330	−0.429688
$221$	25.3990	1.70852
$222$	−26.6219	−1.78675
$223$	−3.11207	−0.208400	−0.104200	−	0.994556i	$-0.533228\pi$
$223$	−3.11207	−0.208400	−0.104200	+	0.994556i	$0.533228\pi$
$224$	−31.3773	−2.09649
$225$	1.41950	0.0946332
$226$	−15.1707	−1.00914
$227$	−3.57158	−0.237054	−0.118527	−	0.992951i	$-0.537817\pi$
$227$	−3.57158	−0.237054	−0.118527	+	0.992951i	$0.537817\pi$
$228$	3.88839	0.257515
$229$	5.99340	0.396055	0.198027	−	0.980196i	$-0.436547\pi$
$229$	5.99340	0.396055	0.198027	+	0.980196i	$0.436547\pi$
$230$	−8.16284	−0.538242
$231$	45.9889	3.02584
$232$	−4.89027	−0.321062
$233$	−4.75231	−0.311334	−0.155667	−	0.987810i	$-0.549753\pi$
$233$	−4.75231	−0.311334	−0.155667	+	0.987810i	$0.549753\pi$
$234$	11.6860	0.763939
$235$	−8.76271	−0.571616
$236$	−10.3586	−0.674288
$237$	16.5899	1.07763
$238$	−49.2516	−3.19250
$239$	−15.9181	−1.02965	−0.514827	−	0.857294i	$-0.672144\pi$
$239$	−15.9181	−1.02965	−0.514827	+	0.857294i	$0.672144\pi$
$240$	−10.2123	−0.659199
$241$	12.3726	0.796987	0.398493	−	0.917171i	$-0.369533\pi$
$241$	12.3726	0.796987	0.398493	+	0.917171i	$0.369533\pi$
$242$	−19.1437	−1.23061
$243$	−13.6689	−0.876862
$244$	−9.56590	−0.612394
$245$	15.3444	0.980321
$246$	−22.0067	−1.40310
$247$	−6.01668	−0.382832
$248$	10.7597	0.683239
$249$	−32.3496	−2.05007
$250$	−1.83770	−0.116227
$251$	−28.0777	−1.77225	−0.886125	−	0.463447i	$-0.846612\pi$
$251$	−28.0777	−1.77225	−0.886125	+	0.463447i	$0.846612\pi$
$252$	−9.24066	−0.582107
$253$	−20.5564	−1.29237
$254$	−3.25453	−0.204207
$255$	−11.9192	−0.746408
$256$	20.9775	1.31110
$257$	−8.05379	−0.502382	−0.251191	−	0.967938i	$-0.580822\pi$
$257$	−8.05379	−0.502382	−0.251191	+	0.967938i	$0.580822\pi$
$258$	19.4825	1.21293
$259$	−32.5734	−2.02401
$260$	−6.16935	−0.382607
$261$	−6.06475	−0.375398
$262$	13.3925	0.827391
$263$	28.3461	1.74790	0.873949	−	0.486018i	$-0.161551\pi$
$263$	28.3461	1.74790	0.873949	+	0.486018i	$0.161551\pi$
$264$	−11.1358	−0.685363
$265$	−8.23538	−0.505895
$266$	11.6670	0.715352
$267$	2.10226	0.128656
$268$	−8.79594	−0.537298
$269$	−4.54674	−0.277220	−0.138610	−	0.990347i	$-0.544263\pi$
$269$	−4.54674	−0.277220	−0.138610	+	0.990347i	$0.544263\pi$
$270$	6.10600	0.371600
$271$	12.5691	0.763521	0.381760	−	0.924261i	$-0.375318\pi$
$271$	12.5691	0.763521	0.381760	+	0.924261i	$0.375318\pi$
$272$	27.5420	1.66998
$273$	44.5172	2.69430
$274$	−1.38733	−0.0838115
$275$	−4.62787	−0.279071
$276$	12.8598	0.774071
$277$	16.6171	0.998426	0.499213	−	0.866479i	$-0.333622\pi$
$277$	16.6171	0.998426	0.499213	+	0.866479i	$0.333622\pi$
$278$	11.5759	0.694278
$279$	13.3438	0.798871
$280$	−5.41053	−0.323341
$281$	−32.3979	−1.93270	−0.966348	−	0.257237i	$-0.917188\pi$
$281$	−32.3979	−1.93270	−0.966348	+	0.257237i	$0.917188\pi$
$282$	33.8533	2.01593
$283$	1.59565	0.0948517	0.0474258	−	0.998875i	$-0.484898\pi$
$283$	1.59565	0.0948517	0.0474258	+	0.998875i	$0.484898\pi$
$284$	8.05612	0.478043
$285$	2.82349	0.167249
$286$	−38.0990	−2.25284
$287$	−26.9264	−1.58942
$288$	9.42250	0.555226
$289$	15.1454	0.890908
$290$	7.85151	0.461057
$291$	−1.41707	−0.0830698
$292$	15.1876	0.888786
$293$	12.2986	0.718491	0.359246	−	0.933243i	$-0.383034\pi$
$293$	12.2986	0.718491	0.359246	+	0.933243i	$0.383034\pi$
$294$	−59.2807	−3.45732
$295$	−7.52173	−0.437932
$296$	7.88737	0.458444
$297$	15.3767	0.892246
$298$	32.5484	1.88548
$299$	−19.8986	−1.15076
$300$	2.89514	0.167151
$301$	23.8379	1.37399
$302$	−19.1104	−1.09968
$303$	28.2558	1.62325
$304$	−6.52433	−0.374196
$305$	−6.94612	−0.397734
$306$	14.7901	0.845492
$307$	−25.7367	−1.46887	−0.734436	−	0.678678i	$-0.762553\pi$
$307$	−25.7367	−1.46887	−0.734436	+	0.678678i	$0.762553\pi$
$308$	30.1265	1.71662
$309$	−23.9709	−1.36366
$310$	−17.2751	−0.981157
$311$	8.88817	0.504002	0.252001	−	0.967727i	$-0.418911\pi$
$311$	8.88817	0.504002	0.252001	+	0.967727i	$0.418911\pi$
$312$	−10.7795	−0.610267
$313$	12.2041	0.689818	0.344909	−	0.938636i	$-0.387910\pi$
$313$	12.2041	0.689818	0.344909	+	0.938636i	$0.387910\pi$
$314$	22.1581	1.25045
$315$	−6.70996	−0.378063
$316$	10.8678	0.611359
$317$	14.6948	0.825341	0.412671	−	0.910880i	$-0.364596\pi$
$317$	14.6948	0.825341	0.412671	+	0.910880i	$0.364596\pi$
$318$	31.8160	1.78415
$319$	19.7724	1.10704
$320$	−2.48300	−0.138804
$321$	−17.9097	−0.999620
$322$	38.5857	2.15030
$323$	−7.61483	−0.423700
$324$	−15.4841	−0.860227
$325$	−4.47978	−0.248493
$326$	−12.4342	−0.688664
$327$	26.7365	1.47853
$328$	6.52001	0.360007
$329$	41.4213	2.28363
$330$	17.8790	0.984207
$331$	−18.4337	−1.01321	−0.506603	−	0.862179i	$-0.669099\pi$
$331$	−18.4337	−1.01321	−0.506603	+	0.862179i	$0.669099\pi$
$332$	−21.1917	−1.16305
$333$	9.78166	0.536032
$334$	9.34880	0.511544
$335$	−6.38703	−0.348961
$336$	48.2733	2.63352
$337$	−31.8582	−1.73543	−0.867714	−	0.497064i	$-0.834411\pi$
$337$	−31.8582	−1.73543	−0.867714	+	0.497064i	$0.834411\pi$
$338$	−12.9896	−0.706542
$339$	17.3546	0.942574
$340$	−7.80805	−0.423451
$341$	−43.5036	−2.35585
$342$	−3.50357	−0.189451
$343$	−39.4441	−2.12978
$344$	−5.77216	−0.311214
$345$	9.33796	0.502739
$346$	−3.00274	−0.161428
$347$	5.94469	0.319128	0.159564	−	0.987188i	$-0.448991\pi$
$347$	5.94469	0.319128	0.159564	+	0.987188i	$0.448991\pi$
$348$	−12.3694	−0.663068
$349$	12.4803	0.668053	0.334026	−	0.942564i	$-0.391593\pi$
$349$	12.4803	0.668053	0.334026	+	0.942564i	$0.391593\pi$
$350$	8.68681	0.464330
$351$	14.8846	0.794482
$352$	−30.7194	−1.63735
$353$	17.7880	0.946761	0.473380	−	0.880858i	$-0.343034\pi$
$353$	17.7880	0.946761	0.473380	+	0.880858i	$0.343034\pi$
$354$	29.0590	1.54447
$355$	5.84983	0.310476
$356$	1.37716	0.0729891
$357$	56.3418	2.98193
$358$	16.3506	0.864154
$359$	6.80968	0.359401	0.179700	−	0.983721i	$-0.442487\pi$
$359$	6.80968	0.359401	0.179700	+	0.983721i	$0.442487\pi$
$360$	1.62476	0.0856324
$361$	−17.1961	−0.905060
$362$	13.1369	0.690458
$363$	21.8997	1.14943
$364$	29.1625	1.52853
$365$	11.0282	0.577244
$366$	26.8352	1.40270
$367$	−10.9282	−0.570447	−0.285224	−	0.958461i	$-0.592068\pi$
$367$	−10.9282	−0.570447	−0.285224	+	0.958461i	$0.592068\pi$
$368$	−21.5775	−1.12480
$369$	8.08590	0.420935
$370$	−12.6635	−0.658343
$371$	38.9285	2.02107
$372$	27.2153	1.41105
$373$	−6.11907	−0.316834	−0.158417	−	0.987372i	$-0.550639\pi$
$373$	−6.11907	−0.316834	−0.158417	+	0.987372i	$0.550639\pi$
$374$	−48.2188	−2.49334
$375$	2.10226	0.108560
$376$	−10.0298	−0.517249
$377$	19.1397	0.985742
$378$	−28.8630	−1.48455
$379$	28.8933	1.48415	0.742074	−	0.670318i	$-0.233842\pi$
$379$	28.8933	1.48415	0.742074	+	0.670318i	$0.233842\pi$
$380$	1.84962	0.0948837
$381$	3.72305	0.190738
$382$	16.0626	0.821832
$383$	−15.9953	−0.817322	−0.408661	−	0.912686i	$-0.634004\pi$
$383$	−15.9953	−0.817322	−0.408661	+	0.912686i	$0.634004\pi$
$384$	−18.3166	−0.934713
$385$	21.8759	1.11490
$386$	−6.21497	−0.316334
$387$	−7.15844	−0.363884
$388$	−0.928296	−0.0471271
$389$	30.0561	1.52390	0.761952	−	0.647633i	$-0.224241\pi$
$389$	30.0561	1.52390	0.761952	+	0.647633i	$0.224241\pi$
$390$	17.3069	0.876367
$391$	−25.1840	−1.27361
$392$	17.5633	0.887080
$393$	−15.3205	−0.772816
$394$	−46.8493	−2.36023
$395$	7.89145	0.397062
$396$	−9.04689	−0.454624
$397$	25.5045	1.28004	0.640018	−	0.768360i	$-0.278927\pi$
$397$	25.5045	1.28004	0.640018	+	0.768360i	$0.278927\pi$
$398$	12.9865	0.650954
$399$	−13.3466	−0.668167
$400$	−4.85775	−0.242888
$401$	−19.4677	−0.972172	−0.486086	−	0.873911i	$-0.661576\pi$
$401$	−19.4677	−0.972172	−0.486086	+	0.873911i	$0.661576\pi$
$402$	24.6752	1.23069
$403$	−42.1114	−2.09772
$404$	18.5099	0.920903
$405$	−11.2435	−0.558695
$406$	−37.1140	−1.84194
$407$	−31.8903	−1.58075
$408$	−13.6427	−0.675415
$409$	23.2110	1.14771	0.573854	−	0.818957i	$-0.305448\pi$
$409$	23.2110	1.14771	0.573854	+	0.818957i	$0.305448\pi$
$410$	−10.4681	−0.516984
$411$	1.58705	0.0782833
$412$	−15.7029	−0.773628
$413$	35.5552	1.74956
$414$	−11.5871	−0.569477
$415$	−15.3880	−0.755368
$416$	−29.7363	−1.45794
$417$	−13.2424	−0.648483
$418$	11.4224	0.558688
$419$	−22.3096	−1.08990	−0.544948	−	0.838470i	$-0.683451\pi$
$419$	−22.3096	−1.08990	−0.544948	+	0.838470i	$0.683451\pi$
$420$	−13.6853	−0.667775
$421$	−29.2036	−1.42330	−0.711648	−	0.702536i	$-0.752051\pi$
$421$	−29.2036	−1.42330	−0.711648	+	0.702536i	$0.752051\pi$
$422$	26.1561	1.27326
$423$	−12.4387	−0.604788
$424$	−9.42623	−0.457778
$425$	−5.66969	−0.275021
$426$	−22.5998	−1.09496
$427$	32.8343	1.58896
$428$	−11.7323	−0.567104
$429$	43.5837	2.10424
$430$	9.26742	0.446915
$431$	14.8466	0.715134	0.357567	−	0.933888i	$-0.383606\pi$
$431$	14.8466	0.715134	0.357567	+	0.933888i	$0.383606\pi$
$432$	16.1405	0.776560
$433$	−13.0139	−0.625408	−0.312704	−	0.949851i	$-0.601235\pi$
$433$	−13.0139	−0.625408	−0.312704	+	0.949851i	$0.601235\pi$
$434$	81.6590	3.91976
$435$	−8.98182	−0.430645
$436$	17.5147	0.838800
$437$	5.96576	0.285381
$438$	−42.6057	−2.03578
$439$	−12.1861	−0.581609	−0.290804	−	0.956783i	$-0.593923\pi$
$439$	−12.1861	−0.581609	−0.290804	+	0.956783i	$0.593923\pi$
$440$	−5.29707	−0.252528
$441$	21.7814	1.03721
$442$	−46.6758	−2.22014
$443$	−1.82580	−0.0867465	−0.0433732	−	0.999059i	$-0.513810\pi$
$443$	−1.82580	−0.0867465	−0.0433732	+	0.999059i	$0.513810\pi$
$444$	19.9502	0.946795
$445$	1.00000	0.0474045
$446$	5.71906	0.270806
$447$	−37.2341	−1.76111
$448$	11.7371	0.554528
$449$	−10.9646	−0.517449	−0.258725	−	0.965951i	$-0.583302\pi$
$449$	−10.9646	−0.517449	−0.258725	+	0.965951i	$0.583302\pi$
$450$	−2.60862	−0.122971
$451$	−26.3618	−1.24133
$452$	11.3687	0.534740
$453$	21.8615	1.02714
$454$	6.56351	0.308041
$455$	21.1759	0.992740
$456$	3.23178	0.151342
$457$	4.14517	0.193903	0.0969514	−	0.995289i	$-0.469091\pi$
$457$	4.14517	0.193903	0.0969514	+	0.995289i	$0.469091\pi$
$458$	−11.0141	−0.514655
$459$	18.8383	0.879295
$460$	6.11715	0.285213
$461$	−35.6431	−1.66006	−0.830031	−	0.557717i	$-0.811678\pi$
$461$	−35.6431	−1.66006	−0.830031	+	0.557717i	$0.811678\pi$
$462$	−84.5139	−3.93194
$463$	−3.08964	−0.143588	−0.0717939	−	0.997419i	$-0.522872\pi$
$463$	−3.08964	−0.143588	−0.0717939	+	0.997419i	$0.522872\pi$
$464$	20.7545	0.963506
$465$	19.7620	0.916440
$466$	8.73335	0.404564
$467$	−5.13078	−0.237424	−0.118712	−	0.992929i	$-0.537877\pi$
$467$	−5.13078	−0.237424	−0.118712	+	0.992929i	$0.537877\pi$
$468$	−8.75738	−0.404810
$469$	30.1914	1.39411
$470$	16.1033	0.742789
$471$	−25.3480	−1.16797
$472$	−8.60939	−0.396279
$473$	23.3381	1.07309
$474$	−30.4873	−1.40033
$475$	1.34308	0.0616245
$476$	36.9086	1.69170
$477$	−11.6901	−0.535253
$478$	29.2527	1.33799
$479$	6.76545	0.309121	0.154561	−	0.987983i	$-0.450604\pi$
$479$	6.76545	0.309121	0.154561	+	0.987983i	$0.450604\pi$
$480$	13.9546	0.636938
$481$	−30.8698	−1.40754
$482$	−22.7371	−1.03565
$483$	−44.1405	−2.00846
$484$	14.3461	0.652096
$485$	−0.674067	−0.0306078
$486$	25.1194	1.13944
$487$	−0.747648	−0.0338792	−0.0169396	−	0.999857i	$-0.505392\pi$
$487$	−0.747648	−0.0338792	−0.0169396	+	0.999857i	$0.505392\pi$
$488$	−7.95055	−0.359905
$489$	14.2242	0.643240
$490$	−28.1985	−1.27388
$491$	22.1110	0.997857	0.498928	−	0.866643i	$-0.333727\pi$
$491$	22.1110	0.997857	0.498928	+	0.866643i	$0.333727\pi$
$492$	16.4916	0.743500
$493$	24.2235	1.09097
$494$	11.0569	0.497472
$495$	−6.56926	−0.295266
$496$	−45.6646	−2.05040
$497$	−27.6521	−1.24036
$498$	59.4490	2.66397
$499$	−35.1425	−1.57319	−0.786597	−	0.617467i	$-0.788159\pi$
$499$	−35.1425	−1.57319	−0.786597	+	0.617467i	$0.788159\pi$
$500$	1.37716	0.0615883
$501$	−10.6947	−0.477802
$502$	51.5985	2.30295
$503$	−14.8767	−0.663320	−0.331660	−	0.943399i	$-0.607609\pi$
$503$	−14.8767	−0.663320	−0.331660	+	0.943399i	$0.607609\pi$
$504$	−7.68023	−0.342105
$505$	13.4407	0.598103
$506$	37.7766	1.67937
$507$	14.8596	0.659938
$508$	2.43891	0.108209
$509$	−13.5925	−0.602476	−0.301238	−	0.953549i	$-0.597400\pi$
$509$	−13.5925	−0.602476	−0.301238	+	0.953549i	$0.597400\pi$
$510$	21.9039	0.969922
$511$	−52.1303	−2.30611
$512$	−21.1249	−0.933599
$513$	−4.46254	−0.197026
$514$	14.8005	0.652822
$515$	−11.4024	−0.502451
$516$	−14.6000	−0.642730
$517$	40.5527	1.78351
$518$	59.8602	2.63011
$519$	3.43502	0.150780
$520$	−5.12756	−0.224858
$521$	21.2471	0.930851	0.465425	−	0.885087i	$-0.345901\pi$
$521$	21.2471	0.930851	0.465425	+	0.885087i	$0.345901\pi$
$522$	11.1452	0.487813
$523$	37.9681	1.66023	0.830115	−	0.557592i	$-0.188275\pi$
$523$	37.9681	1.66023	0.830115	+	0.557592i	$0.188275\pi$
$524$	−10.0362	−0.438433
$525$	−9.93737	−0.433702
$526$	−52.0918	−2.27131
$527$	−53.2971	−2.32166
$528$	47.2610	2.05677
$529$	−3.26981	−0.142166
$530$	15.1342	0.657387
$531$	−10.6771	−0.463346
$532$	−8.74316	−0.379064
$533$	−25.5182	−1.10532
$534$	−3.86333	−0.167183
$535$	−8.51924	−0.368319
$536$	−7.31061	−0.315770
$537$	−18.7044	−0.807154
$538$	8.35556	0.360234
$539$	−71.0121	−3.05871
$540$	−4.57577	−0.196910
$541$	24.0786	1.03522	0.517609	−	0.855617i	$-0.326822\pi$
$541$	24.0786	1.03522	0.517609	+	0.855617i	$0.326822\pi$
$542$	−23.0984	−0.992159
$543$	−15.0280	−0.644915
$544$	−37.6349	−1.61358
$545$	12.7180	0.544779
$546$	−81.8094	−3.50112
$547$	−7.90482	−0.337986	−0.168993	−	0.985617i	$-0.554052\pi$
$547$	−7.90482	−0.337986	−0.168993	+	0.985617i	$0.554052\pi$
$548$	1.03965	0.0444116
$549$	−9.86001	−0.420815
$550$	8.50466	0.362640
$551$	−5.73823	−0.244457
$552$	10.6883	0.454922
$553$	−37.3028	−1.58628
$554$	−30.5374	−1.29741
$555$	14.4865	0.614919
$556$	−8.67488	−0.367897
$557$	−28.0206	−1.18727	−0.593635	−	0.804734i	$-0.702308\pi$
$557$	−28.0206	−1.18727	−0.593635	+	0.804734i	$0.702308\pi$
$558$	−24.5219	−1.03810
$559$	22.5912	0.955507
$560$	22.9626	0.970345
$561$	55.1604	2.32887
$562$	59.5378	2.51145
$563$	−41.0773	−1.73120	−0.865601	−	0.500735i	$-0.833063\pi$
$563$	−41.0773	−1.73120	−0.865601	+	0.500735i	$0.833063\pi$
$564$	−25.3693	−1.06824
$565$	8.25522	0.347300
$566$	−2.93234	−0.123255
$567$	53.1480	2.23201
$568$	6.69572	0.280946
$569$	−30.8521	−1.29339	−0.646693	−	0.762751i	$-0.723848\pi$
$569$	−30.8521	−1.29339	−0.646693	+	0.762751i	$0.723848\pi$
$570$	−5.18875	−0.217333
$571$	27.2466	1.14023	0.570117	−	0.821564i	$-0.306898\pi$
$571$	27.2466	1.14023	0.570117	+	0.821564i	$0.306898\pi$
$572$	28.5510	1.19378
$573$	−18.3749	−0.767623
$574$	49.4828	2.06537
$575$	4.44187	0.185239
$576$	−3.52462	−0.146859
$577$	26.1936	1.09045	0.545227	−	0.838288i	$-0.316443\pi$
$577$	26.1936	1.09045	0.545227	+	0.838288i	$0.316443\pi$
$578$	−27.8328	−1.15769
$579$	7.10968	0.295468
$580$	−5.88384	−0.244313
$581$	72.7390	3.01772
$582$	2.60415	0.107945
$583$	38.1123	1.57845
$584$	12.6229	0.522341
$585$	−6.35903	−0.262914
$586$	−22.6012	−0.933645
$587$	−16.6599	−0.687627	−0.343814	−	0.939038i	$-0.611719\pi$
$587$	−16.6599	−0.687627	−0.343814	+	0.939038i	$0.611719\pi$
$588$	44.4243	1.83203
$589$	12.6254	0.520219
$590$	13.8227	0.569072
$591$	53.5937	2.20455
$592$	−33.4744	−1.37579
$593$	−30.7704	−1.26359	−0.631794	−	0.775136i	$-0.717681\pi$
$593$	−30.7704	−1.26359	−0.631794	+	0.775136i	$0.717681\pi$
$594$	−28.2578	−1.15943
$595$	26.8006	1.09872
$596$	−24.3914	−0.999113
$597$	−14.8560	−0.608017
$598$	36.5677	1.49536
$599$	−24.7821	−1.01257	−0.506285	−	0.862366i	$-0.668981\pi$
$599$	−24.7821	−1.01257	−0.506285	+	0.862366i	$0.668981\pi$
$600$	2.40625	0.0982348
$601$	2.25624	0.0920340	0.0460170	−	0.998941i	$-0.485347\pi$
$601$	2.25624	0.0920340	0.0460170	+	0.998941i	$0.485347\pi$
$602$	−43.8070	−1.78544
$603$	−9.06638	−0.369212
$604$	14.3211	0.582719
$605$	10.4172	0.423519
$606$	−51.9258	−2.10934
$607$	20.1084	0.816175	0.408088	−	0.912943i	$-0.366196\pi$
$607$	20.1084	0.816175	0.408088	+	0.912943i	$0.366196\pi$
$608$	8.91521	0.361560
$609$	42.4570	1.72044
$610$	12.7649	0.516837
$611$	39.2550	1.58809
$612$	−11.0835	−0.448025
$613$	−14.9346	−0.603204	−0.301602	−	0.953434i	$-0.597521\pi$
$613$	−14.9346	−0.603204	−0.301602	+	0.953434i	$0.597521\pi$
$614$	47.2964	1.90873
$615$	11.9751	0.482884
$616$	25.0392	1.00886
$617$	17.1073	0.688714	0.344357	−	0.938839i	$-0.388097\pi$
$617$	17.1073	0.688714	0.344357	+	0.938839i	$0.388097\pi$
$618$	44.0514	1.77201
$619$	39.8548	1.60190	0.800950	−	0.598732i	$-0.204328\pi$
$619$	39.8548	1.60190	0.800950	+	0.598732i	$0.204328\pi$
$620$	12.9457	0.519914
$621$	−14.7587	−0.592245
$622$	−16.3338	−0.654927
$623$	−4.72699	−0.189383
$624$	45.7486	1.83141
$625$	1.00000	0.0400000
$626$	−22.4276	−0.896387
$627$	−13.0668	−0.521836
$628$	−16.6050	−0.662613
$629$	−39.0695	−1.55780
$630$	12.3309	0.491275
$631$	−1.26371	−0.0503077	−0.0251538	−	0.999684i	$-0.508008\pi$
$631$	−1.26371	−0.0503077	−0.0251538	+	0.999684i	$0.508008\pi$
$632$	9.03257	0.359296
$633$	−29.9215	−1.18927
$634$	−27.0047	−1.07249
$635$	1.77097	0.0702790
$636$	−23.8426	−0.945420
$637$	−68.7397	−2.72356
$638$	−36.3358	−1.43855
$639$	8.30382	0.328494
$640$	−8.71279	−0.344403
$641$	10.4319	0.412037	0.206018	−	0.978548i	$-0.433949\pi$
$641$	10.4319	0.412037	0.206018	+	0.978548i	$0.433949\pi$
$642$	32.9127	1.29896
$643$	35.6759	1.40692	0.703460	−	0.710735i	$-0.251637\pi$
$643$	35.6759	1.40692	0.703460	+	0.710735i	$0.251637\pi$
$644$	−28.9157	−1.13944
$645$	−10.6016	−0.417436
$646$	13.9938	0.550579
$647$	−27.5263	−1.08217	−0.541085	−	0.840968i	$-0.681986\pi$
$647$	−27.5263	−1.08217	−0.541085	+	0.840968i	$0.681986\pi$
$648$	−12.8694	−0.505556
$649$	34.8096	1.36640
$650$	8.23250	0.322905
$651$	−93.4147	−3.66121
$652$	9.31803	0.364922
$653$	−1.79316	−0.0701718	−0.0350859	−	0.999384i	$-0.511170\pi$
$653$	−1.79316	−0.0701718	−0.0350859	+	0.999384i	$0.511170\pi$
$654$	−49.1338	−1.92128
$655$	−7.28762	−0.284751
$656$	−27.6713	−1.08038
$657$	15.6545	0.610742
$658$	−76.1200	−2.96747
$659$	5.13756	0.200131	0.100065	−	0.994981i	$-0.468095\pi$
$659$	5.13756	0.200131	0.100065	+	0.994981i	$0.468095\pi$
$660$	−13.3983	−0.521530
$661$	−18.0419	−0.701748	−0.350874	−	0.936423i	$-0.614115\pi$
$661$	−18.0419	−0.701748	−0.350874	+	0.936423i	$0.614115\pi$
$662$	33.8756	1.31661
$663$	53.3952	2.07370
$664$	−17.6132	−0.683523
$665$	−6.34870	−0.246192
$666$	−17.9758	−0.696548
$667$	−18.9777	−0.734819
$668$	−7.00590	−0.271066
$669$	−6.54238	−0.252943
$670$	11.7375	0.453458
$671$	32.1458	1.24097
$672$	−65.9633	−2.54459
$673$	−1.02851	−0.0396461	−0.0198231	−	0.999804i	$-0.506310\pi$
$673$	−1.02851	−0.0396461	−0.0198231	+	0.999804i	$0.506310\pi$
$674$	58.5460	2.25511
$675$	−3.32263	−0.127888
$676$	9.73428	0.374395
$677$	36.0528	1.38562	0.692812	−	0.721119i	$-0.256372\pi$
$677$	36.0528	1.38562	0.692812	+	0.721119i	$0.256372\pi$
$678$	−31.8927	−1.22483
$679$	3.18631	0.122279
$680$	−6.48955	−0.248863
$681$	−7.50839	−0.287722
$682$	79.9467	3.06132
$683$	14.0836	0.538895	0.269447	−	0.963015i	$-0.413159\pi$
$683$	14.0836	0.538895	0.269447	+	0.963015i	$0.413159\pi$
$684$	2.62554	0.100390
$685$	0.754924	0.0288442
$686$	72.4866	2.76755
$687$	12.5997	0.480708
$688$	24.4973	0.933952
$689$	36.8926	1.40550
$690$	−17.1604	−0.653286
$691$	15.6187	0.594163	0.297081	−	0.954852i	$-0.403987\pi$
$691$	15.6187	0.594163	0.297081	+	0.954852i	$0.403987\pi$
$692$	2.25022	0.0855406
$693$	31.0528	1.17960
$694$	−10.9246	−0.414692
$695$	−6.29913	−0.238939
$696$	−10.2806	−0.389686
$697$	−32.2963	−1.22331
$698$	−22.9350	−0.868103
$699$	−9.99060	−0.377879
$700$	−6.50981	−0.246048
$701$	23.8768	0.901814	0.450907	−	0.892571i	$-0.351101\pi$
$701$	23.8768	0.901814	0.450907	+	0.892571i	$0.351101\pi$
$702$	−27.3535	−1.03239
$703$	9.25504	0.349060
$704$	11.4910	0.433084
$705$	−18.4215	−0.693794
$706$	−32.6891	−1.23027
$707$	−63.5340	−2.38944
$708$	−21.7765	−0.818410
$709$	−13.7608	−0.516798	−0.258399	−	0.966038i	$-0.583195\pi$
$709$	−13.7608	−0.516798	−0.258399	+	0.966038i	$0.583195\pi$
$710$	−10.7502	−0.403450
$711$	11.2019	0.420104
$712$	1.14460	0.0428958
$713$	41.7551	1.56374
$714$	−103.540	−3.87487
$715$	20.7318	0.775326
$716$	−12.2529	−0.457914
$717$	−33.4639	−1.24973
$718$	−12.5142	−0.467025
$719$	−18.7708	−0.700032	−0.350016	−	0.936744i	$-0.613824\pi$
$719$	−18.7708	−0.700032	−0.350016	+	0.936744i	$0.613824\pi$
$720$	−6.89557	−0.256983
$721$	53.8992	2.00731
$722$	31.6014	1.17608
$723$	26.0103	0.967335
$724$	−9.84462	−0.365873
$725$	−4.27246	−0.158675
$726$	−40.2451	−1.49364
$727$	26.9412	0.999192	0.499596	−	0.866258i	$-0.333482\pi$
$727$	26.9412	0.999192	0.499596	+	0.866258i	$0.333482\pi$
$728$	24.2379	0.898318
$729$	4.99491	0.184997
$730$	−20.2666	−0.750101
$731$	28.5919	1.05751
$732$	−20.1100	−0.743287
$733$	17.5711	0.649004	0.324502	−	0.945885i	$-0.394803\pi$
$733$	17.5711	0.649004	0.324502	+	0.945885i	$0.394803\pi$
$734$	20.0828	0.741269
$735$	32.2580	1.18985
$736$	29.4847	1.08682
$737$	29.5584	1.08880
$738$	−14.8595	−0.546986
$739$	11.5994	0.426689	0.213345	−	0.976977i	$-0.431564\pi$
$739$	11.5994	0.426689	0.213345	+	0.976977i	$0.431564\pi$
$740$	9.48989	0.348855
$741$	−12.6486	−0.464659
$742$	−71.5392	−2.62628
$743$	8.10055	0.297180	0.148590	−	0.988899i	$-0.452526\pi$
$743$	8.10055	0.297180	0.148590	+	0.988899i	$0.452526\pi$
$744$	22.6196	0.829275
$745$	−17.7115	−0.648898
$746$	11.2450	0.411710
$747$	−21.8433	−0.799203
$748$	36.1347	1.32121
$749$	40.2704	1.47145
$750$	−3.86333	−0.141069
$751$	2.23998	0.0817381	0.0408690	−	0.999165i	$-0.486987\pi$
$751$	2.23998	0.0817381	0.0408690	+	0.999165i	$0.486987\pi$
$752$	42.5671	1.55226
$753$	−59.0267	−2.15105
$754$	−35.1730	−1.28093
$755$	10.3991	0.378461
$756$	21.6296	0.786663
$757$	−13.5325	−0.491846	−0.245923	−	0.969289i	$-0.579091\pi$
$757$	−13.5325	−0.491846	−0.245923	+	0.969289i	$0.579091\pi$
$758$	−53.0973	−1.92858
$759$	−43.2149	−1.56860
$760$	1.53729	0.0557633
$761$	37.5201	1.36010	0.680051	−	0.733165i	$-0.261958\pi$
$761$	37.5201	1.36010	0.680051	+	0.733165i	$0.261958\pi$
$762$	−6.84186	−0.247855
$763$	−60.1178	−2.17641
$764$	−12.0371	−0.435487
$765$	−8.04812	−0.290980
$766$	29.3946	1.06207
$767$	33.6957	1.21668
$768$	44.1002	1.59133
$769$	7.95922	0.287017	0.143508	−	0.989649i	$-0.454162\pi$
$769$	7.95922	0.287017	0.143508	+	0.989649i	$0.454162\pi$
$770$	−40.2014	−1.44876
$771$	−16.9312	−0.609761
$772$	4.65744	0.167625
$773$	40.5378	1.45804	0.729021	−	0.684491i	$-0.239976\pi$
$773$	40.5378	1.45804	0.729021	+	0.684491i	$0.239976\pi$
$774$	13.1551	0.472850
$775$	9.40035	0.337670
$776$	−0.771539	−0.0276966
$777$	−68.4777	−2.45662
$778$	−55.2342	−1.98024
$779$	7.65057	0.274110
$780$	−12.9696	−0.464386
$781$	−27.0722	−0.968721
$782$	46.2808	1.65500
$783$	14.1958	0.507316
$784$	−74.5395	−2.66213
$785$	−12.0575	−0.430350
$786$	28.1545	1.00424
$787$	−39.0735	−1.39282	−0.696409	−	0.717645i	$-0.745220\pi$
$787$	−39.0735	−1.39282	−0.696409	+	0.717645i	$0.745220\pi$
$788$	35.1084	1.25068
$789$	59.5910	2.12149
$790$	−14.5021	−0.515963
$791$	−39.0224	−1.38748
$792$	−7.51919	−0.267183
$793$	31.1171	1.10500
$794$	−46.8698	−1.66335
$795$	−17.3129	−0.614025
$796$	−9.73194	−0.344939
$797$	−1.92101	−0.0680456	−0.0340228	−	0.999421i	$-0.510832\pi$
$797$	−1.92101	−0.0680456	−0.0340228	+	0.999421i	$0.510832\pi$
$798$	24.5272	0.868252
$799$	49.6819	1.75762
$800$	6.63791	0.234685
$801$	1.41950	0.0501555
$802$	35.7759	1.26329
$803$	−51.0372	−1.80106
$804$	−18.4914	−0.652140
$805$	−20.9967	−0.740035
$806$	77.3884	2.72589
$807$	−9.55843	−0.336473
$808$	15.3842	0.541216
$809$	−35.3476	−1.24276	−0.621378	−	0.783511i	$-0.713427\pi$
$809$	−35.3476	−1.24276	−0.621378	+	0.783511i	$0.713427\pi$
$810$	20.6623	0.725998
$811$	−42.5750	−1.49501	−0.747506	−	0.664255i	$-0.768749\pi$
$811$	−42.5750	−1.49501	−0.747506	+	0.664255i	$0.768749\pi$
$812$	27.8129	0.976040
$813$	26.4236	0.926716
$814$	58.6050	2.05410
$815$	6.76614	0.237007
$816$	57.9004	2.02692
$817$	−6.77304	−0.236959
$818$	−42.6549	−1.49139
$819$	30.0591	1.05035
$820$	7.84471	0.273949
$821$	−15.8671	−0.553766	−0.276883	−	0.960904i	$-0.589301\pi$
$821$	−15.8671	−0.553766	−0.276883	+	0.960904i	$0.589301\pi$
$822$	−2.91652	−0.101725
$823$	−13.2314	−0.461217	−0.230609	−	0.973047i	$-0.574072\pi$
$823$	−13.2314	−0.461217	−0.230609	+	0.973047i	$0.574072\pi$
$824$	−13.0513	−0.454662
$825$	−9.72899	−0.338720
$826$	−65.3399	−2.27346
$827$	−26.0034	−0.904228	−0.452114	−	0.891960i	$-0.649330\pi$
$827$	−26.0034	−0.904228	−0.452114	+	0.891960i	$0.649330\pi$
$828$	8.68328	0.301765
$829$	−4.11404	−0.142886	−0.0714432	−	0.997445i	$-0.522760\pi$
$829$	−4.11404	−0.142886	−0.0714432	+	0.997445i	$0.522760\pi$
$830$	28.2786	0.981565
$831$	34.9335	1.21183
$832$	11.1233	0.385631
$833$	−86.9983	−3.01431
$834$	24.3356	0.842674
$835$	−5.08722	−0.176050
$836$	−8.55983	−0.296048
$837$	−31.2338	−1.07960
$838$	40.9985	1.41627
$839$	34.8946	1.20469	0.602347	−	0.798234i	$-0.294232\pi$
$839$	34.8946	1.20469	0.602347	+	0.798234i	$0.294232\pi$
$840$	−11.3743	−0.392452
$841$	−10.7461	−0.370555
$842$	53.6675	1.84951
$843$	−68.1088	−2.34579
$844$	−19.6011	−0.674698
$845$	7.06839	0.243160
$846$	22.8586	0.785894
$847$	−49.2420	−1.69198
$848$	40.0054	1.37379
$849$	3.35448	0.115125
$850$	10.4192	0.357376
$851$	30.6086	1.04925
$852$	16.9361	0.580220
$853$	2.21046	0.0756847	0.0378423	−	0.999284i	$-0.487952\pi$
$853$	2.21046	0.0756847	0.0378423	+	0.999284i	$0.487952\pi$
$854$	−60.3397	−2.06478
$855$	1.90649	0.0652007
$856$	−9.75115	−0.333287
$857$	0.0315501	0.00107773	0.000538865	−	1.00000i	$-0.499828\pi$
$857$	0.0315501	0.00107773	0.000538865		1.00000i	$0.499828\pi$
$858$	−80.0939	−2.73436
$859$	47.3550	1.61573	0.807867	−	0.589365i	$-0.200622\pi$
$859$	47.3550	1.61573	0.807867	+	0.589365i	$0.200622\pi$
$860$	−6.94491	−0.236820
$861$	−56.6063	−1.92914
$862$	−27.2836	−0.929283
$863$	2.08780	0.0710696	0.0355348	−	0.999368i	$-0.488687\pi$
$863$	2.08780	0.0710696	0.0355348	+	0.999368i	$0.488687\pi$
$864$	−22.0553	−0.750336
$865$	1.63396	0.0555564
$866$	23.9157	0.812689
$867$	31.8396	1.08133
$868$	−61.1944	−2.07707
$869$	−36.5206	−1.23888
$870$	16.5059	0.559603
$871$	28.6125	0.969496
$872$	14.5570	0.492964
$873$	−0.956838	−0.0323840
$874$	−10.9633	−0.370839
$875$	−4.72699	−0.159801
$876$	31.9283	1.07876
$877$	−49.8979	−1.68493	−0.842467	−	0.538748i	$-0.818898\pi$
$877$	−49.8979	−1.68493	−0.842467	+	0.538748i	$0.818898\pi$
$878$	22.3944	0.755773
$879$	25.8548	0.872062
$880$	22.4811	0.757836
$881$	35.1975	1.18583	0.592917	−	0.805264i	$-0.297976\pi$
$881$	35.1975	1.18583	0.592917	+	0.805264i	$0.297976\pi$
$882$	−40.0278	−1.34781
$883$	−47.7239	−1.60604	−0.803019	−	0.595953i	$-0.796774\pi$
$883$	−47.7239	−1.60604	−0.803019	+	0.595953i	$0.796774\pi$
$884$	34.9783	1.17645
$885$	−15.8126	−0.531536
$886$	3.35528	0.112723
$887$	−20.4982	−0.688263	−0.344132	−	0.938921i	$-0.611827\pi$
$887$	−20.4982	−0.688263	−0.344132	+	0.938921i	$0.611827\pi$
$888$	16.5813	0.556432
$889$	−8.37138	−0.280767
$890$	−1.83770	−0.0616000
$891$	52.0336	1.74319
$892$	−4.28581	−0.143499
$893$	−11.7690	−0.393834
$894$	68.4253	2.28848
$895$	−8.89727	−0.297403
$896$	41.1853	1.37590
$897$	−41.8320	−1.39673
$898$	20.1496	0.672401
$899$	−40.1626	−1.33950
$900$	1.95487	0.0651624
$901$	46.6921	1.55554
$902$	48.4452	1.61305
$903$	50.1135	1.66767
$904$	9.44895	0.314267
$905$	−7.14852	−0.237625
$906$	−40.1750	−1.33473
$907$	16.1289	0.535551	0.267775	−	0.963481i	$-0.413712\pi$
$907$	16.1289	0.535551	0.267775	+	0.963481i	$0.413712\pi$
$908$	−4.91862	−0.163230
$909$	19.0790	0.632811
$910$	−38.9150	−1.29002
$911$	−5.86877	−0.194441	−0.0972204	−	0.995263i	$-0.530995\pi$
$911$	−5.86877	−0.194441	−0.0972204	+	0.995263i	$0.530995\pi$
$912$	−13.7158	−0.454177
$913$	71.2138	2.35683
$914$	−7.61759	−0.251968
$915$	−14.6026	−0.482746
$916$	8.25384	0.272715
$917$	34.4485	1.13759
$918$	−34.6192	−1.14260
$919$	−52.0408	−1.71667	−0.858333	−	0.513093i	$-0.828500\pi$
$919$	−52.0408	−1.71667	−0.858333	+	0.513093i	$0.828500\pi$
$920$	5.08417	0.167620
$921$	−54.1052	−1.78283
$922$	65.5014	2.15717
$923$	−26.2059	−0.862578
$924$	63.3338	2.08353
$925$	6.89093	0.226572
$926$	5.67784	0.186585
$927$	−16.1857	−0.531609
$928$	−28.3602	−0.930969
$929$	−9.81663	−0.322073	−0.161037	−	0.986948i	$-0.551484\pi$
$929$	−9.81663	−0.322073	−0.161037	+	0.986948i	$0.551484\pi$
$930$	−36.3167	−1.19087
$931$	20.6087	0.675424
$932$	−6.54468	−0.214378
$933$	18.6853	0.611728
$934$	9.42886	0.308522
$935$	26.2386	0.858095
$936$	−7.27857	−0.237907
$937$	37.5704	1.22737	0.613686	−	0.789550i	$-0.289686\pi$
$937$	37.5704	1.22737	0.613686	+	0.789550i	$0.289686\pi$
$938$	−55.4829	−1.81158
$939$	25.6563	0.837261
$940$	−12.0676	−0.393603
$941$	−0.0461835	−0.00150554	−0.000752769	−	1.00000i	$-0.500240\pi$
$941$	−0.0461835	−0.00150554	−0.000752769		1.00000i	$0.500240\pi$
$942$	46.5821	1.51773
$943$	25.3023	0.823955
$944$	36.5387	1.18923
$945$	15.7060	0.510917
$946$	−42.8885	−1.39442
$947$	−26.0877	−0.847735	−0.423867	−	0.905724i	$-0.639328\pi$
$947$	−26.0877	−0.847735	−0.423867	+	0.905724i	$0.639328\pi$
$948$	22.8469	0.742031
$949$	−49.4040	−1.60372
$950$	−2.46817	−0.0800782
$951$	30.8923	1.00175
$952$	30.6760	0.994216
$953$	35.0541	1.13551	0.567757	−	0.823196i	$-0.307811\pi$
$953$	35.0541	1.13551	0.567757	+	0.823196i	$0.307811\pi$
$954$	21.4829	0.695536
$955$	−8.74055	−0.282838
$956$	−21.9217	−0.708997
$957$	41.5667	1.34366
$958$	−12.4329	−0.401688
$959$	−3.56852	−0.115234
$960$	−5.21992	−0.168472
$961$	57.3665	1.85053
$962$	56.7296	1.82904
$963$	−12.0931	−0.389693
$964$	17.0389	0.548788
$965$	3.38192	0.108868
$966$	81.1171	2.60990
$967$	−14.8997	−0.479141	−0.239570	−	0.970879i	$-0.577007\pi$
$967$	−14.8997	−0.479141	−0.239570	+	0.970879i	$0.577007\pi$
$968$	11.9236	0.383238
$969$	−16.0083	−0.514262
$970$	1.23874	0.0397734
$971$	−9.37217	−0.300767	−0.150384	−	0.988628i	$-0.548051\pi$
$971$	−9.37217	−0.300767	−0.150384	+	0.988628i	$0.548051\pi$
$972$	−18.8242	−0.603788
$973$	29.7759	0.954572
$974$	1.37396	0.0440244
$975$	−9.41765	−0.301606
$976$	33.7426	1.08007
$977$	−10.5242	−0.336698	−0.168349	−	0.985727i	$-0.553844\pi$
$977$	−10.5242	−0.336698	−0.168349	+	0.985727i	$0.553844\pi$
$978$	−26.1398	−0.835860
$979$	−4.62787	−0.147907
$980$	21.1317	0.675027
$981$	18.0532	0.576393
$982$	−40.6335	−1.29667
$983$	−1.32593	−0.0422905	−0.0211453	−	0.999776i	$-0.506731\pi$
$983$	−1.32593	−0.0422905	−0.0211453	+	0.999776i	$0.506731\pi$
$984$	13.7068	0.436955
$985$	25.4934	0.812287
$986$	−44.5157	−1.41767
$987$	87.0783	2.77173
$988$	−8.28590	−0.263610
$989$	−22.4001	−0.712280
$990$	12.0724	0.383684
$991$	−3.04920	−0.0968609	−0.0484305	−	0.998827i	$-0.515422\pi$
$991$	−3.04920	−0.0968609	−0.0484305	+	0.998827i	$0.515422\pi$
$992$	62.3986	1.98116
$993$	−38.7524	−1.22977
$994$	50.8163	1.61180
$995$	−7.06669	−0.224029
$996$	−44.5505	−1.41164
$997$	−25.7222	−0.814630	−0.407315	−	0.913288i	$-0.633535\pi$
$997$	−25.7222	−0.814630	−0.407315	+	0.913288i	$0.633535\pi$
$998$	64.5815	2.04429
$999$	−22.8960	−0.724397

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	445.2.a.f.1.3	✓	7
3.2	odd	2		4005.2.a.o.1.5		7
4.3	odd	2		7120.2.a.bj.1.1		7
5.4	even	2		2225.2.a.k.1.5		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
445.2.a.f.1.3	✓	7	1.1	even	1	trivial
2225.2.a.k.1.5		7	5.4	even	2
4005.2.a.o.1.5		7	3.2	odd	2
7120.2.a.bj.1.1		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)

\(f(q)\)	\(=\)	\(q-1.83770 q^{2} +2.10226 q^{3} +1.37716 q^{4} +1.00000 q^{5} -3.86333 q^{6} -4.72699 q^{7} +1.14460 q^{8} +1.41950 q^{9} +O(q^{10})\)	\(q-1.83770 q^{2} +2.10226 q^{3} +1.37716 q^{4} +1.00000 q^{5} -3.86333 q^{6} -4.72699 q^{7} +1.14460 q^{8} +1.41950 q^{9} -1.83770 q^{10} -4.62787 q^{11} +2.89514 q^{12} -4.47978 q^{13} +8.68681 q^{14} +2.10226 q^{15} -4.85775 q^{16} -5.66969 q^{17} -2.60862 q^{18} +1.34308 q^{19} +1.37716 q^{20} -9.93737 q^{21} +8.50466 q^{22} +4.44187 q^{23} +2.40625 q^{24} +1.00000 q^{25} +8.23250 q^{26} -3.32263 q^{27} -6.50981 q^{28} -4.27246 q^{29} -3.86333 q^{30} +9.40035 q^{31} +6.63791 q^{32} -9.72899 q^{33} +10.4192 q^{34} -4.72699 q^{35} +1.95487 q^{36} +6.89093 q^{37} -2.46817 q^{38} -9.41765 q^{39} +1.14460 q^{40} +5.69631 q^{41} +18.2619 q^{42} -5.04294 q^{43} -6.37330 q^{44} +1.41950 q^{45} -8.16284 q^{46} -8.76271 q^{47} -10.2123 q^{48} +15.3444 q^{49} -1.83770 q^{50} -11.9192 q^{51} -6.16935 q^{52} -8.23538 q^{53} +6.10600 q^{54} -4.62787 q^{55} -5.41053 q^{56} +2.82349 q^{57} +7.85151 q^{58} -7.52173 q^{59} +2.89514 q^{60} -6.94612 q^{61} -17.2751 q^{62} -6.70996 q^{63} -2.48300 q^{64} -4.47978 q^{65} +17.8790 q^{66} -6.38703 q^{67} -7.80805 q^{68} +9.33796 q^{69} +8.68681 q^{70} +5.84983 q^{71} +1.62476 q^{72} +11.0282 q^{73} -12.6635 q^{74} +2.10226 q^{75} +1.84962 q^{76} +21.8759 q^{77} +17.3069 q^{78} +7.89145 q^{79} -4.85775 q^{80} -11.2435 q^{81} -10.4681 q^{82} -15.3880 q^{83} -13.6853 q^{84} -5.66969 q^{85} +9.26742 q^{86} -8.98182 q^{87} -5.29707 q^{88} +1.00000 q^{89} -2.60862 q^{90} +21.1759 q^{91} +6.11715 q^{92} +19.7620 q^{93} +16.1033 q^{94} +1.34308 q^{95} +13.9546 q^{96} -0.674067 q^{97} -28.1985 q^{98} -6.56926 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 4 q^{2} - 8 q^{3} + 8 q^{4} + 7 q^{5} - 2 q^{6} - 16 q^{7} - 12 q^{8} + 11 q^{9}+O(q^{10}) \) 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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