LMFDB - Embedded newform 8045.2.a.c.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8045,2,Mod(1,8045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8045, 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("8045.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8045 = 5 \cdot 1609 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8045.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:	$64.2396484261$
Analytic rank:	$1$
Dimension:	$127$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Character	$\chi$	$=$	8045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.71791 q^{2} -0.717872 q^{3} +5.38703 q^{4} +1.00000 q^{5} +1.95111 q^{6} -2.87590 q^{7} -9.20565 q^{8} -2.48466 q^{9} +O(q^{10})$	\(q-2.71791 q^{2} -0.717872 q^{3} +5.38703 q^{4} +1.00000 q^{5} +1.95111 q^{6} -2.87590 q^{7} -9.20565 q^{8} -2.48466 q^{9} -2.71791 q^{10} +1.47572 q^{11} -3.86720 q^{12} -3.21810 q^{13} +7.81643 q^{14} -0.717872 q^{15} +14.2461 q^{16} -5.13134 q^{17} +6.75308 q^{18} +5.27011 q^{19} +5.38703 q^{20} +2.06453 q^{21} -4.01087 q^{22} +6.64876 q^{23} +6.60848 q^{24} +1.00000 q^{25} +8.74652 q^{26} +3.93728 q^{27} -15.4926 q^{28} -5.49479 q^{29} +1.95111 q^{30} +3.28570 q^{31} -20.3082 q^{32} -1.05938 q^{33} +13.9465 q^{34} -2.87590 q^{35} -13.3849 q^{36} -4.17425 q^{37} -14.3237 q^{38} +2.31019 q^{39} -9.20565 q^{40} +5.05368 q^{41} -5.61120 q^{42} -7.61426 q^{43} +7.94974 q^{44} -2.48466 q^{45} -18.0707 q^{46} +0.565484 q^{47} -10.2268 q^{48} +1.27080 q^{49} -2.71791 q^{50} +3.68364 q^{51} -17.3360 q^{52} -2.83733 q^{53} -10.7012 q^{54} +1.47572 q^{55} +26.4745 q^{56} -3.78326 q^{57} +14.9344 q^{58} +4.93372 q^{59} -3.86720 q^{60} -6.48798 q^{61} -8.93024 q^{62} +7.14563 q^{63} +26.7038 q^{64} -3.21810 q^{65} +2.87929 q^{66} +13.5180 q^{67} -27.6427 q^{68} -4.77296 q^{69} +7.81643 q^{70} +0.0585361 q^{71} +22.8729 q^{72} -0.489824 q^{73} +11.3452 q^{74} -0.717872 q^{75} +28.3902 q^{76} -4.24402 q^{77} -6.27888 q^{78} -7.78877 q^{79} +14.2461 q^{80} +4.62752 q^{81} -13.7354 q^{82} +15.0802 q^{83} +11.1217 q^{84} -5.13134 q^{85} +20.6949 q^{86} +3.94456 q^{87} -13.5849 q^{88} +15.9426 q^{89} +6.75308 q^{90} +9.25494 q^{91} +35.8171 q^{92} -2.35871 q^{93} -1.53694 q^{94} +5.27011 q^{95} +14.5787 q^{96} -6.62362 q^{97} -3.45391 q^{98} -3.66666 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 127 q - 20 q^{2} - 31 q^{3} + 116 q^{4} + 127 q^{5} - 17 q^{6} - 63 q^{7} - 57 q^{8} + 122 q^{9}+O(q^{10}) $ 127 * q - 20 * q^2 - 31 * q^3 + 116 * q^4 + 127 * q^5 - 17 * q^6 - 63 * q^7 - 57 * q^8 + 122 * q^9	\( 127 q - 20 q^{2} - 31 q^{3} + 116 q^{4} + 127 q^{5} - 17 q^{6} - 63 q^{7} - 57 q^{8} + 122 q^{9} - 20 q^{10} - 32 q^{11} - 65 q^{12} - 49 q^{13} - 4 q^{14} - 31 q^{15} + 98 q^{16} - 53 q^{17} - 60 q^{18} - 126 q^{19} + 116 q^{20} - 24 q^{21} - 46 q^{22} - 121 q^{23} - 51 q^{24} + 127 q^{25} - 9 q^{26} - 112 q^{27} - 123 q^{28} - 6 q^{29} - 17 q^{30} - 54 q^{31} - 119 q^{32} - 57 q^{33} - 53 q^{34} - 63 q^{35} + 133 q^{36} - 61 q^{37} - 27 q^{38} - 33 q^{39} - 57 q^{40} - 8 q^{41} - 46 q^{42} - 208 q^{43} - 54 q^{44} + 122 q^{45} - 34 q^{46} - 116 q^{47} - 106 q^{48} + 110 q^{49} - 20 q^{50} - 54 q^{51} - 142 q^{52} - 60 q^{53} - 62 q^{54} - 32 q^{55} + 33 q^{56} - 56 q^{57} - 87 q^{58} - 53 q^{59} - 65 q^{60} - 76 q^{61} - 84 q^{62} - 215 q^{63} + 67 q^{64} - 49 q^{65} + 9 q^{66} - 145 q^{67} - 133 q^{68} + q^{69} - 4 q^{70} - 4 q^{71} - 167 q^{72} - 155 q^{73} - 14 q^{74} - 31 q^{75} - 199 q^{76} - 97 q^{77} - 24 q^{78} - 73 q^{79} + 98 q^{80} + 127 q^{81} - 69 q^{82} - 225 q^{83} - 59 q^{84} - 53 q^{85} + 30 q^{86} - 179 q^{87} - 119 q^{88} - 25 q^{89} - 60 q^{90} - 160 q^{91} - 188 q^{92} - 44 q^{93} - 32 q^{94} - 126 q^{95} - 43 q^{96} - 72 q^{97} - 111 q^{98} - 141 q^{99}+O(q^{100}) \) 127 * q - 20 * q^2 - 31 * q^3 + 116 * q^4 + 127 * q^5 - 17 * q^6 - 63 * q^7 - 57 * q^8 + 122 * q^9 - 20 * q^10 - 32 * q^11 - 65 * q^12 - 49 * q^13 - 4 * q^14 - 31 * q^15 + 98 * q^16 - 53 * q^17 - 60 * q^18 - 126 * q^19 + 116 * q^20 - 24 * q^21 - 46 * q^22 - 121 * q^23 - 51 * q^24 + 127 * q^25 - 9 * q^26 - 112 * q^27 - 123 * q^28 - 6 * q^29 - 17 * q^30 - 54 * q^31 - 119 * q^32 - 57 * q^33 - 53 * q^34 - 63 * q^35 + 133 * q^36 - 61 * q^37 - 27 * q^38 - 33 * q^39 - 57 * q^40 - 8 * q^41 - 46 * q^42 - 208 * q^43 - 54 * q^44 + 122 * q^45 - 34 * q^46 - 116 * q^47 - 106 * q^48 + 110 * q^49 - 20 * q^50 - 54 * q^51 - 142 * q^52 - 60 * q^53 - 62 * q^54 - 32 * q^55 + 33 * q^56 - 56 * q^57 - 87 * q^58 - 53 * q^59 - 65 * q^60 - 76 * q^61 - 84 * q^62 - 215 * q^63 + 67 * q^64 - 49 * q^65 + 9 * q^66 - 145 * q^67 - 133 * q^68 + q^69 - 4 * q^70 - 4 * q^71 - 167 * q^72 - 155 * q^73 - 14 * q^74 - 31 * q^75 - 199 * q^76 - 97 * q^77 - 24 * q^78 - 73 * q^79 + 98 * q^80 + 127 * q^81 - 69 * q^82 - 225 * q^83 - 59 * q^84 - 53 * q^85 + 30 * q^86 - 179 * q^87 - 119 * q^88 - 25 * q^89 - 60 * q^90 - 160 * q^91 - 188 * q^92 - 44 * q^93 - 32 * q^94 - 126 * q^95 - 43 * q^96 - 72 * q^97 - 111 * q^98 - 141 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.71791	−1.92185	−0.960926	−	0.276805i	$-0.910725\pi$
$2$	−2.71791	−1.92185	−0.960926	+	0.276805i	$0.910725\pi$
$3$	−0.717872	−0.414463	−0.207232	−	0.978292i	$-0.566445\pi$
$3$	−0.717872	−0.414463	−0.207232	+	0.978292i	$0.566445\pi$
$4$	5.38703	2.69352
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	1.95111	0.796537
$7$	−2.87590	−1.08699	−0.543494	−	0.839413i	$-0.682899\pi$
$7$	−2.87590	−1.08699	−0.543494	+	0.839413i	$0.682899\pi$
$8$	−9.20565	−3.25469
$9$	−2.48466	−0.828220
$10$	−2.71791	−0.859479
$11$	1.47572	0.444946	0.222473	−	0.974939i	$-0.428587\pi$
$11$	1.47572	0.444946	0.222473	+	0.974939i	$0.428587\pi$
$12$	−3.86720	−1.11636
$13$	−3.21810	−0.892541	−0.446271	−	0.894898i	$-0.647248\pi$
$13$	−3.21810	−0.892541	−0.446271	+	0.894898i	$0.647248\pi$
$14$	7.81643	2.08903
$15$	−0.717872	−0.185354
$16$	14.2461	3.56152
$17$	−5.13134	−1.24453	−0.622266	−	0.782806i	$-0.713788\pi$
$17$	−5.13134	−1.24453	−0.622266	+	0.782806i	$0.713788\pi$
$18$	6.75308	1.59172
$19$	5.27011	1.20905	0.604523	−	0.796588i	$-0.293364\pi$
$19$	5.27011	1.20905	0.604523	+	0.796588i	$0.293364\pi$
$20$	5.38703	1.20458
$21$	2.06453	0.450517
$22$	−4.01087	−0.855120
$23$	6.64876	1.38636	0.693181	−	0.720763i	$-0.256209\pi$
$23$	6.64876	1.38636	0.693181	+	0.720763i	$0.256209\pi$
$24$	6.60848	1.34895
$25$	1.00000	0.200000
$26$	8.74652	1.71533
$27$	3.93728	0.757730
$28$	−15.4926	−2.92782
$29$	−5.49479	−1.02036	−0.510179	−	0.860068i	$-0.670421\pi$
$29$	−5.49479	−1.02036	−0.510179	+	0.860068i	$0.670421\pi$
$30$	1.95111	0.356222
$31$	3.28570	0.590130	0.295065	−	0.955477i	$-0.404659\pi$
$31$	3.28570	0.590130	0.295065	+	0.955477i	$0.404659\pi$
$32$	−20.3082	−3.59002
$33$	−1.05938	−0.184414
$34$	13.9465	2.39181
$35$	−2.87590	−0.486116
$36$	−13.3849	−2.23082
$37$	−4.17425	−0.686243	−0.343121	−	0.939291i	$-0.611484\pi$
$37$	−4.17425	−0.686243	−0.343121	+	0.939291i	$0.611484\pi$
$38$	−14.3237	−2.32361
$39$	2.31019	0.369926
$40$	−9.20565	−1.45554
$41$	5.05368	0.789252	0.394626	−	0.918842i	$-0.370874\pi$
$41$	5.05368	0.789252	0.394626	+	0.918842i	$0.370874\pi$
$42$	−5.61120	−0.865826
$43$	−7.61426	−1.16116	−0.580582	−	0.814202i	$-0.697175\pi$
$43$	−7.61426	−1.16116	−0.580582	+	0.814202i	$0.697175\pi$
$44$	7.94974	1.19847
$45$	−2.48466	−0.370391
$46$	−18.0707	−2.66438
$47$	0.565484	0.0824844	0.0412422	−	0.999149i	$-0.486868\pi$
$47$	0.565484	0.0824844	0.0412422	+	0.999149i	$0.486868\pi$
$48$	−10.2268	−1.47612
$49$	1.27080	0.181542
$50$	−2.71791	−0.384370
$51$	3.68364	0.515813
$52$	−17.3360	−2.40408
$53$	−2.83733	−0.389738	−0.194869	−	0.980829i	$-0.562428\pi$
$53$	−2.83733	−0.389738	−0.194869	+	0.980829i	$0.562428\pi$
$54$	−10.7012	−1.45625
$55$	1.47572	0.198986
$56$	26.4745	3.53781
$57$	−3.78326	−0.501105
$58$	14.9344	1.96098
$59$	4.93372	0.642316	0.321158	−	0.947026i	$-0.395928\pi$
$59$	4.93372	0.642316	0.321158	+	0.947026i	$0.395928\pi$
$60$	−3.86720	−0.499253
$61$	−6.48798	−0.830700	−0.415350	−	0.909662i	$-0.636341\pi$
$61$	−6.48798	−0.830700	−0.415350	+	0.909662i	$0.636341\pi$
$62$	−8.93024	−1.13414
$63$	7.14563	0.900265
$64$	26.7038	3.33797
$65$	−3.21810	−0.399157
$66$	2.87929	0.354416
$67$	13.5180	1.65149	0.825744	−	0.564045i	$-0.190756\pi$
$67$	13.5180	1.65149	0.825744	+	0.564045i	$0.190756\pi$
$68$	−27.6427	−3.35217
$69$	−4.77296	−0.574596
$70$	7.81643	0.934243
$71$	0.0585361	0.00694696	0.00347348	−	0.999994i	$-0.498894\pi$
$71$	0.0585361	0.00694696	0.00347348	+	0.999994i	$0.498894\pi$
$72$	22.8729	2.69560
$73$	−0.489824	−0.0573296	−0.0286648	−	0.999589i	$-0.509126\pi$
$73$	−0.489824	−0.0573296	−0.0286648	+	0.999589i	$0.509126\pi$
$74$	11.3452	1.31886
$75$	−0.717872	−0.0828927
$76$	28.3902	3.25658
$77$	−4.24402	−0.483650
$78$	−6.27888	−0.710943
$79$	−7.78877	−0.876305	−0.438153	−	0.898901i	$-0.644367\pi$
$79$	−7.78877	−0.876305	−0.438153	+	0.898901i	$0.644367\pi$
$80$	14.2461	1.59276
$81$	4.62752	0.514169
$82$	−13.7354	−1.51683
$83$	15.0802	1.65526	0.827632	−	0.561271i	$-0.189687\pi$
$83$	15.0802	1.65526	0.827632	+	0.561271i	$0.189687\pi$
$84$	11.1217	1.21347
$85$	−5.13134	−0.556571
$86$	20.6949	2.23158
$87$	3.94456	0.422901
$88$	−13.5849	−1.44816
$89$	15.9426	1.68992	0.844958	−	0.534833i	$-0.179626\pi$
$89$	15.9426	1.68992	0.844958	+	0.534833i	$0.179626\pi$
$90$	6.75308	0.711837
$91$	9.25494	0.970182
$92$	35.8171	3.73419
$93$	−2.35871	−0.244587
$94$	−1.53694	−0.158523
$95$	5.27011	0.540702
$96$	14.5787	1.48793
$97$	−6.62362	−0.672527	−0.336263	−	0.941768i	$-0.609163\pi$
$97$	−6.62362	−0.672527	−0.336263	+	0.941768i	$0.609163\pi$
$98$	−3.45391	−0.348898
$99$	−3.66666	−0.368513
$100$	5.38703	0.538703
$101$	−11.1182	−1.10630	−0.553151	−	0.833081i	$-0.686575\pi$
$101$	−11.1182	−1.10630	−0.553151	+	0.833081i	$0.686575\pi$
$102$	−10.0118	−0.991316
$103$	−1.74232	−0.171676	−0.0858379	−	0.996309i	$-0.527357\pi$
$103$	−1.74232	−0.171676	−0.0858379	+	0.996309i	$0.527357\pi$
$104$	29.6247	2.90495
$105$	2.06453	0.201477
$106$	7.71161	0.749018
$107$	0.356426	0.0344570	0.0172285	−	0.999852i	$-0.494516\pi$
$107$	0.356426	0.0344570	0.0172285	+	0.999852i	$0.494516\pi$
$108$	21.2103	2.04096
$109$	11.2425	1.07683	0.538417	−	0.842679i	$-0.319023\pi$
$109$	11.2425	1.07683	0.538417	+	0.842679i	$0.319023\pi$
$110$	−4.01087	−0.382421
$111$	2.99658	0.284423
$112$	−40.9702	−3.87132
$113$	−5.53906	−0.521071	−0.260535	−	0.965464i	$-0.583899\pi$
$113$	−5.53906	−0.521071	−0.260535	+	0.965464i	$0.583899\pi$
$114$	10.2826	0.963050
$115$	6.64876	0.620000
$116$	−29.6006	−2.74835
$117$	7.99590	0.739221
$118$	−13.4094	−1.23444
$119$	14.7572	1.35279
$120$	6.60848	0.603269
$121$	−8.82226	−0.802023
$122$	17.6337	1.59648
$123$	−3.62789	−0.327116
$124$	17.7002	1.58952
$125$	1.00000	0.0894427
$126$	−19.4212	−1.73018
$127$	20.1623	1.78912	0.894558	−	0.446953i	$-0.147491\pi$
$127$	20.1623	1.78912	0.894558	+	0.446953i	$0.147491\pi$
$128$	−31.9620	−2.82507
$129$	5.46606	0.481260
$130$	8.74652	0.767120
$131$	−20.8585	−1.82242	−0.911209	−	0.411944i	$-0.864850\pi$
$131$	−20.8585	−1.82242	−0.911209	+	0.411944i	$0.864850\pi$
$132$	−5.70689	−0.496721
$133$	−15.1563	−1.31422
$134$	−36.7407	−3.17392
$135$	3.93728	0.338867
$136$	47.2373	4.05056
$137$	21.5229	1.83883	0.919413	−	0.393294i	$-0.128665\pi$
$137$	21.5229	1.83883	0.919413	+	0.393294i	$0.128665\pi$
$138$	12.9725	1.10429
$139$	20.2179	1.71486	0.857431	−	0.514598i	$-0.172059\pi$
$139$	20.2179	1.71486	0.857431	+	0.514598i	$0.172059\pi$
$140$	−15.4926	−1.30936
$141$	−0.405945	−0.0341867
$142$	−0.159096	−0.0133510
$143$	−4.74901	−0.397132
$144$	−35.3966	−2.94972
$145$	−5.49479	−0.456318
$146$	1.33130	0.110179
$147$	−0.912269	−0.0752427
$148$	−22.4868	−1.84841
$149$	−1.75329	−0.143635	−0.0718177	−	0.997418i	$-0.522880\pi$
$149$	−1.75329	−0.143635	−0.0718177	+	0.997418i	$0.522880\pi$
$150$	1.95111	0.159307
$151$	8.99962	0.732379	0.366189	−	0.930540i	$-0.380662\pi$
$151$	8.99962	0.732379	0.366189	+	0.930540i	$0.380662\pi$
$152$	−48.5148	−3.93507
$153$	12.7496	1.03075
$154$	11.5349	0.929505
$155$	3.28570	0.263914
$156$	12.4450	0.996401
$157$	−18.8994	−1.50834	−0.754168	−	0.656681i	$-0.771960\pi$
$157$	−18.8994	−1.50834	−0.754168	+	0.656681i	$0.771960\pi$
$158$	21.1692	1.68413
$159$	2.03684	0.161532
$160$	−20.3082	−1.60551
$161$	−19.1212	−1.50696
$162$	−12.5772	−0.988156
$163$	7.33502	0.574523	0.287262	−	0.957852i	$-0.407255\pi$
$163$	7.33502	0.574523	0.287262	+	0.957852i	$0.407255\pi$
$164$	27.2243	2.12586
$165$	−1.05938	−0.0824723
$166$	−40.9866	−3.18117
$167$	−3.71045	−0.287124	−0.143562	−	0.989641i	$-0.545856\pi$
$167$	−3.71045	−0.287124	−0.143562	+	0.989641i	$0.545856\pi$
$168$	−19.0053	−1.46629
$169$	−2.64381	−0.203370
$170$	13.9465	1.06965
$171$	−13.0944	−1.00136
$172$	−41.0183	−3.12761
$173$	10.1159	0.769100	0.384550	−	0.923104i	$-0.374357\pi$
$173$	10.1159	0.769100	0.384550	+	0.923104i	$0.374357\pi$
$174$	−10.7209	−0.812753
$175$	−2.87590	−0.217398
$176$	21.0232	1.58468
$177$	−3.54178	−0.266216
$178$	−43.3306	−3.24777
$179$	−17.8431	−1.33366	−0.666829	−	0.745211i	$-0.732349\pi$
$179$	−17.8431	−1.33366	−0.666829	+	0.745211i	$0.732349\pi$
$180$	−13.3849	−0.997655
$181$	−15.0417	−1.11804	−0.559021	−	0.829154i	$-0.688823\pi$
$181$	−15.0417	−1.11804	−0.559021	+	0.829154i	$0.688823\pi$
$182$	−25.1541	−1.86455
$183$	4.65753	0.344295
$184$	−61.2062	−4.51218
$185$	−4.17425	−0.306897
$186$	6.41077	0.470060
$187$	−7.57240	−0.553749
$188$	3.04628	0.222173
$189$	−11.3232	−0.823643
$190$	−14.3237	−1.03915
$191$	22.4121	1.62168	0.810842	−	0.585265i	$-0.199009\pi$
$191$	22.4121	1.62168	0.810842	+	0.585265i	$0.199009\pi$
$192$	−19.1699	−1.38347
$193$	15.1520	1.09066	0.545331	−	0.838220i	$-0.316404\pi$
$193$	15.1520	1.09066	0.545331	+	0.838220i	$0.316404\pi$
$194$	18.0024	1.29250
$195$	2.31019	0.165436
$196$	6.84582	0.488987
$197$	7.37401	0.525376	0.262688	−	0.964881i	$-0.415391\pi$
$197$	7.37401	0.525376	0.262688	+	0.964881i	$0.415391\pi$
$198$	9.96564	0.708228
$199$	14.4167	1.02197	0.510985	−	0.859589i	$-0.329281\pi$
$199$	14.4167	1.02197	0.510985	+	0.859589i	$0.329281\pi$
$200$	−9.20565	−0.650938
$201$	−9.70420	−0.684481
$202$	30.2182	2.12615
$203$	15.8025	1.10912
$204$	19.8439	1.38935
$205$	5.05368	0.352964
$206$	4.73547	0.329936
$207$	−16.5199	−1.14821
$208$	−45.8453	−3.17880
$209$	7.77719	0.537960
$210$	−5.61120	−0.387209
$211$	−18.5750	−1.27875	−0.639377	−	0.768893i	$-0.720808\pi$
$211$	−18.5750	−1.27875	−0.639377	+	0.768893i	$0.720808\pi$
$212$	−15.2848	−1.04976
$213$	−0.0420214	−0.00287926
$214$	−0.968733	−0.0662212
$215$	−7.61426	−0.519288
$216$	−36.2452	−2.46618
$217$	−9.44935	−0.641464
$218$	−30.5560	−2.06951
$219$	0.351631	0.0237610
$220$	7.94974	0.535971
$221$	16.5132	1.11080
$222$	−8.14443	−0.546618
$223$	−2.51190	−0.168209	−0.0841045	−	0.996457i	$-0.526803\pi$
$223$	−2.51190	−0.168209	−0.0841045	+	0.996457i	$0.526803\pi$
$224$	58.4044	3.90231
$225$	−2.48466	−0.165644
$226$	15.0547	1.00142
$227$	−17.6943	−1.17441	−0.587205	−	0.809438i	$-0.699772\pi$
$227$	−17.6943	−1.17441	−0.587205	+	0.809438i	$0.699772\pi$
$228$	−20.3806	−1.34973
$229$	−12.0879	−0.798793	−0.399397	−	0.916778i	$-0.630780\pi$
$229$	−12.0879	−0.798793	−0.399397	+	0.916778i	$0.630780\pi$
$230$	−18.0707	−1.19155
$231$	3.04666	0.200455
$232$	50.5832	3.32095
$233$	−5.75667	−0.377132	−0.188566	−	0.982060i	$-0.560384\pi$
$233$	−5.75667	−0.377132	−0.188566	+	0.982060i	$0.560384\pi$
$234$	−21.7321	−1.42067
$235$	0.565484	0.0368881
$236$	26.5781	1.73009
$237$	5.59134	0.363196
$238$	−40.1087	−2.59986
$239$	3.12360	0.202049	0.101024	−	0.994884i	$-0.467788\pi$
$239$	3.12360	0.202049	0.101024	+	0.994884i	$0.467788\pi$
$240$	−10.2268	−0.660140
$241$	−8.82057	−0.568183	−0.284091	−	0.958797i	$-0.591692\pi$
$241$	−8.82057	−0.568183	−0.284091	+	0.958797i	$0.591692\pi$
$242$	23.9781	1.54137
$243$	−15.1338	−0.970834
$244$	−34.9509	−2.23751
$245$	1.27080	0.0811882
$246$	9.86028	0.628669
$247$	−16.9598	−1.07912
$248$	−30.2470	−1.92069
$249$	−10.8256	−0.686046
$250$	−2.71791	−0.171896
$251$	25.3336	1.59904	0.799522	−	0.600637i	$-0.205086\pi$
$251$	25.3336	1.59904	0.799522	+	0.600637i	$0.205086\pi$
$252$	38.4938	2.42488
$253$	9.81169	0.616856
$254$	−54.7993	−3.43842
$255$	3.68364	0.230678
$256$	33.4623	2.09139
$257$	−10.8008	−0.673736	−0.336868	−	0.941552i	$-0.609368\pi$
$257$	−10.8008	−0.673736	−0.336868	+	0.941552i	$0.609368\pi$
$258$	−14.8563	−0.924910
$259$	12.0047	0.745938
$260$	−17.3360	−1.07514
$261$	13.6527	0.845081
$262$	56.6916	3.50242
$263$	10.2746	0.633558	0.316779	−	0.948499i	$-0.397399\pi$
$263$	10.2746	0.633558	0.316779	+	0.948499i	$0.397399\pi$
$264$	9.75225	0.600209
$265$	−2.83733	−0.174296
$266$	41.1935	2.52573
$267$	−11.4448	−0.700408
$268$	72.8220	4.44831
$269$	−2.14882	−0.131016	−0.0655081	−	0.997852i	$-0.520867\pi$
$269$	−2.14882	−0.131016	−0.0655081	+	0.997852i	$0.520867\pi$
$270$	−10.7012	−0.651253
$271$	−17.2528	−1.04803	−0.524017	−	0.851708i	$-0.675567\pi$
$271$	−17.2528	−1.04803	−0.524017	+	0.851708i	$0.675567\pi$
$272$	−73.1013	−4.43242
$273$	−6.64386	−0.402105
$274$	−58.4973	−3.53395
$275$	1.47572	0.0889891
$276$	−25.7121	−1.54768
$277$	0.307079	0.0184506	0.00922529	−	0.999957i	$-0.497063\pi$
$277$	0.307079	0.0184506	0.00922529	+	0.999957i	$0.497063\pi$
$278$	−54.9505	−3.29571
$279$	−8.16385	−0.488757
$280$	26.4745	1.58216
$281$	22.9010	1.36616	0.683079	−	0.730344i	$-0.260640\pi$
$281$	22.9010	1.36616	0.683079	+	0.730344i	$0.260640\pi$
$282$	1.10332	0.0657019
$283$	−3.09720	−0.184110	−0.0920548	−	0.995754i	$-0.529344\pi$
$283$	−3.09720	−0.184110	−0.0920548	+	0.995754i	$0.529344\pi$
$284$	0.315336	0.0187118
$285$	−3.78326	−0.224101
$286$	12.9074	0.763230
$287$	−14.5339	−0.857907
$288$	50.4590	2.97333
$289$	9.33060	0.548859
$290$	14.9344	0.876975
$291$	4.75491	0.278738
$292$	−2.63870	−0.154418
$293$	20.8642	1.21890	0.609451	−	0.792824i	$-0.291390\pi$
$293$	20.8642	1.21890	0.609451	+	0.792824i	$0.291390\pi$
$294$	2.47946	0.144605
$295$	4.93372	0.287252
$296$	38.4267	2.23351
$297$	5.81032	0.337149
$298$	4.76529	0.276046
$299$	−21.3964	−1.23739
$300$	−3.86720	−0.223273
$301$	21.8978	1.26217
$302$	−24.4602	−1.40752
$303$	7.98143	0.458521
$304$	75.0783	4.30604
$305$	−6.48798	−0.371500
$306$	−34.6523	−1.98094
$307$	−33.0332	−1.88530	−0.942651	−	0.333779i	$-0.891676\pi$
$307$	−33.0332	−1.88530	−0.942651	+	0.333779i	$0.891676\pi$
$308$	−22.8627	−1.30272
$309$	1.25076	0.0711533
$310$	−8.93024	−0.507204
$311$	29.8148	1.69064	0.845321	−	0.534259i	$-0.179409\pi$
$311$	29.8148	1.69064	0.845321	+	0.534259i	$0.179409\pi$
$312$	−21.2668	−1.20399
$313$	7.21110	0.407595	0.203798	−	0.979013i	$-0.434672\pi$
$313$	7.21110	0.407595	0.203798	+	0.979013i	$0.434672\pi$
$314$	51.3669	2.89880
$315$	7.14563	0.402611
$316$	−41.9584	−2.36034
$317$	−12.2535	−0.688224	−0.344112	−	0.938929i	$-0.611820\pi$
$317$	−12.2535	−0.688224	−0.344112	+	0.938929i	$0.611820\pi$
$318$	−5.53595	−0.310441
$319$	−8.10876	−0.454004
$320$	26.7038	1.49279
$321$	−0.255868	−0.0142812
$322$	51.9696	2.89615
$323$	−27.0427	−1.50470
$324$	24.9286	1.38492
$325$	−3.21810	−0.178508
$326$	−19.9359	−1.10415
$327$	−8.07065	−0.446308
$328$	−46.5224	−2.56877
$329$	−1.62628	−0.0896595
$330$	2.87929	0.158500
$331$	−22.5526	−1.23961	−0.619803	−	0.784758i	$-0.712787\pi$
$331$	−22.5526	−1.23961	−0.619803	+	0.784758i	$0.712787\pi$
$332$	81.2374	4.45848
$333$	10.3716	0.568360
$334$	10.0847	0.551809
$335$	13.5180	0.738568
$336$	29.4114	1.60452
$337$	−34.8793	−1.89999	−0.949997	−	0.312259i	$-0.898914\pi$
$337$	−34.8793	−1.89999	−0.949997	+	0.312259i	$0.898914\pi$
$338$	7.18563	0.390847
$339$	3.97633	0.215965
$340$	−27.6427	−1.49913
$341$	4.84877	0.262576
$342$	35.5895	1.92446
$343$	16.4766	0.889653
$344$	70.0942	3.77923
$345$	−4.77296	−0.256967
$346$	−27.4942	−1.47810
$347$	−17.0771	−0.916746	−0.458373	−	0.888760i	$-0.651568\pi$
$347$	−17.0771	−0.916746	−0.458373	+	0.888760i	$0.651568\pi$
$348$	21.2495	1.13909
$349$	1.70279	0.0911481	0.0455741	−	0.998961i	$-0.485488\pi$
$349$	1.70279	0.0911481	0.0455741	+	0.998961i	$0.485488\pi$
$350$	7.81643	0.417806
$351$	−12.6706	−0.676306
$352$	−29.9692	−1.59736
$353$	−24.7763	−1.31871	−0.659354	−	0.751833i	$-0.729170\pi$
$353$	−24.7763	−1.31871	−0.659354	+	0.751833i	$0.729170\pi$
$354$	9.62623	0.511628
$355$	0.0585361	0.00310677
$356$	85.8835	4.55182
$357$	−10.5938	−0.560682
$358$	48.4960	2.56309
$359$	−1.35213	−0.0713628	−0.0356814	−	0.999363i	$-0.511360\pi$
$359$	−1.35213	−0.0713628	−0.0356814	+	0.999363i	$0.511360\pi$
$360$	22.8729	1.20551
$361$	8.77404	0.461791
$362$	40.8820	2.14871
$363$	6.33325	0.332409
$364$	49.8567	2.61320
$365$	−0.489824	−0.0256386
$366$	−12.6588	−0.661684
$367$	30.7969	1.60759	0.803793	−	0.594909i	$-0.202812\pi$
$367$	30.7969	1.60759	0.803793	+	0.594909i	$0.202812\pi$
$368$	94.7187	4.93755
$369$	−12.5567	−0.653674
$370$	11.3452	0.589811
$371$	8.15988	0.423640
$372$	−12.7065	−0.658799
$373$	9.82560	0.508750	0.254375	−	0.967106i	$-0.418130\pi$
$373$	9.82560	0.508750	0.254375	+	0.967106i	$0.418130\pi$
$374$	20.5811	1.06422
$375$	−0.717872	−0.0370707
$376$	−5.20565	−0.268461
$377$	17.6828	0.910711
$378$	30.7755	1.58292
$379$	−29.2757	−1.50379	−0.751896	−	0.659281i	$-0.770861\pi$
$379$	−29.2757	−1.50379	−0.751896	+	0.659281i	$0.770861\pi$
$380$	28.3902	1.45639
$381$	−14.4739	−0.741523
$382$	−60.9141	−3.11664
$383$	4.70626	0.240479	0.120239	−	0.992745i	$-0.461634\pi$
$383$	4.70626	0.240479	0.120239	+	0.992745i	$0.461634\pi$
$384$	22.9446	1.17089
$385$	−4.24402	−0.216295
$386$	−41.1817	−2.09609
$387$	18.9188	0.961699
$388$	−35.6817	−1.81146
$389$	7.11282	0.360634	0.180317	−	0.983609i	$-0.442288\pi$
$389$	7.11282	0.360634	0.180317	+	0.983609i	$0.442288\pi$
$390$	−6.27888	−0.317943
$391$	−34.1170	−1.72537
$392$	−11.6985	−0.590864
$393$	14.9737	0.755326
$394$	−20.0419	−1.00970
$395$	−7.78877	−0.391896
$396$	−19.7524	−0.992596
$397$	1.37127	0.0688220	0.0344110	−	0.999408i	$-0.489044\pi$
$397$	1.37127	0.0688220	0.0344110	+	0.999408i	$0.489044\pi$
$398$	−39.1832	−1.96408
$399$	10.8803	0.544695
$400$	14.2461	0.712303
$401$	−4.27511	−0.213489	−0.106744	−	0.994287i	$-0.534043\pi$
$401$	−4.27511	−0.213489	−0.106744	+	0.994287i	$0.534043\pi$
$402$	26.3751	1.31547
$403$	−10.5737	−0.526715
$404$	−59.8941	−2.97984
$405$	4.62752	0.229943
$406$	−42.9497	−2.13156
$407$	−6.16002	−0.305341
$408$	−33.9103	−1.67881
$409$	−26.2346	−1.29722	−0.648610	−	0.761121i	$-0.724649\pi$
$409$	−26.2346	−1.29722	−0.648610	+	0.761121i	$0.724649\pi$
$410$	−13.7354	−0.678345
$411$	−15.4507	−0.762126
$412$	−9.38593	−0.462412
$413$	−14.1889	−0.698189
$414$	44.8996	2.20670
$415$	15.0802	0.740257
$416$	65.3539	3.20424
$417$	−14.5139	−0.710748
$418$	−21.1377	−1.03388
$419$	34.7146	1.69592	0.847960	−	0.530061i	$-0.177831\pi$
$419$	34.7146	1.69592	0.847960	+	0.530061i	$0.177831\pi$
$420$	11.1217	0.542682
$421$	−11.5980	−0.565251	−0.282626	−	0.959230i	$-0.591205\pi$
$421$	−11.5980	−0.565251	−0.282626	+	0.959230i	$0.591205\pi$
$422$	50.4851	2.45758
$423$	−1.40504	−0.0683152
$424$	26.1195	1.26847
$425$	−5.13134	−0.248906
$426$	0.114210	0.00553351
$427$	18.6588	0.902961
$428$	1.92008	0.0928104
$429$	3.40918	0.164597
$430$	20.6949	0.997995
$431$	10.0411	0.483662	0.241831	−	0.970318i	$-0.422252\pi$
$431$	10.0411	0.483662	0.241831	+	0.970318i	$0.422252\pi$
$432$	56.0908	2.69867
$433$	−13.3481	−0.641471	−0.320735	−	0.947169i	$-0.603930\pi$
$433$	−13.3481	−0.641471	−0.320735	+	0.947169i	$0.603930\pi$
$434$	25.6825	1.23280
$435$	3.94456	0.189127
$436$	60.5636	2.90047
$437$	35.0397	1.67618
$438$	−0.955701	−0.0456652
$439$	15.8810	0.757959	0.378979	−	0.925405i	$-0.376275\pi$
$439$	15.8810	0.757959	0.378979	+	0.925405i	$0.376275\pi$
$440$	−13.5849	−0.647637
$441$	−3.15750	−0.150357
$442$	−44.8813	−2.13479
$443$	8.58916	0.408083	0.204042	−	0.978962i	$-0.434592\pi$
$443$	8.58916	0.408083	0.204042	+	0.978962i	$0.434592\pi$
$444$	16.1427	0.766097
$445$	15.9426	0.755753
$446$	6.82711	0.323273
$447$	1.25864	0.0595316
$448$	−76.7973	−3.62833
$449$	0.464825	0.0219365	0.0109682	−	0.999940i	$-0.496509\pi$
$449$	0.464825	0.0219365	0.0109682	+	0.999940i	$0.496509\pi$
$450$	6.75308	0.318343
$451$	7.45780	0.351174
$452$	−29.8391	−1.40351
$453$	−6.46057	−0.303544
$454$	48.0914	2.25704
$455$	9.25494	0.433878
$456$	34.8274	1.63094
$457$	−23.3290	−1.09129	−0.545643	−	0.838018i	$-0.683714\pi$
$457$	−23.3290	−1.09129	−0.545643	+	0.838018i	$0.683714\pi$
$458$	32.8539	1.53516
$459$	−20.2035	−0.943019
$460$	35.8171	1.66998
$461$	−13.4674	−0.627239	−0.313619	−	0.949549i	$-0.601542\pi$
$461$	−13.4674	−0.627239	−0.313619	+	0.949549i	$0.601542\pi$
$462$	−8.28054	−0.385246
$463$	15.5267	0.721588	0.360794	−	0.932645i	$-0.382506\pi$
$463$	15.5267	0.721588	0.360794	+	0.932645i	$0.382506\pi$
$464$	−78.2792	−3.63402
$465$	−2.35871	−0.109383
$466$	15.6461	0.724793
$467$	−36.5298	−1.69040	−0.845200	−	0.534451i	$-0.820519\pi$
$467$	−36.5298	−1.69040	−0.845200	+	0.534451i	$0.820519\pi$
$468$	43.0742	1.99110
$469$	−38.8764	−1.79515
$470$	−1.53694	−0.0708935
$471$	13.5673	0.625150
$472$	−45.4181	−2.09054
$473$	−11.2365	−0.516655
$474$	−15.1967	−0.698010
$475$	5.27011	0.241809
$476$	79.4975	3.64376
$477$	7.04981	0.322789
$478$	−8.48965	−0.388308
$479$	35.1883	1.60779	0.803897	−	0.594769i	$-0.202756\pi$
$479$	35.1883	1.60779	0.803897	+	0.594769i	$0.202756\pi$
$480$	14.5787	0.665423
$481$	13.4332	0.612500
$482$	23.9735	1.09196
$483$	13.7265	0.624579
$484$	−47.5258	−2.16026
$485$	−6.62362	−0.300763
$486$	41.1323	1.86580
$487$	24.2275	1.09785	0.548927	−	0.835871i	$-0.315037\pi$
$487$	24.2275	1.09785	0.548927	+	0.835871i	$0.315037\pi$
$488$	59.7260	2.70367
$489$	−5.26560	−0.238119
$490$	−3.45391	−0.156032
$491$	−13.2943	−0.599962	−0.299981	−	0.953945i	$-0.596980\pi$
$491$	−13.2943	−0.599962	−0.299981	+	0.953945i	$0.596980\pi$
$492$	−19.5436	−0.881092
$493$	28.1956	1.26987
$494$	46.0951	2.07392
$495$	−3.66666	−0.164804
$496$	46.8083	2.10176
$497$	−0.168344	−0.00755126
$498$	29.4231	1.31848
$499$	7.60925	0.340637	0.170318	−	0.985389i	$-0.445520\pi$
$499$	7.60925	0.340637	0.170318	+	0.985389i	$0.445520\pi$
$500$	5.38703	0.240915
$501$	2.66363	0.119002
$502$	−68.8545	−3.07313
$503$	−27.5386	−1.22788	−0.613942	−	0.789351i	$-0.710417\pi$
$503$	−27.5386	−1.22788	−0.613942	+	0.789351i	$0.710417\pi$
$504$	−65.7802	−2.93008
$505$	−11.1182	−0.494753
$506$	−26.6673	−1.18551
$507$	1.89791	0.0842893
$508$	108.615	4.81901
$509$	19.0218	0.843128	0.421564	−	0.906799i	$-0.361481\pi$
$509$	19.0218	0.843128	0.421564	+	0.906799i	$0.361481\pi$
$510$	−10.0118	−0.443330
$511$	1.40869	0.0623166
$512$	−27.0235	−1.19428
$513$	20.7499	0.916130
$514$	29.3556	1.29482
$515$	−1.74232	−0.0767758
$516$	29.4458	1.29628
$517$	0.834495	0.0367011
$518$	−32.6278	−1.43358
$519$	−7.26194	−0.318764
$520$	29.6247	1.29913
$521$	−16.6901	−0.731205	−0.365602	−	0.930771i	$-0.619137\pi$
$521$	−16.6901	−0.731205	−0.365602	+	0.930771i	$0.619137\pi$
$522$	−37.1068	−1.62412
$523$	13.2317	0.578581	0.289291	−	0.957241i	$-0.406581\pi$
$523$	13.2317	0.578581	0.289291	+	0.957241i	$0.406581\pi$
$524$	−112.366	−4.90872
$525$	2.06453	0.0901033
$526$	−27.9254	−1.21760
$527$	−16.8600	−0.734435
$528$	−15.0919	−0.656792
$529$	21.2060	0.922000
$530$	7.71161	0.334971
$531$	−12.2586	−0.531979
$532$	−81.6475	−3.53987
$533$	−16.2633	−0.704440
$534$	31.1058	1.34608
$535$	0.356426	0.0154096
$536$	−124.442	−5.37508
$537$	12.8091	0.552752
$538$	5.84031	0.251794
$539$	1.87534	0.0807765
$540$	21.2103	0.912745
$541$	−44.8979	−1.93031	−0.965155	−	0.261680i	$-0.915724\pi$
$541$	−44.8979	−1.93031	−0.965155	+	0.261680i	$0.915724\pi$
$542$	46.8916	2.01417
$543$	10.7980	0.463387
$544$	104.208	4.46789
$545$	11.2425	0.481574
$546$	18.0574	0.772786
$547$	37.4367	1.60068	0.800338	−	0.599549i	$-0.204653\pi$
$547$	37.4367	1.60068	0.800338	+	0.599549i	$0.204653\pi$
$548$	115.945	4.95291
$549$	16.1204	0.688003
$550$	−4.01087	−0.171024
$551$	−28.9582	−1.23366
$552$	43.9382	1.87013
$553$	22.3997	0.952533
$554$	−0.834613	−0.0354593
$555$	2.99658	0.127198
$556$	108.915	4.61901
$557$	10.2891	0.435963	0.217981	−	0.975953i	$-0.430053\pi$
$557$	10.2891	0.435963	0.217981	+	0.975953i	$0.430053\pi$
$558$	22.1886	0.939319
$559$	24.5035	1.03639
$560$	−40.9702	−1.73131
$561$	5.43601	0.229509
$562$	−62.2428	−2.62555
$563$	3.58295	0.151003	0.0755017	−	0.997146i	$-0.475944\pi$
$563$	3.58295	0.151003	0.0755017	+	0.997146i	$0.475944\pi$
$564$	−2.18684	−0.0920826
$565$	−5.53906	−0.233030
$566$	8.41792	0.353832
$567$	−13.3083	−0.558895
$568$	−0.538863	−0.0226102
$569$	−12.3081	−0.515983	−0.257991	−	0.966147i	$-0.583061\pi$
$569$	−12.3081	−0.515983	−0.257991	+	0.966147i	$0.583061\pi$
$570$	10.2826	0.430689
$571$	−38.7661	−1.62231	−0.811156	−	0.584829i	$-0.801161\pi$
$571$	−38.7661	−1.62231	−0.811156	+	0.584829i	$0.801161\pi$
$572$	−25.5831	−1.06968
$573$	−16.0890	−0.672129
$574$	39.5017	1.64877
$575$	6.64876	0.277272
$576$	−66.3498	−2.76457
$577$	−12.5584	−0.522814	−0.261407	−	0.965229i	$-0.584186\pi$
$577$	−12.5584	−0.522814	−0.261407	+	0.965229i	$0.584186\pi$
$578$	−25.3597	−1.05483
$579$	−10.8772	−0.452040
$580$	−29.6006	−1.22910
$581$	−43.3691	−1.79925
$582$	−12.9234	−0.535693
$583$	−4.18710	−0.173412
$584$	4.50915	0.186590
$585$	7.99590	0.330590
$586$	−56.7071	−2.34255
$587$	−20.6765	−0.853412	−0.426706	−	0.904390i	$-0.640326\pi$
$587$	−20.6765	−0.853412	−0.426706	+	0.904390i	$0.640326\pi$
$588$	−4.91442	−0.202667
$589$	17.3160	0.713494
$590$	−13.4094	−0.552056
$591$	−5.29359	−0.217749
$592$	−59.4667	−2.44407
$593$	11.8506	0.486646	0.243323	−	0.969945i	$-0.421763\pi$
$593$	11.8506	0.486646	0.243323	+	0.969945i	$0.421763\pi$
$594$	−15.7919	−0.647950
$595$	14.7572	0.604986
$596$	−9.44505	−0.386884
$597$	−10.3493	−0.423569
$598$	58.1535	2.37807
$599$	−14.5125	−0.592967	−0.296483	−	0.955038i	$-0.595814\pi$
$599$	−14.5125	−0.592967	−0.296483	+	0.955038i	$0.595814\pi$
$600$	6.60848	0.269790
$601$	−2.99946	−0.122351	−0.0611753	−	0.998127i	$-0.519485\pi$
$601$	−2.99946	−0.122351	−0.0611753	+	0.998127i	$0.519485\pi$
$602$	−59.5163	−2.42570
$603$	−33.5877	−1.36780
$604$	48.4812	1.97267
$605$	−8.82226	−0.358676
$606$	−21.6928	−0.881210
$607$	−29.0838	−1.18047	−0.590237	−	0.807230i	$-0.700966\pi$
$607$	−29.0838	−1.18047	−0.590237	+	0.807230i	$0.700966\pi$
$608$	−107.026	−4.34050
$609$	−11.3441	−0.459688
$610$	17.6337	0.713969
$611$	−1.81979	−0.0736207
$612$	68.6827	2.77633
$613$	14.4041	0.581774	0.290887	−	0.956757i	$-0.406050\pi$
$613$	14.4041	0.581774	0.290887	+	0.956757i	$0.406050\pi$
$614$	89.7812	3.62327
$615$	−3.62789	−0.146291
$616$	39.0689	1.57413
$617$	34.9634	1.40757	0.703787	−	0.710411i	$-0.251491\pi$
$617$	34.9634	1.40757	0.703787	+	0.710411i	$0.251491\pi$
$618$	−3.39946	−0.136746
$619$	−19.6329	−0.789112	−0.394556	−	0.918872i	$-0.629102\pi$
$619$	−19.6329	−0.789112	−0.394556	+	0.918872i	$0.629102\pi$
$620$	17.7002	0.710857
$621$	26.1780	1.05049
$622$	−81.0339	−3.24916
$623$	−45.8494	−1.83692
$624$	32.9110	1.31750
$625$	1.00000	0.0400000
$626$	−19.5991	−0.783338
$627$	−5.58302	−0.222965
$628$	−101.812	−4.06273
$629$	21.4195	0.854051
$630$	−19.4212	−0.773759
$631$	−19.9774	−0.795288	−0.397644	−	0.917540i	$-0.630172\pi$
$631$	−19.9774	−0.795288	−0.397644	+	0.917540i	$0.630172\pi$
$632$	71.7007	2.85210
$633$	13.3344	0.529997
$634$	33.3038	1.32266
$635$	20.1623	0.800117
$636$	10.9725	0.435089
$637$	−4.08956	−0.162034
$638$	22.0389	0.872528
$639$	−0.145442	−0.00575361
$640$	−31.9620	−1.26341
$641$	6.85743	0.270852	0.135426	−	0.990787i	$-0.456760\pi$
$641$	6.85743	0.270852	0.135426	+	0.990787i	$0.456760\pi$
$642$	0.695426	0.0274463
$643$	−32.6598	−1.28798	−0.643988	−	0.765036i	$-0.722721\pi$
$643$	−32.6598	−1.28798	−0.643988	+	0.765036i	$0.722721\pi$
$644$	−103.006	−4.05902
$645$	5.46606	0.215226
$646$	73.4996	2.89180
$647$	−34.2535	−1.34664	−0.673322	−	0.739349i	$-0.735133\pi$
$647$	−34.2535	−1.34664	−0.673322	+	0.739349i	$0.735133\pi$
$648$	−42.5993	−1.67346
$649$	7.28078	0.285796
$650$	8.74652	0.343067
$651$	6.78342	0.265863
$652$	39.5140	1.54749
$653$	23.4719	0.918526	0.459263	−	0.888300i	$-0.348113\pi$
$653$	23.4719	0.918526	0.459263	+	0.888300i	$0.348113\pi$
$654$	21.9353	0.857738
$655$	−20.8585	−0.815010
$656$	71.9950	2.81093
$657$	1.21705	0.0474815
$658$	4.42007	0.172312
$659$	1.27010	0.0494762	0.0247381	−	0.999694i	$-0.492125\pi$
$659$	1.27010	0.0494762	0.0247381	+	0.999694i	$0.492125\pi$
$660$	−5.70689	−0.222141
$661$	13.9696	0.543354	0.271677	−	0.962389i	$-0.412422\pi$
$661$	13.9696	0.543354	0.271677	+	0.962389i	$0.412422\pi$
$662$	61.2961	2.38234
$663$	−11.8543	−0.460384
$664$	−138.823	−5.38737
$665$	−15.1563	−0.587736
$666$	−28.1891	−1.09230
$667$	−36.5336	−1.41459
$668$	−19.9883	−0.773372
$669$	1.80322	0.0697165
$670$	−36.7407	−1.41942
$671$	−9.57442	−0.369616
$672$	−41.9268	−1.61736
$673$	−24.3171	−0.937357	−0.468679	−	0.883369i	$-0.655270\pi$
$673$	−24.3171	−0.937357	−0.468679	+	0.883369i	$0.655270\pi$
$674$	94.7987	3.65151
$675$	3.93728	0.151546
$676$	−14.2423	−0.547780
$677$	−30.4809	−1.17148	−0.585738	−	0.810500i	$-0.699195\pi$
$677$	−30.4809	−1.17148	−0.585738	+	0.810500i	$0.699195\pi$
$678$	−10.8073	−0.415052
$679$	19.0489	0.731028
$680$	47.2373	1.81147
$681$	12.7022	0.486750
$682$	−13.1785	−0.504632
$683$	38.5407	1.47472	0.737360	−	0.675500i	$-0.236072\pi$
$683$	38.5407	1.47472	0.737360	+	0.675500i	$0.236072\pi$
$684$	−70.5401	−2.69717
$685$	21.5229	0.822348
$686$	−44.7819	−1.70978
$687$	8.67759	0.331071
$688$	−108.473	−4.13550
$689$	9.13083	0.347857
$690$	12.9725	0.493853
$691$	−6.97119	−0.265197	−0.132598	−	0.991170i	$-0.542332\pi$
$691$	−6.97119	−0.265197	−0.132598	+	0.991170i	$0.542332\pi$
$692$	54.4949	2.07158
$693$	10.5449	0.400569
$694$	46.4140	1.76185
$695$	20.2179	0.766910
$696$	−36.3122	−1.37641
$697$	−25.9321	−0.982249
$698$	−4.62802	−0.175173
$699$	4.13255	0.156307
$700$	−15.4926	−0.585564
$701$	−14.2661	−0.538823	−0.269412	−	0.963025i	$-0.586829\pi$
$701$	−14.2661	−0.538823	−0.269412	+	0.963025i	$0.586829\pi$
$702$	34.4375	1.29976
$703$	−21.9988	−0.829699
$704$	39.4072	1.48522
$705$	−0.405945	−0.0152888
$706$	67.3397	2.53436
$707$	31.9748	1.20254
$708$	−19.0797	−0.717058
$709$	7.27646	0.273273	0.136637	−	0.990621i	$-0.456371\pi$
$709$	7.27646	0.273273	0.136637	+	0.990621i	$0.456371\pi$
$710$	−0.159096	−0.00597076
$711$	19.3524	0.725774
$712$	−146.762	−5.50015
$713$	21.8458	0.818133
$714$	28.7929	1.07755
$715$	−4.74901	−0.177603
$716$	−96.1215	−3.59223
$717$	−2.24234	−0.0837418
$718$	3.67497	0.137149
$719$	−18.0564	−0.673391	−0.336695	−	0.941614i	$-0.609309\pi$
$719$	−18.0564	−0.673391	−0.336695	+	0.941614i	$0.609309\pi$
$720$	−35.3966	−1.31915
$721$	5.01074	0.186610
$722$	−23.8470	−0.887495
$723$	6.33203	0.235491
$724$	−81.0302	−3.01146
$725$	−5.49479	−0.204071
$726$	−17.2132	−0.638842
$727$	40.7644	1.51187	0.755934	−	0.654648i	$-0.227183\pi$
$727$	40.7644	1.51187	0.755934	+	0.654648i	$0.227183\pi$
$728$	−85.1978	−3.15764
$729$	−3.01842	−0.111794
$730$	1.33130	0.0492736
$731$	39.0713	1.44510
$732$	25.0903	0.927364
$733$	2.72022	0.100474	0.0502368	−	0.998737i	$-0.484002\pi$
$733$	2.72022	0.100474	0.0502368	+	0.998737i	$0.484002\pi$
$734$	−83.7033	−3.08954
$735$	−0.912269	−0.0336495
$736$	−135.024	−4.97707
$737$	19.9488	0.734822
$738$	34.1279	1.25627
$739$	15.4760	0.569292	0.284646	−	0.958633i	$-0.408124\pi$
$739$	15.4760	0.569292	0.284646	+	0.958633i	$0.408124\pi$
$740$	−22.4868	−0.826633
$741$	12.1749	0.447257
$742$	−22.1778	−0.814174
$743$	−42.8089	−1.57051	−0.785253	−	0.619175i	$-0.787467\pi$
$743$	−42.8089	−1.57051	−0.785253	+	0.619175i	$0.787467\pi$
$744$	21.7135	0.796055
$745$	−1.75329	−0.0642357
$746$	−26.7051	−0.977743
$747$	−37.4691	−1.37092
$748$	−40.7928	−1.49153
$749$	−1.02504	−0.0374543
$750$	1.95111	0.0712445
$751$	−11.8855	−0.433709	−0.216854	−	0.976204i	$-0.569580\pi$
$751$	−11.8855	−0.433709	−0.216854	+	0.976204i	$0.569580\pi$
$752$	8.05593	0.293769
$753$	−18.1863	−0.662745
$754$	−48.0603	−1.75025
$755$	8.99962	0.327530
$756$	−60.9986	−2.21850
$757$	−28.8647	−1.04911	−0.524553	−	0.851378i	$-0.675768\pi$
$757$	−28.8647	−1.04911	−0.524553	+	0.851378i	$0.675768\pi$
$758$	79.5688	2.89007
$759$	−7.04354	−0.255664
$760$	−48.5148	−1.75982
$761$	−46.5776	−1.68844	−0.844218	−	0.536001i	$-0.819934\pi$
$761$	−46.5776	−1.68844	−0.844218	+	0.536001i	$0.819934\pi$
$762$	39.3389	1.42510
$763$	−32.3322	−1.17050
$764$	120.735	4.36803
$765$	12.7496	0.460964
$766$	−12.7912	−0.462165
$767$	−15.8772	−0.573293
$768$	−24.0216	−0.866806
$769$	−18.5243	−0.668005	−0.334002	−	0.942572i	$-0.608399\pi$
$769$	−18.5243	−0.668005	−0.334002	+	0.942572i	$0.608399\pi$
$770$	11.5349	0.415687
$771$	7.75360	0.279239
$772$	81.6242	2.93772
$773$	1.14052	0.0410215	0.0205108	−	0.999790i	$-0.493471\pi$
$773$	1.14052	0.0410215	0.0205108	+	0.999790i	$0.493471\pi$
$774$	−51.4197	−1.84824
$775$	3.28570	0.118026
$776$	60.9747	2.18887
$777$	−8.61785	−0.309164
$778$	−19.3320	−0.693085
$779$	26.6334	0.954241
$780$	12.4450	0.445604
$781$	0.0863828	0.00309102
$782$	92.7270	3.31591
$783$	−21.6345	−0.773156
$784$	18.1038	0.646566
$785$	−18.8994	−0.674548
$786$	−40.6973	−1.45162
$787$	−3.66717	−0.130721	−0.0653603	−	0.997862i	$-0.520820\pi$
$787$	−3.66717	−0.130721	−0.0653603	+	0.997862i	$0.520820\pi$
$788$	39.7240	1.41511
$789$	−7.37583	−0.262586
$790$	21.1692	0.753165
$791$	15.9298	0.566398
$792$	33.7540	1.19940
$793$	20.8790	0.741434
$794$	−3.72698	−0.132266
$795$	2.03684	0.0722393
$796$	77.6631	2.75269
$797$	12.1520	0.430444	0.215222	−	0.976565i	$-0.430952\pi$
$797$	12.1520	0.430444	0.215222	+	0.976565i	$0.430952\pi$
$798$	−29.5716	−1.04682
$799$	−2.90169	−0.102654
$800$	−20.3082	−0.718004
$801$	−39.6120	−1.39962
$802$	11.6194	0.410294
$803$	−0.722843	−0.0255086
$804$	−52.2768	−1.84366
$805$	−19.1212	−0.673932
$806$	28.7384	1.01227
$807$	1.54258	0.0543014
$808$	102.350	3.60067
$809$	12.4994	0.439454	0.219727	−	0.975561i	$-0.429483\pi$
$809$	12.4994	0.439454	0.219727	+	0.975561i	$0.429483\pi$
$810$	−12.5772	−0.441917
$811$	−32.0473	−1.12533	−0.562667	−	0.826684i	$-0.690225\pi$
$811$	−32.0473	−1.12533	−0.562667	+	0.826684i	$0.690225\pi$
$812$	85.1284	2.98742
$813$	12.3853	0.434372
$814$	16.7424	0.586820
$815$	7.33502	0.256935
$816$	52.4774	1.83708
$817$	−40.1280	−1.40390
$818$	71.3034	2.49306
$819$	−22.9954	−0.803524
$820$	27.2243	0.950715
$821$	−37.5290	−1.30977	−0.654885	−	0.755728i	$-0.727283\pi$
$821$	−37.5290	−1.30977	−0.654885	+	0.755728i	$0.727283\pi$
$822$	41.9936	1.46469
$823$	−53.5896	−1.86802	−0.934008	−	0.357253i	$-0.883713\pi$
$823$	−53.5896	−1.86802	−0.934008	+	0.357253i	$0.883713\pi$
$824$	16.0392	0.558752
$825$	−1.05938	−0.0368827
$826$	38.5641	1.34182
$827$	1.32719	0.0461511	0.0230755	−	0.999734i	$-0.492654\pi$
$827$	1.32719	0.0461511	0.0230755	+	0.999734i	$0.492654\pi$
$828$	−88.9933	−3.09273
$829$	−14.8791	−0.516772	−0.258386	−	0.966042i	$-0.583191\pi$
$829$	−14.8791	−0.516772	−0.258386	+	0.966042i	$0.583191\pi$
$830$	−40.9866	−1.42266
$831$	−0.220443	−0.00764709
$832$	−85.9355	−2.97928
$833$	−6.52088	−0.225935
$834$	39.4474	1.36595
$835$	−3.71045	−0.128406
$836$	41.8960	1.44900
$837$	12.9367	0.447159
$838$	−94.3512	−3.25931
$839$	17.4649	0.602957	0.301478	−	0.953473i	$-0.402520\pi$
$839$	17.4649	0.602957	0.301478	+	0.953473i	$0.402520\pi$
$840$	−19.0053	−0.655745
$841$	1.19275	0.0411293
$842$	31.5223	1.08633
$843$	−16.4400	−0.566222
$844$	−100.064	−3.44435
$845$	−2.64381	−0.0909497
$846$	3.81876	0.131292
$847$	25.3719	0.871790
$848$	−40.4208	−1.38806
$849$	2.22339	0.0763067
$850$	13.9465	0.478361
$851$	−27.7536	−0.951381
$852$	−0.226371	−0.00775533
$853$	−23.2554	−0.796251	−0.398125	−	0.917331i	$-0.630339\pi$
$853$	−23.2554	−0.796251	−0.398125	+	0.917331i	$0.630339\pi$
$854$	−50.7128	−1.73536
$855$	−13.0944	−0.447820
$856$	−3.28113	−0.112147
$857$	−0.295914	−0.0101082	−0.00505411	−	0.999987i	$-0.501609\pi$
$857$	−0.295914	−0.0101082	−0.00505411	+	0.999987i	$0.501609\pi$
$858$	−9.26585	−0.316331
$859$	−22.8268	−0.778841	−0.389421	−	0.921060i	$-0.627325\pi$
$859$	−22.8268	−0.778841	−0.389421	+	0.921060i	$0.627325\pi$
$860$	−41.0183	−1.39871
$861$	10.4334	0.355571
$862$	−27.2908	−0.929527
$863$	−15.1899	−0.517071	−0.258535	−	0.966002i	$-0.583240\pi$
$863$	−15.1899	−0.517071	−0.258535	+	0.966002i	$0.583240\pi$
$864$	−79.9592	−2.72027
$865$	10.1159	0.343952
$866$	36.2790	1.23281
$867$	−6.69817	−0.227482
$868$	−50.9040	−1.72779
$869$	−11.4940	−0.389908
$870$	−10.7209	−0.363474
$871$	−43.5024	−1.47402
$872$	−103.494	−3.50476
$873$	16.4574	0.557000
$874$	−95.2347	−3.22136
$875$	−2.87590	−0.0972231
$876$	1.89425	0.0640007
$877$	−55.1645	−1.86277	−0.931387	−	0.364031i	$-0.881400\pi$
$877$	−55.1645	−1.86277	−0.931387	+	0.364031i	$0.881400\pi$
$878$	−43.1631	−1.45668
$879$	−14.9778	−0.505190
$880$	21.0232	0.708691
$881$	−18.2998	−0.616536	−0.308268	−	0.951300i	$-0.599749\pi$
$881$	−18.2998	−0.616536	−0.308268	+	0.951300i	$0.599749\pi$
$882$	8.58179	0.288964
$883$	5.27613	0.177556	0.0887779	−	0.996051i	$-0.471704\pi$
$883$	5.27613	0.177556	0.0887779	+	0.996051i	$0.471704\pi$
$884$	88.9570	2.99195
$885$	−3.54178	−0.119056
$886$	−23.3446	−0.784276
$887$	31.2842	1.05042	0.525209	−	0.850973i	$-0.323987\pi$
$887$	31.2842	1.05042	0.525209	+	0.850973i	$0.323987\pi$
$888$	−27.5854	−0.925707
$889$	−57.9848	−1.94475
$890$	−43.3306	−1.45245
$891$	6.82891	0.228777
$892$	−13.5317	−0.453074
$893$	2.98016	0.0997274
$894$	−3.42087	−0.114411
$895$	−17.8431	−0.596430
$896$	91.9195	3.07081
$897$	15.3599	0.512851
$898$	−1.26335	−0.0421586
$899$	−18.0543	−0.602143
$900$	−13.3849	−0.446165
$901$	14.5593	0.485041
$902$	−20.2696	−0.674905
$903$	−15.7198	−0.523123
$904$	50.9907	1.69592
$905$	−15.0417	−0.500003
$906$	17.5592	0.583367
$907$	23.1938	0.770136	0.385068	−	0.922888i	$-0.374178\pi$
$907$	23.1938	0.770136	0.385068	+	0.922888i	$0.374178\pi$
$908$	−95.3196	−3.16329
$909$	27.6249	0.916261
$910$	−25.1541	−0.833850
$911$	45.9021	1.52080	0.760402	−	0.649452i	$-0.225002\pi$
$911$	45.9021	1.52080	0.760402	+	0.649452i	$0.225002\pi$
$912$	−53.8966	−1.78469
$913$	22.2541	0.736503
$914$	63.4062	2.09729
$915$	4.65753	0.153973
$916$	−65.1181	−2.15156
$917$	59.9870	1.98095
$918$	54.9113	1.81234
$919$	−23.9239	−0.789177	−0.394588	−	0.918858i	$-0.629113\pi$
$919$	−23.9239	−0.789177	−0.394588	+	0.918858i	$0.629113\pi$
$920$	−61.2062	−2.01791
$921$	23.7136	0.781389
$922$	36.6032	1.20546
$923$	−0.188375	−0.00620045
$924$	16.4125	0.539930
$925$	−4.17425	−0.137249
$926$	−42.2002	−1.38679
$927$	4.32907	0.142185
$928$	111.589	3.66310
$929$	19.5277	0.640682	0.320341	−	0.947302i	$-0.396203\pi$
$929$	19.5277	0.640682	0.320341	+	0.947302i	$0.396203\pi$
$930$	6.41077	0.210217
$931$	6.69723	0.219493
$932$	−31.0114	−1.01581
$933$	−21.4032	−0.700709
$934$	99.2848	3.24870
$935$	−7.57240	−0.247644
$936$	−73.6074	−2.40593
$937$	3.22512	0.105360	0.0526801	−	0.998611i	$-0.483224\pi$
$937$	3.22512	0.105360	0.0526801	+	0.998611i	$0.483224\pi$
$938$	105.663	3.45001
$939$	−5.17664	−0.168933
$940$	3.04628	0.0993588
$941$	25.0882	0.817851	0.408926	−	0.912568i	$-0.365904\pi$
$941$	25.0882	0.817851	0.408926	+	0.912568i	$0.365904\pi$
$942$	−36.8748	−1.20145
$943$	33.6007	1.09419
$944$	70.2861	2.28762
$945$	−11.3232	−0.368345
$946$	30.5398	0.992934
$947$	−35.9751	−1.16903	−0.584517	−	0.811382i	$-0.698716\pi$
$947$	−35.9751	−1.16903	−0.584517	+	0.811382i	$0.698716\pi$
$948$	30.1207	0.978275
$949$	1.57631	0.0511691
$950$	−14.3237	−0.464721
$951$	8.79642	0.285244
$952$	−135.850	−4.40291
$953$	−10.4894	−0.339787	−0.169893	−	0.985462i	$-0.554342\pi$
$953$	−10.4894	−0.339787	−0.169893	+	0.985462i	$0.554342\pi$
$954$	−19.1607	−0.620352
$955$	22.4121	0.725239
$956$	16.8269	0.544221
$957$	5.82105	0.188168
$958$	−95.6386	−3.08994
$959$	−61.8977	−1.99878
$960$	−19.1699	−0.618705
$961$	−20.2042	−0.651747
$962$	−36.5102	−1.17714
$963$	−0.885597	−0.0285380
$964$	−47.5167	−1.53041
$965$	15.1520	0.487759
$966$	−37.3075	−1.20035
$967$	−43.2466	−1.39072	−0.695358	−	0.718664i	$-0.744754\pi$
$967$	−43.2466	−1.39072	−0.695358	+	0.718664i	$0.744754\pi$
$968$	81.2146	2.61034
$969$	19.4132	0.623641
$970$	18.0024	0.578022
$971$	−19.0107	−0.610084	−0.305042	−	0.952339i	$-0.598670\pi$
$971$	−19.0107	−0.610084	−0.305042	+	0.952339i	$0.598670\pi$
$972$	−81.5263	−2.61496
$973$	−58.1447	−1.86403
$974$	−65.8482	−2.10991
$975$	2.31019	0.0739851
$976$	−92.4281	−2.95855
$977$	−27.2884	−0.873034	−0.436517	−	0.899696i	$-0.643788\pi$
$977$	−27.2884	−0.873034	−0.436517	+	0.899696i	$0.643788\pi$
$978$	14.3114	0.457629
$979$	23.5268	0.751920
$980$	6.84582	0.218682
$981$	−27.9337	−0.891855
$982$	36.1326	1.15304
$983$	21.4292	0.683486	0.341743	−	0.939793i	$-0.388983\pi$
$983$	21.4292	0.683486	0.341743	+	0.939793i	$0.388983\pi$
$984$	33.3971	1.06466
$985$	7.37401	0.234955
$986$	−76.6332	−2.44050
$987$	1.16746	0.0371606
$988$	−91.3628	−2.90664
$989$	−50.6254	−1.60979
$990$	9.96564	0.316729
$991$	15.9163	0.505596	0.252798	−	0.967519i	$-0.418649\pi$
$991$	15.9163	0.505596	0.252798	+	0.967519i	$0.418649\pi$
$992$	−66.7267	−2.11858
$993$	16.1899	0.513771
$994$	0.457544	0.0145124
$995$	14.4167	0.457039
$996$	−58.3180	−1.84788
$997$	48.1387	1.52457	0.762284	−	0.647243i	$-0.224078\pi$
$997$	48.1387	1.52457	0.762284	+	0.647243i	$0.224078\pi$
$998$	−20.6813	−0.654654
$999$	−16.4352	−0.519987

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8045.2.a.c.1.4	✓	127

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8045.2.a.c.1.4	✓	127	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8045 = 5 \cdot 1609 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8045.a (trivial)

\(f(q)\)	\(=\)	\(q-2.71791 q^{2} -0.717872 q^{3} +5.38703 q^{4} +1.00000 q^{5} +1.95111 q^{6} -2.87590 q^{7} -9.20565 q^{8} -2.48466 q^{9} +O(q^{10})\)	\(q-2.71791 q^{2} -0.717872 q^{3} +5.38703 q^{4} +1.00000 q^{5} +1.95111 q^{6} -2.87590 q^{7} -9.20565 q^{8} -2.48466 q^{9} -2.71791 q^{10} +1.47572 q^{11} -3.86720 q^{12} -3.21810 q^{13} +7.81643 q^{14} -0.717872 q^{15} +14.2461 q^{16} -5.13134 q^{17} +6.75308 q^{18} +5.27011 q^{19} +5.38703 q^{20} +2.06453 q^{21} -4.01087 q^{22} +6.64876 q^{23} +6.60848 q^{24} +1.00000 q^{25} +8.74652 q^{26} +3.93728 q^{27} -15.4926 q^{28} -5.49479 q^{29} +1.95111 q^{30} +3.28570 q^{31} -20.3082 q^{32} -1.05938 q^{33} +13.9465 q^{34} -2.87590 q^{35} -13.3849 q^{36} -4.17425 q^{37} -14.3237 q^{38} +2.31019 q^{39} -9.20565 q^{40} +5.05368 q^{41} -5.61120 q^{42} -7.61426 q^{43} +7.94974 q^{44} -2.48466 q^{45} -18.0707 q^{46} +0.565484 q^{47} -10.2268 q^{48} +1.27080 q^{49} -2.71791 q^{50} +3.68364 q^{51} -17.3360 q^{52} -2.83733 q^{53} -10.7012 q^{54} +1.47572 q^{55} +26.4745 q^{56} -3.78326 q^{57} +14.9344 q^{58} +4.93372 q^{59} -3.86720 q^{60} -6.48798 q^{61} -8.93024 q^{62} +7.14563 q^{63} +26.7038 q^{64} -3.21810 q^{65} +2.87929 q^{66} +13.5180 q^{67} -27.6427 q^{68} -4.77296 q^{69} +7.81643 q^{70} +0.0585361 q^{71} +22.8729 q^{72} -0.489824 q^{73} +11.3452 q^{74} -0.717872 q^{75} +28.3902 q^{76} -4.24402 q^{77} -6.27888 q^{78} -7.78877 q^{79} +14.2461 q^{80} +4.62752 q^{81} -13.7354 q^{82} +15.0802 q^{83} +11.1217 q^{84} -5.13134 q^{85} +20.6949 q^{86} +3.94456 q^{87} -13.5849 q^{88} +15.9426 q^{89} +6.75308 q^{90} +9.25494 q^{91} +35.8171 q^{92} -2.35871 q^{93} -1.53694 q^{94} +5.27011 q^{95} +14.5787 q^{96} -6.62362 q^{97} -3.45391 q^{98} -3.66666 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 127 q - 20 q^{2} - 31 q^{3} + 116 q^{4} + 127 q^{5} - 17 q^{6} - 63 q^{7} - 57 q^{8} + 122 q^{9}+O(q^{10}) \) 127 * q - 20 * q^2 - 31 * q^3 + 116 * q^4 + 127 * q^5 - 17 * q^6 - 63 * q^7 - 57 * q^8 + 122 * q^9	\( 127 q - 20 q^{2} - 31 q^{3} + 116 q^{4} + 127 q^{5} - 17 q^{6} - 63 q^{7} - 57 q^{8} + 122 q^{9} - 20 q^{10} - 32 q^{11} - 65 q^{12} - 49 q^{13} - 4 q^{14} - 31 q^{15} + 98 q^{16} - 53 q^{17} - 60 q^{18} - 126 q^{19} + 116 q^{20} - 24 q^{21} - 46 q^{22} - 121 q^{23} - 51 q^{24} + 127 q^{25} - 9 q^{26} - 112 q^{27} - 123 q^{28} - 6 q^{29} - 17 q^{30} - 54 q^{31} - 119 q^{32} - 57 q^{33} - 53 q^{34} - 63 q^{35} + 133 q^{36} - 61 q^{37} - 27 q^{38} - 33 q^{39} - 57 q^{40} - 8 q^{41} - 46 q^{42} - 208 q^{43} - 54 q^{44} + 122 q^{45} - 34 q^{46} - 116 q^{47} - 106 q^{48} + 110 q^{49} - 20 q^{50} - 54 q^{51} - 142 q^{52} - 60 q^{53} - 62 q^{54} - 32 q^{55} + 33 q^{56} - 56 q^{57} - 87 q^{58} - 53 q^{59} - 65 q^{60} - 76 q^{61} - 84 q^{62} - 215 q^{63} + 67 q^{64} - 49 q^{65} + 9 q^{66} - 145 q^{67} - 133 q^{68} + q^{69} - 4 q^{70} - 4 q^{71} - 167 q^{72} - 155 q^{73} - 14 q^{74} - 31 q^{75} - 199 q^{76} - 97 q^{77} - 24 q^{78} - 73 q^{79} + 98 q^{80} + 127 q^{81} - 69 q^{82} - 225 q^{83} - 59 q^{84} - 53 q^{85} + 30 q^{86} - 179 q^{87} - 119 q^{88} - 25 q^{89} - 60 q^{90} - 160 q^{91} - 188 q^{92} - 44 q^{93} - 32 q^{94} - 126 q^{95} - 43 q^{96} - 72 q^{97} - 111 q^{98} - 141 q^{99}+O(q^{100}) \) 127 * q - 20 * q^2 - 31 * q^3 + 116 * q^4 + 127 * q^5 - 17 * q^6 - 63 * q^7 - 57 * q^8 + 122 * q^9 - 20 * q^10 - 32 * q^11 - 65 * q^12 - 49 * q^13 - 4 * q^14 - 31 * q^15 + 98 * q^16 - 53 * q^17 - 60 * q^18 - 126 * q^19 + 116 * q^20 - 24 * q^21 - 46 * q^22 - 121 * q^23 - 51 * q^24 + 127 * q^25 - 9 * q^26 - 112 * q^27 - 123 * q^28 - 6 * q^29 - 17 * q^30 - 54 * q^31 - 119 * q^32 - 57 * q^33 - 53 * q^34 - 63 * q^35 + 133 * q^36 - 61 * q^37 - 27 * q^38 - 33 * q^39 - 57 * q^40 - 8 * q^41 - 46 * q^42 - 208 * q^43 - 54 * q^44 + 122 * q^45 - 34 * q^46 - 116 * q^47 - 106 * q^48 + 110 * q^49 - 20 * q^50 - 54 * q^51 - 142 * q^52 - 60 * q^53 - 62 * q^54 - 32 * q^55 + 33 * q^56 - 56 * q^57 - 87 * q^58 - 53 * q^59 - 65 * q^60 - 76 * q^61 - 84 * q^62 - 215 * q^63 + 67 * q^64 - 49 * q^65 + 9 * q^66 - 145 * q^67 - 133 * q^68 + q^69 - 4 * q^70 - 4 * q^71 - 167 * q^72 - 155 * q^73 - 14 * q^74 - 31 * q^75 - 199 * q^76 - 97 * q^77 - 24 * q^78 - 73 * q^79 + 98 * q^80 + 127 * q^81 - 69 * q^82 - 225 * q^83 - 59 * q^84 - 53 * q^85 + 30 * q^86 - 179 * q^87 - 119 * q^88 - 25 * q^89 - 60 * q^90 - 160 * q^91 - 188 * q^92 - 44 * q^93 - 32 * q^94 - 126 * q^95 - 43 * q^96 - 72 * q^97 - 111 * q^98 - 141 * q^99

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