LMFDB - Embedded newform 8043.2.a.m.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8043,2,Mod(1,8043)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8043, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8043.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8043 = 3 \cdot 7 \cdot 383 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8043.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:	$64.2236783457$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.48119$ of defining polynomial
Character	$\chi$	$=$	8043.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.193937 q^{2} +1.00000 q^{3} -1.96239 q^{4} -0.806063 q^{5} -0.193937 q^{6} -1.00000 q^{7} +0.768452 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.193937 q^{2} +1.00000 q^{3} -1.96239 q^{4} -0.806063 q^{5} -0.193937 q^{6} -1.00000 q^{7} +0.768452 q^{8} +1.00000 q^{9} +0.156325 q^{10} -2.15633 q^{11} -1.96239 q^{12} +4.80606 q^{13} +0.193937 q^{14} -0.806063 q^{15} +3.77575 q^{16} -6.96239 q^{17} -0.193937 q^{18} -5.92478 q^{19} +1.58181 q^{20} -1.00000 q^{21} +0.418190 q^{22} +5.73813 q^{23} +0.768452 q^{24} -4.35026 q^{25} -0.932071 q^{26} +1.00000 q^{27} +1.96239 q^{28} -7.27504 q^{29} +0.156325 q^{30} -3.03761 q^{31} -2.26916 q^{32} -2.15633 q^{33} +1.35026 q^{34} +0.806063 q^{35} -1.96239 q^{36} +10.0000 q^{37} +1.14903 q^{38} +4.80606 q^{39} -0.619421 q^{40} +11.5066 q^{41} +0.193937 q^{42} +2.38787 q^{43} +4.23155 q^{44} -0.806063 q^{45} -1.11283 q^{46} -4.31265 q^{47} +3.77575 q^{48} +1.00000 q^{49} +0.843675 q^{50} -6.96239 q^{51} -9.43136 q^{52} +1.03761 q^{53} -0.193937 q^{54} +1.73813 q^{55} -0.768452 q^{56} -5.92478 q^{57} +1.41090 q^{58} -1.92478 q^{59} +1.58181 q^{60} -13.8945 q^{61} +0.589104 q^{62} -1.00000 q^{63} -7.11142 q^{64} -3.87399 q^{65} +0.418190 q^{66} -15.5369 q^{67} +13.6629 q^{68} +5.73813 q^{69} -0.156325 q^{70} +10.8872 q^{71} +0.768452 q^{72} +1.22425 q^{73} -1.93937 q^{74} -4.35026 q^{75} +11.6267 q^{76} +2.15633 q^{77} -0.932071 q^{78} -7.76845 q^{79} -3.04349 q^{80} +1.00000 q^{81} -2.23155 q^{82} +7.22425 q^{83} +1.96239 q^{84} +5.61213 q^{85} -0.463096 q^{86} -7.27504 q^{87} -1.65703 q^{88} +6.73084 q^{89} +0.156325 q^{90} -4.80606 q^{91} -11.2605 q^{92} -3.03761 q^{93} +0.836381 q^{94} +4.77575 q^{95} -2.26916 q^{96} +13.1187 q^{97} -0.193937 q^{98} -2.15633 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} + 3 q^{3} + 5 q^{4} - 2 q^{5} - q^{6} - 3 q^{7} - 9 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q - q^2 + 3 * q^3 + 5 * q^4 - 2 * q^5 - q^6 - 3 * q^7 - 9 * q^8 + 3 * q^9	\( 3 q - q^{2} + 3 q^{3} + 5 q^{4} - 2 q^{5} - q^{6} - 3 q^{7} - 9 q^{8} + 3 q^{9} - 10 q^{10} + 4 q^{11} + 5 q^{12} + 14 q^{13} + q^{14} - 2 q^{15} + 13 q^{16} - 10 q^{17} - q^{18} + 4 q^{19} + 6 q^{20} - 3 q^{21} + 8 q^{23} - 9 q^{24} - 3 q^{25} + 6 q^{26} + 3 q^{27} - 5 q^{28} + 10 q^{29} - 10 q^{30} - 20 q^{31} - 29 q^{32} + 4 q^{33} - 6 q^{34} + 2 q^{35} + 5 q^{36} + 30 q^{37} - 20 q^{38} + 14 q^{39} - 14 q^{40} + 14 q^{41} + q^{42} + 8 q^{43} + 24 q^{44} - 2 q^{45} - 36 q^{46} + 8 q^{47} + 13 q^{48} + 3 q^{49} + 13 q^{50} - 10 q^{51} + 14 q^{52} + 14 q^{53} - q^{54} - 4 q^{55} + 9 q^{56} + 4 q^{57} - 10 q^{58} + 16 q^{59} + 6 q^{60} - 22 q^{61} + 16 q^{62} - 3 q^{63} + 13 q^{64} - 20 q^{65} - 24 q^{67} + 10 q^{68} + 8 q^{69} + 10 q^{70} - 9 q^{72} + 2 q^{73} - 10 q^{74} - 3 q^{75} + 60 q^{76} - 4 q^{77} + 6 q^{78} - 12 q^{79} + 34 q^{80} + 3 q^{81} - 18 q^{82} + 20 q^{83} - 5 q^{84} + 16 q^{85} - 24 q^{86} + 10 q^{87} - 28 q^{88} - 2 q^{89} - 10 q^{90} - 14 q^{91} + 24 q^{92} - 20 q^{93} + 16 q^{95} - 29 q^{96} + 18 q^{97} - q^{98} + 4 q^{99}+O(q^{100}) \) 3 * q - q^2 + 3 * q^3 + 5 * q^4 - 2 * q^5 - q^6 - 3 * q^7 - 9 * q^8 + 3 * q^9 - 10 * q^10 + 4 * q^11 + 5 * q^12 + 14 * q^13 + q^14 - 2 * q^15 + 13 * q^16 - 10 * q^17 - q^18 + 4 * q^19 + 6 * q^20 - 3 * q^21 + 8 * q^23 - 9 * q^24 - 3 * q^25 + 6 * q^26 + 3 * q^27 - 5 * q^28 + 10 * q^29 - 10 * q^30 - 20 * q^31 - 29 * q^32 + 4 * q^33 - 6 * q^34 + 2 * q^35 + 5 * q^36 + 30 * q^37 - 20 * q^38 + 14 * q^39 - 14 * q^40 + 14 * q^41 + q^42 + 8 * q^43 + 24 * q^44 - 2 * q^45 - 36 * q^46 + 8 * q^47 + 13 * q^48 + 3 * q^49 + 13 * q^50 - 10 * q^51 + 14 * q^52 + 14 * q^53 - q^54 - 4 * q^55 + 9 * q^56 + 4 * q^57 - 10 * q^58 + 16 * q^59 + 6 * q^60 - 22 * q^61 + 16 * q^62 - 3 * q^63 + 13 * q^64 - 20 * q^65 - 24 * q^67 + 10 * q^68 + 8 * q^69 + 10 * q^70 - 9 * q^72 + 2 * q^73 - 10 * q^74 - 3 * q^75 + 60 * q^76 - 4 * q^77 + 6 * q^78 - 12 * q^79 + 34 * q^80 + 3 * q^81 - 18 * q^82 + 20 * q^83 - 5 * q^84 + 16 * q^85 - 24 * q^86 + 10 * q^87 - 28 * q^88 - 2 * q^89 - 10 * q^90 - 14 * q^91 + 24 * q^92 - 20 * q^93 + 16 * q^95 - 29 * q^96 + 18 * q^97 - q^98 + 4 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−0.193937	−0.137134	−0.0685669	−	0.997647i	$-0.521843\pi$
$2$	−0.193937	−0.137134	−0.0685669	+	0.997647i	$0.521843\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.96239	−0.981194
$5$	−0.806063	−0.360483	−0.180241	−	0.983622i	$-0.557688\pi$
$5$	−0.806063	−0.360483	−0.180241	+	0.983622i	$0.557688\pi$
$6$	−0.193937	−0.0791743
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0.768452	0.271689
$9$	1.00000	0.333333
$10$	0.156325	0.0494344
$11$	−2.15633	−0.650157	−0.325078	−	0.945687i	$-0.605391\pi$
$11$	−2.15633	−0.650157	−0.325078	+	0.945687i	$0.605391\pi$
$12$	−1.96239	−0.566493
$13$	4.80606	1.33296	0.666481	−	0.745522i	$-0.267800\pi$
$13$	4.80606	1.33296	0.666481	+	0.745522i	$0.267800\pi$
$14$	0.193937	0.0518317
$15$	−0.806063	−0.208125
$16$	3.77575	0.943937
$17$	−6.96239	−1.68863	−0.844314	−	0.535849i	$-0.819992\pi$
$17$	−6.96239	−1.68863	−0.844314	+	0.535849i	$0.819992\pi$
$18$	−0.193937	−0.0457113
$19$	−5.92478	−1.35924	−0.679619	−	0.733566i	$-0.737855\pi$
$19$	−5.92478	−1.35924	−0.679619	+	0.733566i	$0.737855\pi$
$20$	1.58181	0.353703
$21$	−1.00000	−0.218218
$22$	0.418190	0.0891585
$23$	5.73813	1.19648	0.598242	−	0.801316i	$-0.295866\pi$
$23$	5.73813	1.19648	0.598242	+	0.801316i	$0.295866\pi$
$24$	0.768452	0.156860
$25$	−4.35026	−0.870052
$26$	−0.932071	−0.182794
$27$	1.00000	0.192450
$28$	1.96239	0.370857
$29$	−7.27504	−1.35094	−0.675470	−	0.737387i	$-0.736059\pi$
$29$	−7.27504	−1.35094	−0.675470	+	0.737387i	$0.736059\pi$
$30$	0.156325	0.0285409
$31$	−3.03761	−0.545571	−0.272786	−	0.962075i	$-0.587945\pi$
$31$	−3.03761	−0.545571	−0.272786	+	0.962075i	$0.587945\pi$
$32$	−2.26916	−0.401134
$33$	−2.15633	−0.375368
$34$	1.35026	0.231568
$35$	0.806063	0.136250
$36$	−1.96239	−0.327065
$37$	10.0000	1.64399	0.821995	−	0.569495i	$-0.192861\pi$
$37$	10.0000	1.64399	0.821995	+	0.569495i	$0.192861\pi$
$38$	1.14903	0.186397
$39$	4.80606	0.769586
$40$	−0.619421	−0.0979391
$41$	11.5066	1.79703	0.898513	−	0.438946i	$-0.144648\pi$
$41$	11.5066	1.79703	0.898513	+	0.438946i	$0.144648\pi$
$42$	0.193937	0.0299251
$43$	2.38787	0.364147	0.182074	−	0.983285i	$-0.441719\pi$
$43$	2.38787	0.364147	0.182074	+	0.983285i	$0.441719\pi$
$44$	4.23155	0.637930
$45$	−0.806063	−0.120161
$46$	−1.11283	−0.164078
$47$	−4.31265	−0.629065	−0.314532	−	0.949247i	$-0.601848\pi$
$47$	−4.31265	−0.629065	−0.314532	+	0.949247i	$0.601848\pi$
$48$	3.77575	0.544982
$49$	1.00000	0.142857
$50$	0.843675	0.119314
$51$	−6.96239	−0.974929
$52$	−9.43136	−1.30789
$53$	1.03761	0.142527	0.0712634	−	0.997458i	$-0.477297\pi$
$53$	1.03761	0.142527	0.0712634	+	0.997458i	$0.477297\pi$
$54$	−0.193937	−0.0263914
$55$	1.73813	0.234370
$56$	−0.768452	−0.102689
$57$	−5.92478	−0.784756
$58$	1.41090	0.185260
$59$	−1.92478	−0.250585	−0.125292	−	0.992120i	$-0.539987\pi$
$59$	−1.92478	−0.250585	−0.125292	+	0.992120i	$0.539987\pi$
$60$	1.58181	0.204211
$61$	−13.8945	−1.77900	−0.889502	−	0.456932i	$-0.848948\pi$
$61$	−13.8945	−1.77900	−0.889502	+	0.456932i	$0.848948\pi$
$62$	0.589104	0.0748163
$63$	−1.00000	−0.125988
$64$	−7.11142	−0.888927
$65$	−3.87399	−0.480510
$66$	0.418190	0.0514757
$67$	−15.5369	−1.89813	−0.949067	−	0.315073i	$-0.897971\pi$
$67$	−15.5369	−1.89813	−0.949067	+	0.315073i	$0.897971\pi$
$68$	13.6629	1.65687
$69$	5.73813	0.690790
$70$	−0.156325	−0.0186844
$71$	10.8872	1.29207	0.646034	−	0.763308i	$-0.276426\pi$
$71$	10.8872	1.29207	0.646034	+	0.763308i	$0.276426\pi$
$72$	0.768452	0.0905629
$73$	1.22425	0.143288	0.0716440	−	0.997430i	$-0.477175\pi$
$73$	1.22425	0.143288	0.0716440	+	0.997430i	$0.477175\pi$
$74$	−1.93937	−0.225447
$75$	−4.35026	−0.502325
$76$	11.6267	1.33368
$77$	2.15633	0.245736
$78$	−0.932071	−0.105536
$79$	−7.76845	−0.874019	−0.437010	−	0.899457i	$-0.643962\pi$
$79$	−7.76845	−0.874019	−0.437010	+	0.899457i	$0.643962\pi$
$80$	−3.04349	−0.340273
$81$	1.00000	0.111111
$82$	−2.23155	−0.246433
$83$	7.22425	0.792965	0.396482	−	0.918042i	$-0.370231\pi$
$83$	7.22425	0.792965	0.396482	+	0.918042i	$0.370231\pi$
$84$	1.96239	0.214114
$85$	5.61213	0.608721
$86$	−0.463096	−0.0499369
$87$	−7.27504	−0.779966
$88$	−1.65703	−0.176640
$89$	6.73084	0.713468	0.356734	−	0.934206i	$-0.383890\pi$
$89$	6.73084	0.713468	0.356734	+	0.934206i	$0.383890\pi$
$90$	0.156325	0.0164781
$91$	−4.80606	−0.503812
$92$	−11.2605	−1.17398
$93$	−3.03761	−0.314986
$94$	0.836381	0.0862661
$95$	4.77575	0.489981
$96$	−2.26916	−0.231595
$97$	13.1187	1.33200	0.666002	−	0.745950i	$-0.268004\pi$
$97$	13.1187	1.33200	0.666002	+	0.745950i	$0.268004\pi$
$98$	−0.193937	−0.0195906
$99$	−2.15633	−0.216719
$100$	8.53690	0.853690
$101$	6.18664	0.615594	0.307797	−	0.951452i	$-0.400408\pi$
$101$	6.18664	0.615594	0.307797	+	0.951452i	$0.400408\pi$
$102$	1.35026	0.133696
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	3.69323	0.362151
$105$	0.806063	0.0786637
$106$	−0.201231	−0.0195453
$107$	−12.0811	−1.16792	−0.583962	−	0.811781i	$-0.698498\pi$
$107$	−12.0811	−1.16792	−0.583962	+	0.811781i	$0.698498\pi$
$108$	−1.96239	−0.188831
$109$	17.8496	1.70968	0.854839	−	0.518894i	$-0.173656\pi$
$109$	17.8496	1.70968	0.854839	+	0.518894i	$0.173656\pi$
$110$	−0.337088	−0.0321401
$111$	10.0000	0.949158
$112$	−3.77575	−0.356774
$113$	11.7988	1.10993	0.554967	−	0.831872i	$-0.312731\pi$
$113$	11.7988	1.10993	0.554967	+	0.831872i	$0.312731\pi$
$114$	1.14903	0.107617
$115$	−4.62530	−0.431312
$116$	14.2765	1.32554
$117$	4.80606	0.444321
$118$	0.373285	0.0343636
$119$	6.96239	0.638241
$120$	−0.619421	−0.0565452
$121$	−6.35026	−0.577297
$122$	2.69464	0.243962
$123$	11.5066	1.03751
$124$	5.96097	0.535311
$125$	7.53690	0.674121
$126$	0.193937	0.0172772
$127$	−6.61942	−0.587379	−0.293689	−	0.955901i	$-0.594883\pi$
$127$	−6.61942	−0.587379	−0.293689	+	0.955901i	$0.594883\pi$
$128$	5.91748	0.523037
$129$	2.38787	0.210241
$130$	0.751309	0.0658941
$131$	−12.6253	−1.10308	−0.551539	−	0.834149i	$-0.685959\pi$
$131$	−12.6253	−1.10308	−0.551539	+	0.834149i	$0.685959\pi$
$132$	4.23155	0.368309
$133$	5.92478	0.513743
$134$	3.01317	0.260299
$135$	−0.806063	−0.0693749
$136$	−5.35026	−0.458781
$137$	5.22425	0.446338	0.223169	−	0.974780i	$-0.428360\pi$
$137$	5.22425	0.446338	0.223169	+	0.974780i	$0.428360\pi$
$138$	−1.11283	−0.0947307
$139$	2.26187	0.191849	0.0959244	−	0.995389i	$-0.469419\pi$
$139$	2.26187	0.191849	0.0959244	+	0.995389i	$0.469419\pi$
$140$	−1.58181	−0.133687
$141$	−4.31265	−0.363191
$142$	−2.11142	−0.177186
$143$	−10.3634	−0.866634
$144$	3.77575	0.314646
$145$	5.86414	0.486991
$146$	−0.237428	−0.0196496
$147$	1.00000	0.0824786
$148$	−19.6239	−1.61307
$149$	−12.0508	−0.987239	−0.493619	−	0.869678i	$-0.664326\pi$
$149$	−12.0508	−0.987239	−0.493619	+	0.869678i	$0.664326\pi$
$150$	0.843675	0.0688858
$151$	−17.8437	−1.45210	−0.726049	−	0.687643i	$-0.758645\pi$
$151$	−17.8437	−1.45210	−0.726049	+	0.687643i	$0.758645\pi$
$152$	−4.55291	−0.369290
$153$	−6.96239	−0.562876
$154$	−0.418190	−0.0336987
$155$	2.44851	0.196669
$156$	−9.43136	−0.755113
$157$	9.05808	0.722913	0.361457	−	0.932389i	$-0.382280\pi$
$157$	9.05808	0.722913	0.361457	+	0.932389i	$0.382280\pi$
$158$	1.50659	0.119858
$159$	1.03761	0.0822879
$160$	1.82909	0.144602
$161$	−5.73813	−0.452228
$162$	−0.193937	−0.0152371
$163$	20.3938	1.59736	0.798681	−	0.601755i	$-0.205531\pi$
$163$	20.3938	1.59736	0.798681	+	0.601755i	$0.205531\pi$
$164$	−22.5804	−1.76323
$165$	1.73813	0.135314
$166$	−1.40105	−0.108742
$167$	−0.775746	−0.0600290	−0.0300145	−	0.999549i	$-0.509555\pi$
$167$	−0.775746	−0.0600290	−0.0300145	+	0.999549i	$0.509555\pi$
$168$	−0.768452	−0.0592874
$169$	10.0982	0.776788
$170$	−1.08840	−0.0834762
$171$	−5.92478	−0.453079
$172$	−4.68594	−0.357299
$173$	−0.574515	−0.0436796	−0.0218398	−	0.999761i	$-0.506952\pi$
$173$	−0.574515	−0.0436796	−0.0218398	+	0.999761i	$0.506952\pi$
$174$	1.41090	0.106960
$175$	4.35026	0.328849
$176$	−8.14174	−0.613706
$177$	−1.92478	−0.144675
$178$	−1.30536	−0.0978406
$179$	14.7816	1.10483	0.552415	−	0.833569i	$-0.313706\pi$
$179$	14.7816	1.10483	0.552415	+	0.833569i	$0.313706\pi$
$180$	1.58181	0.117901
$181$	1.26916	0.0943359	0.0471679	−	0.998887i	$-0.484980\pi$
$181$	1.26916	0.0943359	0.0471679	+	0.998887i	$0.484980\pi$
$182$	0.932071	0.0690897
$183$	−13.8945	−1.02711
$184$	4.40948	0.325071
$185$	−8.06063	−0.592630
$186$	0.589104	0.0431952
$187$	15.0132	1.09787
$188$	8.46310	0.617235
$189$	−1.00000	−0.0727393
$190$	−0.926192	−0.0671930
$191$	5.00729	0.362315	0.181158	−	0.983454i	$-0.442016\pi$
$191$	5.00729	0.362315	0.181158	+	0.983454i	$0.442016\pi$
$192$	−7.11142	−0.513222
$193$	−22.7513	−1.63768	−0.818838	−	0.574025i	$-0.805381\pi$
$193$	−22.7513	−1.63768	−0.818838	+	0.574025i	$0.805381\pi$
$194$	−2.54420	−0.182663
$195$	−3.87399	−0.277422
$196$	−1.96239	−0.140171
$197$	4.26187	0.303645	0.151823	−	0.988408i	$-0.451486\pi$
$197$	4.26187	0.303645	0.151823	+	0.988408i	$0.451486\pi$
$198$	0.418190	0.0297195
$199$	19.9756	1.41603	0.708015	−	0.706197i	$-0.249591\pi$
$199$	19.9756	1.41603	0.708015	+	0.706197i	$0.249591\pi$
$200$	−3.34297	−0.236384
$201$	−15.5369	−1.09589
$202$	−1.19982	−0.0844188
$203$	7.27504	0.510608
$204$	13.6629	0.956595
$205$	−9.27504	−0.647797
$206$	0	0
$207$	5.73813	0.398828
$208$	18.1465	1.25823
$209$	12.7757	0.883717
$210$	−0.156325	−0.0107875
$211$	4.39375	0.302478	0.151239	−	0.988497i	$-0.451674\pi$
$211$	4.39375	0.302478	0.151239	+	0.988497i	$0.451674\pi$
$212$	−2.03620	−0.139847
$213$	10.8872	0.745976
$214$	2.34297	0.160162
$215$	−1.92478	−0.131269
$216$	0.768452	0.0522865
$217$	3.03761	0.206206
$218$	−3.46168	−0.234455
$219$	1.22425	0.0827274
$220$	−3.41090	−0.229963
$221$	−33.4617	−2.25088
$222$	−1.93937	−0.130162
$223$	−8.18664	−0.548218	−0.274109	−	0.961699i	$-0.588383\pi$
$223$	−8.18664	−0.548218	−0.274109	+	0.961699i	$0.588383\pi$
$224$	2.26916	0.151615
$225$	−4.35026	−0.290017
$226$	−2.28821	−0.152210
$227$	12.4387	0.825583	0.412791	−	0.910826i	$-0.364554\pi$
$227$	12.4387	0.825583	0.412791	+	0.910826i	$0.364554\pi$
$228$	11.6267	0.769998
$229$	−5.47627	−0.361882	−0.180941	−	0.983494i	$-0.557914\pi$
$229$	−5.47627	−0.361882	−0.180941	+	0.983494i	$0.557914\pi$
$230$	0.897015	0.0591474
$231$	2.15633	0.141876
$232$	−5.59052	−0.367036
$233$	−9.66291	−0.633038	−0.316519	−	0.948586i	$-0.602514\pi$
$233$	−9.66291	−0.633038	−0.316519	+	0.948586i	$0.602514\pi$
$234$	−0.932071	−0.0609314
$235$	3.47627	0.226767
$236$	3.77716	0.245872
$237$	−7.76845	−0.504615
$238$	−1.35026	−0.0875245
$239$	18.7816	1.21488	0.607441	−	0.794365i	$-0.292196\pi$
$239$	18.7816	1.21488	0.607441	+	0.794365i	$0.292196\pi$
$240$	−3.04349	−0.196456
$241$	10.1055	0.650955	0.325478	−	0.945550i	$-0.394475\pi$
$241$	10.1055	0.650955	0.325478	+	0.945550i	$0.394475\pi$
$242$	1.23155	0.0791669
$243$	1.00000	0.0641500
$244$	27.2663	1.74555
$245$	−0.806063	−0.0514975
$246$	−2.23155	−0.142278
$247$	−28.4749	−1.81181
$248$	−2.33426	−0.148226
$249$	7.22425	0.457818
$250$	−1.46168	−0.0924448
$251$	1.14903	0.0725262	0.0362631	−	0.999342i	$-0.488455\pi$
$251$	1.14903	0.0725262	0.0362631	+	0.999342i	$0.488455\pi$
$252$	1.96239	0.123619
$253$	−12.3733	−0.777902
$254$	1.28375	0.0805495
$255$	5.61213	0.351445
$256$	13.0752	0.817201
$257$	−2.41819	−0.150843	−0.0754213	−	0.997152i	$-0.524030\pi$
$257$	−2.41819	−0.150843	−0.0754213	+	0.997152i	$0.524030\pi$
$258$	−0.463096	−0.0288311
$259$	−10.0000	−0.621370
$260$	7.60228	0.471473
$261$	−7.27504	−0.450314
$262$	2.44851	0.151269
$263$	2.26187	0.139473	0.0697363	−	0.997565i	$-0.477784\pi$
$263$	2.26187	0.139473	0.0697363	+	0.997565i	$0.477784\pi$
$264$	−1.65703	−0.101983
$265$	−0.836381	−0.0513785
$266$	−1.14903	−0.0704516
$267$	6.73084	0.411921
$268$	30.4894	1.86244
$269$	22.2677	1.35769	0.678844	−	0.734282i	$-0.262481\pi$
$269$	22.2677	1.35769	0.678844	+	0.734282i	$0.262481\pi$
$270$	0.156325	0.00951365
$271$	26.5745	1.61429	0.807143	−	0.590355i	$-0.201012\pi$
$271$	26.5745	1.61429	0.807143	+	0.590355i	$0.201012\pi$
$272$	−26.2882	−1.59396
$273$	−4.80606	−0.290876
$274$	−1.01317	−0.0612081
$275$	9.38058	0.565670
$276$	−11.2605	−0.677799
$277$	−18.0000	−1.08152	−0.540758	−	0.841178i	$-0.681862\pi$
$277$	−18.0000	−1.08152	−0.540758	+	0.841178i	$0.681862\pi$
$278$	−0.438658	−0.0263090
$279$	−3.03761	−0.181857
$280$	0.619421	0.0370175
$281$	0.261865	0.0156216	0.00781078	−	0.999969i	$-0.497514\pi$
$281$	0.261865	0.0156216	0.00781078	+	0.999969i	$0.497514\pi$
$282$	0.836381	0.0498057
$283$	17.5271	1.04188	0.520938	−	0.853595i	$-0.325582\pi$
$283$	17.5271	1.04188	0.520938	+	0.853595i	$0.325582\pi$
$284$	−21.3649	−1.26777
$285$	4.77575	0.282891
$286$	2.00985	0.118845
$287$	−11.5066	−0.679212
$288$	−2.26916	−0.133711
$289$	31.4749	1.85146
$290$	−1.13727	−0.0667829
$291$	13.1187	0.769033
$292$	−2.40246	−0.140593
$293$	25.1392	1.46865	0.734323	−	0.678800i	$-0.237500\pi$
$293$	25.1392	1.46865	0.734323	+	0.678800i	$0.237500\pi$
$294$	−0.193937	−0.0113106
$295$	1.55149	0.0903314
$296$	7.68452	0.446654
$297$	−2.15633	−0.125123
$298$	2.33709	0.135384
$299$	27.5778	1.59487
$300$	8.53690	0.492878
$301$	−2.38787	−0.137635
$302$	3.46054	0.199132
$303$	6.18664	0.355413
$304$	−22.3705	−1.28303
$305$	11.1998	0.641300
$306$	1.35026	0.0771893
$307$	0.438658	0.0250356	0.0125178	−	0.999922i	$-0.496015\pi$
$307$	0.438658	0.0250356	0.0125178	+	0.999922i	$0.496015\pi$
$308$	−4.23155	−0.241115
$309$	0	0
$310$	−0.474855	−0.0269700
$311$	−10.7005	−0.606771	−0.303386	−	0.952868i	$-0.598117\pi$
$311$	−10.7005	−0.606771	−0.303386	+	0.952868i	$0.598117\pi$
$312$	3.69323	0.209088
$313$	18.9986	1.07386	0.536932	−	0.843626i	$-0.319583\pi$
$313$	18.9986	1.07386	0.536932	+	0.843626i	$0.319583\pi$
$314$	−1.75669	−0.0991359
$315$	0.806063	0.0454165
$316$	15.2447	0.857583
$317$	8.72496	0.490043	0.245021	−	0.969518i	$-0.421205\pi$
$317$	8.72496	0.490043	0.245021	+	0.969518i	$0.421205\pi$
$318$	−0.201231	−0.0112845
$319$	15.6873	0.878323
$320$	5.73226	0.320443
$321$	−12.0811	−0.674301
$322$	1.11283	0.0620158
$323$	41.2506	2.29524
$324$	−1.96239	−0.109022
$325$	−20.9076	−1.15975
$326$	−3.95509	−0.219052
$327$	17.8496	0.987083
$328$	8.84226	0.488232
$329$	4.31265	0.237764
$330$	−0.337088	−0.0185561
$331$	12.1622	0.668495	0.334248	−	0.942485i	$-0.391518\pi$
$331$	12.1622	0.668495	0.334248	+	0.942485i	$0.391518\pi$
$332$	−14.1768	−0.778053
$333$	10.0000	0.547997
$334$	0.150446	0.00823202
$335$	12.5237	0.684244
$336$	−3.77575	−0.205984
$337$	27.0884	1.47560	0.737799	−	0.675020i	$-0.235865\pi$
$337$	27.0884	1.47560	0.737799	+	0.675020i	$0.235865\pi$
$338$	−1.95842	−0.106524
$339$	11.7988	0.640821
$340$	−11.0132	−0.597273
$341$	6.55008	0.354707
$342$	1.14903	0.0621325
$343$	−1.00000	−0.0539949
$344$	1.83497	0.0989347
$345$	−4.62530	−0.249018
$346$	0.111420	0.00598995
$347$	29.3317	1.57461	0.787304	−	0.616565i	$-0.211476\pi$
$347$	29.3317	1.57461	0.787304	+	0.616565i	$0.211476\pi$
$348$	14.2765	0.765298
$349$	1.89446	0.101408	0.0507041	−	0.998714i	$-0.483853\pi$
$349$	1.89446	0.101408	0.0507041	+	0.998714i	$0.483853\pi$
$350$	−0.843675	−0.0450963
$351$	4.80606	0.256529
$352$	4.89305	0.260800
$353$	−2.96239	−0.157672	−0.0788360	−	0.996888i	$-0.525120\pi$
$353$	−2.96239	−0.157672	−0.0788360	+	0.996888i	$0.525120\pi$
$354$	0.373285	0.0198399
$355$	−8.77575	−0.465768
$356$	−13.2085	−0.700050
$357$	6.96239	0.368489
$358$	−2.86670	−0.151510
$359$	−5.16950	−0.272836	−0.136418	−	0.990651i	$-0.543559\pi$
$359$	−5.16950	−0.272836	−0.136418	+	0.990651i	$0.543559\pi$
$360$	−0.619421	−0.0326464
$361$	16.1030	0.847526
$362$	−0.246136	−0.0129366
$363$	−6.35026	−0.333302
$364$	9.43136	0.494338
$365$	−0.986826	−0.0516528
$366$	2.69464	0.140851
$367$	−36.1378	−1.88638	−0.943188	−	0.332259i	$-0.892189\pi$
$367$	−36.1378	−1.88638	−0.943188	+	0.332259i	$0.892189\pi$
$368$	21.6657	1.12940
$369$	11.5066	0.599009
$370$	1.56325	0.0812696
$371$	−1.03761	−0.0538701
$372$	5.96097	0.309062
$373$	−14.7757	−0.765059	−0.382530	−	0.923943i	$-0.624947\pi$
$373$	−14.7757	−0.765059	−0.382530	+	0.923943i	$0.624947\pi$
$374$	−2.91160	−0.150555
$375$	7.53690	0.389204
$376$	−3.31406	−0.170910
$377$	−34.9643	−1.80075
$378$	0.193937	0.00997502
$379$	22.3185	1.14643	0.573213	−	0.819406i	$-0.305697\pi$
$379$	22.3185	1.14643	0.573213	+	0.819406i	$0.305697\pi$
$380$	−9.37187	−0.480767
$381$	−6.61942	−0.339123
$382$	−0.971097	−0.0496857
$383$	−1.00000	−0.0510976
$383$	−1.00000	−0.0510976
$384$	5.91748	0.301975
$385$	−1.73813	−0.0885836
$386$	4.41231	0.224581
$387$	2.38787	0.121382
$388$	−25.7440	−1.30695
$389$	−23.5223	−1.19263	−0.596314	−	0.802751i	$-0.703369\pi$
$389$	−23.5223	−1.19263	−0.596314	+	0.802751i	$0.703369\pi$
$390$	0.751309	0.0380440
$391$	−39.9511	−2.02042
$392$	0.768452	0.0388127
$393$	−12.6253	−0.636862
$394$	−0.826531	−0.0416401
$395$	6.26187	0.315069
$396$	4.23155	0.212643
$397$	−10.1016	−0.506983	−0.253492	−	0.967338i	$-0.581579\pi$
$397$	−10.1016	−0.506983	−0.253492	+	0.967338i	$0.581579\pi$
$398$	−3.87399	−0.194186
$399$	5.92478	0.296610
$400$	−16.4255	−0.821274
$401$	33.1998	1.65792	0.828960	−	0.559308i	$-0.188933\pi$
$401$	33.1998	1.65792	0.828960	+	0.559308i	$0.188933\pi$
$402$	3.01317	0.150283
$403$	−14.5990	−0.727226
$404$	−12.1406	−0.604017
$405$	−0.806063	−0.0400536
$406$	−1.41090	−0.0700216
$407$	−21.5633	−1.06885
$408$	−5.35026	−0.264877
$409$	14.9829	0.740855	0.370427	−	0.928861i	$-0.379211\pi$
$409$	14.9829	0.740855	0.370427	+	0.928861i	$0.379211\pi$
$410$	1.79877	0.0888349
$411$	5.22425	0.257693
$412$	0	0
$413$	1.92478	0.0947121
$414$	−1.11283	−0.0546928
$415$	−5.82321	−0.285850
$416$	−10.9057	−0.534697
$417$	2.26187	0.110764
$418$	−2.47768	−0.121188
$419$	40.4749	1.97733	0.988663	−	0.150151i	$-0.0479761\pi$
$419$	40.4749	1.97733	0.988663	+	0.150151i	$0.0479761\pi$
$420$	−1.58181	−0.0771844
$421$	17.2243	0.839459	0.419729	−	0.907649i	$-0.362125\pi$
$421$	17.2243	0.839459	0.419729	+	0.907649i	$0.362125\pi$
$422$	−0.852109	−0.0414800
$423$	−4.31265	−0.209688
$424$	0.797355	0.0387230
$425$	30.2882	1.46919
$426$	−2.11142	−0.102299
$427$	13.8945	0.672400
$428$	23.7078	1.14596
$429$	−10.3634	−0.500351
$430$	0.373285	0.0180014
$431$	−12.1260	−0.584089	−0.292045	−	0.956405i	$-0.594336\pi$
$431$	−12.1260	−0.584089	−0.292045	+	0.956405i	$0.594336\pi$
$432$	3.77575	0.181661
$433$	1.22425	0.0588339	0.0294169	−	0.999567i	$-0.490635\pi$
$433$	1.22425	0.0588339	0.0294169	+	0.999567i	$0.490635\pi$
$434$	−0.589104	−0.0282779
$435$	5.86414	0.281164
$436$	−35.0278	−1.67753
$437$	−33.9972	−1.62631
$438$	−0.237428	−0.0113447
$439$	−28.2882	−1.35012	−0.675061	−	0.737762i	$-0.735883\pi$
$439$	−28.2882	−1.35012	−0.675061	+	0.737762i	$0.735883\pi$
$440$	1.33567	0.0636757
$441$	1.00000	0.0476190
$442$	6.48944	0.308671
$443$	−1.90431	−0.0904765	−0.0452382	−	0.998976i	$-0.514405\pi$
$443$	−1.90431	−0.0904765	−0.0452382	+	0.998976i	$0.514405\pi$
$444$	−19.6239	−0.931308
$445$	−5.42548	−0.257193
$446$	1.58769	0.0751793
$447$	−12.0508	−0.569983
$448$	7.11142	0.335983
$449$	25.9756	1.22586	0.612931	−	0.790136i	$-0.289990\pi$
$449$	25.9756	1.22586	0.612931	+	0.790136i	$0.289990\pi$
$450$	0.843675	0.0397712
$451$	−24.8119	−1.16835
$452$	−23.1538	−1.08906
$453$	−17.8437	−0.838369
$454$	−2.41231	−0.113215
$455$	3.87399	0.181616
$456$	−4.55291	−0.213209
$457$	21.8496	1.02208	0.511040	−	0.859557i	$-0.329261\pi$
$457$	21.8496	1.02208	0.511040	+	0.859557i	$0.329261\pi$
$458$	1.06205	0.0496263
$459$	−6.96239	−0.324976
$460$	9.07664	0.423200
$461$	4.09095	0.190535	0.0952673	−	0.995452i	$-0.469629\pi$
$461$	4.09095	0.190535	0.0952673	+	0.995452i	$0.469629\pi$
$462$	−0.418190	−0.0194560
$463$	−10.9927	−0.510874	−0.255437	−	0.966826i	$-0.582219\pi$
$463$	−10.9927	−0.510874	−0.255437	+	0.966826i	$0.582219\pi$
$464$	−27.4687	−1.27520
$465$	2.44851	0.113547
$466$	1.87399	0.0868110
$467$	35.1392	1.62605	0.813024	−	0.582231i	$-0.197820\pi$
$467$	35.1392	1.62605	0.813024	+	0.582231i	$0.197820\pi$
$468$	−9.43136	−0.435965
$469$	15.5369	0.717428
$470$	−0.674176	−0.0310974
$471$	9.05808	0.417374
$472$	−1.47910	−0.0680810
$473$	−5.14903	−0.236753
$474$	1.50659	0.0691998
$475$	25.7743	1.18261
$476$	−13.6629	−0.626239
$477$	1.03761	0.0475090
$478$	−3.64244	−0.166602
$479$	16.6253	0.759629	0.379815	−	0.925063i	$-0.375988\pi$
$479$	16.6253	0.759629	0.379815	+	0.925063i	$0.375988\pi$
$480$	1.82909	0.0834860
$481$	48.0606	2.19138
$482$	−1.95983	−0.0892680
$483$	−5.73813	−0.261094
$484$	12.4617	0.566440
$485$	−10.5745	−0.480164
$486$	−0.193937	−0.00879714
$487$	16.8568	0.763857	0.381928	−	0.924192i	$-0.375260\pi$
$487$	16.8568	0.763857	0.381928	+	0.924192i	$0.375260\pi$
$488$	−10.6772	−0.483335
$489$	20.3938	0.922237
$490$	0.156325	0.00706205
$491$	−5.94921	−0.268484	−0.134242	−	0.990949i	$-0.542860\pi$
$491$	−5.94921	−0.268484	−0.134242	+	0.990949i	$0.542860\pi$
$492$	−22.5804	−1.01800
$493$	50.6516	2.28124
$494$	5.52232	0.248461
$495$	1.73813	0.0781234
$496$	−11.4692	−0.514985
$497$	−10.8872	−0.488356
$498$	−1.40105	−0.0627824
$499$	12.8364	0.574635	0.287318	−	0.957835i	$-0.407236\pi$
$499$	12.8364	0.574635	0.287318	+	0.957835i	$0.407236\pi$
$500$	−14.7903	−0.661444
$501$	−0.775746	−0.0346578
$502$	−0.222839	−0.00994580
$503$	−11.4156	−0.508998	−0.254499	−	0.967073i	$-0.581911\pi$
$503$	−11.4156	−0.508998	−0.254499	+	0.967073i	$0.581911\pi$
$504$	−0.768452	−0.0342296
$505$	−4.98683	−0.221911
$506$	2.39963	0.106677
$507$	10.0982	0.448479
$508$	12.9899	0.576333
$509$	8.88717	0.393917	0.196958	−	0.980412i	$-0.436894\pi$
$509$	8.88717	0.393917	0.196958	+	0.980412i	$0.436894\pi$
$510$	−1.08840	−0.0481950
$511$	−1.22425	−0.0541578
$512$	−14.3707	−0.635103
$513$	−5.92478	−0.261585
$514$	0.468976	0.0206856
$515$	0	0
$516$	−4.68594	−0.206287
$517$	9.29948	0.408991
$518$	1.93937	0.0852108
$519$	−0.574515	−0.0252184
$520$	−2.97698	−0.130549
$521$	23.3766	1.02415	0.512074	−	0.858941i	$-0.328877\pi$
$521$	23.3766	1.02415	0.512074	+	0.858941i	$0.328877\pi$
$522$	1.41090	0.0617532
$523$	−21.9902	−0.961562	−0.480781	−	0.876841i	$-0.659647\pi$
$523$	−21.9902	−0.961562	−0.480781	+	0.876841i	$0.659647\pi$
$524$	24.7757	1.08233
$525$	4.35026	0.189861
$526$	−0.438658	−0.0191264
$527$	21.1490	0.921266
$528$	−8.14174	−0.354324
$529$	9.92619	0.431574
$530$	0.162205	0.00704573
$531$	−1.92478	−0.0835282
$532$	−11.6267	−0.504082
$533$	55.3014	2.39537
$534$	−1.30536	−0.0564883
$535$	9.73813	0.421016
$536$	−11.9394	−0.515702
$537$	14.7816	0.637874
$538$	−4.31853	−0.186185
$539$	−2.15633	−0.0928795
$540$	1.58181	0.0680703
$541$	21.6385	0.930311	0.465155	−	0.885229i	$-0.345998\pi$
$541$	21.6385	0.930311	0.465155	+	0.885229i	$0.345998\pi$
$542$	−5.15377	−0.221373
$543$	1.26916	0.0544648
$544$	15.7988	0.677367
$545$	−14.3879	−0.616309
$546$	0.932071	0.0398890
$547$	19.2447	0.822845	0.411422	−	0.911445i	$-0.365032\pi$
$547$	19.2447	0.822845	0.411422	+	0.911445i	$0.365032\pi$
$548$	−10.2520	−0.437944
$549$	−13.8945	−0.593001
$550$	−1.81924	−0.0775725
$551$	43.1030	1.83625
$552$	4.40948	0.187680
$553$	7.76845	0.330348
$554$	3.49086	0.148312
$555$	−8.06063	−0.342155
$556$	−4.43866	−0.188241
$557$	25.8496	1.09528	0.547640	−	0.836714i	$-0.315526\pi$
$557$	25.8496	1.09528	0.547640	+	0.836714i	$0.315526\pi$
$558$	0.589104	0.0249388
$559$	11.4763	0.485394
$560$	3.04349	0.128611
$561$	15.0132	0.633857
$562$	−0.0507852	−0.00214225
$563$	25.9248	1.09260	0.546300	−	0.837590i	$-0.316036\pi$
$563$	25.9248	1.09260	0.546300	+	0.837590i	$0.316036\pi$
$564$	8.46310	0.356361
$565$	−9.51056	−0.400112
$566$	−3.39914	−0.142876
$567$	−1.00000	−0.0419961
$568$	8.36626	0.351041
$569$	−31.2506	−1.31009	−0.655047	−	0.755588i	$-0.727351\pi$
$569$	−31.2506	−1.31009	−0.655047	+	0.755588i	$0.727351\pi$
$570$	−0.926192	−0.0387939
$571$	0.0811024	0.00339403	0.00169701	−	0.999999i	$-0.499460\pi$
$571$	0.0811024	0.00339403	0.00169701	+	0.999999i	$0.499460\pi$
$572$	20.3371	0.850336
$573$	5.00729	0.209183
$574$	2.23155	0.0931430
$575$	−24.9624	−1.04100
$576$	−7.11142	−0.296309
$577$	−35.1939	−1.46514	−0.732571	−	0.680690i	$-0.761680\pi$
$577$	−35.1939	−1.46514	−0.732571	+	0.680690i	$0.761680\pi$
$578$	−6.10413	−0.253898
$579$	−22.7513	−0.945512
$580$	−11.5077	−0.477832
$581$	−7.22425	−0.299713
$582$	−2.54420	−0.105460
$583$	−2.23743	−0.0926648
$584$	0.940780	0.0389298
$585$	−3.87399	−0.160170
$586$	−4.87541	−0.201401
$587$	−25.5877	−1.05612	−0.528058	−	0.849208i	$-0.677080\pi$
$587$	−25.5877	−1.05612	−0.528058	+	0.849208i	$0.677080\pi$
$588$	−1.96239	−0.0809275
$589$	17.9972	0.741561
$590$	−0.300891	−0.0123875
$591$	4.26187	0.175310
$592$	37.7575	1.55182
$593$	−37.6834	−1.54747	−0.773735	−	0.633509i	$-0.781614\pi$
$593$	−37.6834	−1.54747	−0.773735	+	0.633509i	$0.781614\pi$
$594$	0.418190	0.0171586
$595$	−5.61213	−0.230075
$596$	23.6483	0.968673
$597$	19.9756	0.817545
$598$	−5.34835	−0.218710
$599$	−38.6761	−1.58026	−0.790131	−	0.612938i	$-0.789988\pi$
$599$	−38.6761	−1.58026	−0.790131	+	0.612938i	$0.789988\pi$
$600$	−3.34297	−0.136476
$601$	40.4445	1.64977	0.824884	−	0.565303i	$-0.191241\pi$
$601$	40.4445	1.64977	0.824884	+	0.565303i	$0.191241\pi$
$602$	0.463096	0.0188744
$603$	−15.5369	−0.632712
$604$	35.0162	1.42479
$605$	5.11871	0.208105
$606$	−1.19982	−0.0487392
$607$	2.36344	0.0959289	0.0479644	−	0.998849i	$-0.484727\pi$
$607$	2.36344	0.0959289	0.0479644	+	0.998849i	$0.484727\pi$
$608$	13.4443	0.545237
$609$	7.27504	0.294799
$610$	−2.17205	−0.0879439
$611$	−20.7269	−0.838519
$612$	13.6629	0.552290
$613$	32.8021	1.32486	0.662432	−	0.749122i	$-0.269524\pi$
$613$	32.8021	1.32486	0.662432	+	0.749122i	$0.269524\pi$
$614$	−0.0850719	−0.00343322
$615$	−9.27504	−0.374006
$616$	1.65703	0.0667637
$617$	−34.1866	−1.37630	−0.688151	−	0.725567i	$-0.741577\pi$
$617$	−34.1866	−1.37630	−0.688151	+	0.725567i	$0.741577\pi$
$618$	0	0
$619$	−13.0230	−0.523439	−0.261720	−	0.965144i	$-0.584290\pi$
$619$	−13.0230	−0.523439	−0.261720	+	0.965144i	$0.584290\pi$
$620$	−4.80492	−0.192970
$621$	5.73813	0.230263
$622$	2.07522	0.0832089
$623$	−6.73084	−0.269665
$624$	18.1465	0.726440
$625$	15.6761	0.627043
$626$	−3.68452	−0.147263
$627$	12.7757	0.510214
$628$	−17.7755	−0.709319
$629$	−69.6239	−2.77609
$630$	−0.156325	−0.00622814
$631$	16.4749	0.655854	0.327927	−	0.944703i	$-0.393650\pi$
$631$	16.4749	0.655854	0.327927	+	0.944703i	$0.393650\pi$
$632$	−5.96968	−0.237461
$633$	4.39375	0.174636
$634$	−1.69209	−0.0672014
$635$	5.33567	0.211740
$636$	−2.03620	−0.0807405
$637$	4.80606	0.190423
$638$	−3.04235	−0.120448
$639$	10.8872	0.430690
$640$	−4.76987	−0.188546
$641$	14.1016	0.556979	0.278489	−	0.960439i	$-0.410166\pi$
$641$	14.1016	0.556979	0.278489	+	0.960439i	$0.410166\pi$
$642$	2.34297	0.0924696
$643$	−36.8119	−1.45172	−0.725861	−	0.687842i	$-0.758558\pi$
$643$	−36.8119	−1.45172	−0.725861	+	0.687842i	$0.758558\pi$
$644$	11.2605	0.443724
$645$	−1.92478	−0.0757880
$646$	−8.00000	−0.314756
$647$	10.0752	0.396098	0.198049	−	0.980192i	$-0.436540\pi$
$647$	10.0752	0.396098	0.198049	+	0.980192i	$0.436540\pi$
$648$	0.768452	0.0301876
$649$	4.15045	0.162919
$650$	4.05475	0.159041
$651$	3.03761	0.119053
$652$	−40.0205	−1.56732
$653$	44.7777	1.75229	0.876143	−	0.482052i	$-0.160108\pi$
$653$	44.7777	1.75229	0.876143	+	0.482052i	$0.160108\pi$
$654$	−3.46168	−0.135362
$655$	10.1768	0.397640
$656$	43.4460	1.69628
$657$	1.22425	0.0477627
$658$	−0.836381	−0.0326055
$659$	6.11142	0.238067	0.119034	−	0.992890i	$-0.462020\pi$
$659$	6.11142	0.238067	0.119034	+	0.992890i	$0.462020\pi$
$660$	−3.41090	−0.132769
$661$	13.0132	0.506154	0.253077	−	0.967446i	$-0.418557\pi$
$661$	13.0132	0.506154	0.253077	+	0.967446i	$0.418557\pi$
$662$	−2.35870	−0.0916733
$663$	−33.4617	−1.29954
$664$	5.55149	0.215440
$665$	−4.77575	−0.185195
$666$	−1.93937	−0.0751489
$667$	−41.7452	−1.61638
$668$	1.52232	0.0589002
$669$	−8.18664	−0.316514
$670$	−2.42881	−0.0938331
$671$	29.9610	1.15663
$672$	2.26916	0.0875347
$673$	−48.2374	−1.85942	−0.929708	−	0.368297i	$-0.879941\pi$
$673$	−48.2374	−1.85942	−0.929708	+	0.368297i	$0.879941\pi$
$674$	−5.25343	−0.202355
$675$	−4.35026	−0.167442
$676$	−19.8167	−0.762180
$677$	32.3634	1.24383	0.621914	−	0.783086i	$-0.286355\pi$
$677$	32.3634	1.24383	0.621914	+	0.783086i	$0.286355\pi$
$678$	−2.28821	−0.0878783
$679$	−13.1187	−0.503450
$680$	4.31265	0.165383
$681$	12.4387	0.476650
$682$	−1.27030	−0.0486423
$683$	13.5271	0.517598	0.258799	−	0.965931i	$-0.416673\pi$
$683$	13.5271	0.517598	0.258799	+	0.965931i	$0.416673\pi$
$684$	11.6267	0.444559
$685$	−4.21108	−0.160897
$686$	0.193937	0.00740453
$687$	−5.47627	−0.208933
$688$	9.01600	0.343732
$689$	4.98683	0.189983
$690$	0.897015	0.0341488
$691$	13.9492	0.530653	0.265327	−	0.964159i	$-0.414520\pi$
$691$	13.9492	0.530653	0.265327	+	0.964159i	$0.414520\pi$
$692$	1.12742	0.0428582
$693$	2.15633	0.0819120
$694$	−5.68849	−0.215932
$695$	−1.82321	−0.0691582
$696$	−5.59052	−0.211908
$697$	−80.1133	−3.03451
$698$	−0.367405	−0.0139065
$699$	−9.66291	−0.365485
$700$	−8.53690	−0.322665
$701$	−29.4518	−1.11238	−0.556190	−	0.831055i	$-0.687737\pi$
$701$	−29.4518	−1.11238	−0.556190	+	0.831055i	$0.687737\pi$
$702$	−0.932071	−0.0351788
$703$	−59.2478	−2.23457
$704$	15.3345	0.577942
$705$	3.47627	0.130924
$706$	0.574515	0.0216222
$707$	−6.18664	−0.232673
$708$	3.77716	0.141954
$709$	−22.7757	−0.855361	−0.427681	−	0.903930i	$-0.640669\pi$
$709$	−22.7757	−0.855361	−0.427681	+	0.903930i	$0.640669\pi$
$710$	1.70194	0.0638726
$711$	−7.76845	−0.291340
$712$	5.17233	0.193841
$713$	−17.4302	−0.652767
$714$	−1.35026	−0.0505323
$715$	8.35359	0.312406
$716$	−29.0073	−1.08405
$717$	18.7816	0.701413
$718$	1.00255	0.0374150
$719$	−17.9248	−0.668481	−0.334241	−	0.942488i	$-0.608480\pi$
$719$	−17.9248	−0.668481	−0.334241	+	0.942488i	$0.608480\pi$
$720$	−3.04349	−0.113424
$721$	0	0
$722$	−3.12296	−0.116224
$723$	10.1055	0.375829
$724$	−2.49058	−0.0925618
$725$	31.6483	1.17539
$726$	1.23155	0.0457070
$727$	−34.3634	−1.27447	−0.637235	−	0.770670i	$-0.719922\pi$
$727$	−34.3634	−1.27447	−0.637235	+	0.770670i	$0.719922\pi$
$728$	−3.69323	−0.136880
$729$	1.00000	0.0370370
$730$	0.191382	0.00708335
$731$	−16.6253	−0.614909
$732$	27.2663	1.00779
$733$	35.1002	1.29645	0.648227	−	0.761447i	$-0.275511\pi$
$733$	35.1002	1.29645	0.648227	+	0.761447i	$0.275511\pi$
$734$	7.00843	0.258686
$735$	−0.806063	−0.0297321
$736$	−13.0207	−0.479951
$737$	33.5026	1.23408
$738$	−2.23155	−0.0821444
$739$	−30.4201	−1.11902	−0.559511	−	0.828823i	$-0.689011\pi$
$739$	−30.4201	−1.11902	−0.559511	+	0.828823i	$0.689011\pi$
$740$	15.8181	0.581485
$741$	−28.4749	−1.04605
$742$	0.201231	0.00738741
$743$	−2.83050	−0.103841	−0.0519205	−	0.998651i	$-0.516534\pi$
$743$	−2.83050	−0.103841	−0.0519205	+	0.998651i	$0.516534\pi$
$744$	−2.33426	−0.0855781
$745$	9.71370	0.355882
$746$	2.86556	0.104916
$747$	7.22425	0.264322
$748$	−29.4617	−1.07723
$749$	12.0811	0.441434
$750$	−1.46168	−0.0533731
$751$	22.3390	0.815162	0.407581	−	0.913169i	$-0.366372\pi$
$751$	22.3390	0.815162	0.407581	+	0.913169i	$0.366372\pi$
$752$	−16.2835	−0.593797
$753$	1.14903	0.0418730
$754$	6.78086	0.246944
$755$	14.3831	0.523456
$756$	1.96239	0.0713714
$757$	−45.5487	−1.65549	−0.827747	−	0.561101i	$-0.810378\pi$
$757$	−45.5487	−1.65549	−0.827747	+	0.561101i	$0.810378\pi$
$758$	−4.32838	−0.157214
$759$	−12.3733	−0.449122
$760$	3.66993	0.133122
$761$	25.5125	0.924826	0.462413	−	0.886665i	$-0.346984\pi$
$761$	25.5125	0.924826	0.462413	+	0.886665i	$0.346984\pi$
$762$	1.28375	0.0465053
$763$	−17.8496	−0.646197
$764$	−9.82626	−0.355502
$765$	5.61213	0.202907
$766$	0.193937	0.00700721
$767$	−9.25060	−0.334020
$768$	13.0752	0.471811
$769$	−46.2638	−1.66831	−0.834157	−	0.551527i	$-0.814045\pi$
$769$	−46.2638	−1.66831	−0.834157	+	0.551527i	$0.814045\pi$
$770$	0.337088	0.0121478
$771$	−2.41819	−0.0870890
$772$	44.6469	1.60688
$773$	33.3503	1.19953	0.599763	−	0.800178i	$-0.295262\pi$
$773$	33.3503	1.19953	0.599763	+	0.800178i	$0.295262\pi$
$774$	−0.463096	−0.0166456
$775$	13.2144	0.474675
$776$	10.0811	0.361890
$777$	−10.0000	−0.358748
$778$	4.56184	0.163550
$779$	−68.1740	−2.44259
$780$	7.60228	0.272205
$781$	−23.4763	−0.840047
$782$	7.74798	0.277067
$783$	−7.27504	−0.259989
$784$	3.77575	0.134848
$785$	−7.30139	−0.260598
$786$	2.44851	0.0873354
$787$	−8.84112	−0.315152	−0.157576	−	0.987507i	$-0.550368\pi$
$787$	−8.84112	−0.315152	−0.157576	+	0.987507i	$0.550368\pi$
$788$	−8.36344	−0.297935
$789$	2.26187	0.0805245
$790$	−1.21440	−0.0432066
$791$	−11.7988	−0.419516
$792$	−1.65703	−0.0588801
$793$	−66.7777	−2.37134
$794$	1.95906	0.0695246
$795$	−0.836381	−0.0296634
$796$	−39.1998	−1.38940
$797$	−38.1984	−1.35306	−0.676528	−	0.736417i	$-0.736516\pi$
$797$	−38.1984	−1.35306	−0.676528	+	0.736417i	$0.736516\pi$
$798$	−1.14903	−0.0406753
$799$	30.0263	1.06226
$800$	9.87144	0.349008
$801$	6.73084	0.237823
$802$	−6.43866	−0.227357
$803$	−2.63989	−0.0931597
$804$	30.4894	1.07528
$805$	4.62530	0.163020
$806$	2.83127	0.0997273
$807$	22.2677	0.783862
$808$	4.75414	0.167250
$809$	9.50071	0.334027	0.167014	−	0.985955i	$-0.446588\pi$
$809$	9.50071	0.334027	0.167014	+	0.985955i	$0.446588\pi$
$810$	0.156325	0.00549271
$811$	−25.7090	−0.902764	−0.451382	−	0.892331i	$-0.649069\pi$
$811$	−25.7090	−0.902764	−0.451382	+	0.892331i	$0.649069\pi$
$812$	−14.2765	−0.501005
$813$	26.5745	0.932009
$814$	4.18190	0.146576
$815$	−16.4387	−0.575821
$816$	−26.2882	−0.920272
$817$	−14.1476	−0.494962
$818$	−2.90572	−0.101596
$819$	−4.80606	−0.167937
$820$	18.2012	0.635615
$821$	−10.8002	−0.376929	−0.188465	−	0.982080i	$-0.560351\pi$
$821$	−10.8002	−0.376929	−0.188465	+	0.982080i	$0.560351\pi$
$822$	−1.01317	−0.0353385
$823$	15.4763	0.539469	0.269734	−	0.962935i	$-0.413064\pi$
$823$	15.4763	0.539469	0.269734	+	0.962935i	$0.413064\pi$
$824$	0	0
$825$	9.38058	0.326590
$826$	−0.373285	−0.0129882
$827$	−4.29218	−0.149254	−0.0746269	−	0.997212i	$-0.523777\pi$
$827$	−4.29218	−0.149254	−0.0746269	+	0.997212i	$0.523777\pi$
$828$	−11.2605	−0.391328
$829$	49.3258	1.71316	0.856578	−	0.516017i	$-0.172586\pi$
$829$	49.3258	1.71316	0.856578	+	0.516017i	$0.172586\pi$
$830$	1.12933	0.0391997
$831$	−18.0000	−0.624413
$832$	−34.1779	−1.18491
$833$	−6.96239	−0.241232
$834$	−0.438658	−0.0151895
$835$	0.625301	0.0216394
$836$	−25.0710	−0.867098
$837$	−3.03761	−0.104995
$838$	−7.84955	−0.271158
$839$	40.8773	1.41124	0.705621	−	0.708590i	$-0.250668\pi$
$839$	40.8773	1.41124	0.705621	+	0.708590i	$0.250668\pi$
$840$	0.619421	0.0213721
$841$	23.9262	0.825041
$842$	−3.34041	−0.115118
$843$	0.261865	0.00901911
$844$	−8.62225	−0.296790
$845$	−8.13983	−0.280019
$846$	0.836381	0.0287554
$847$	6.35026	0.218198
$848$	3.91776	0.134536
$849$	17.5271	0.601527
$850$	−5.87399	−0.201476
$851$	57.3813	1.96701
$852$	−21.3649	−0.731948
$853$	−33.9511	−1.16246	−0.581232	−	0.813738i	$-0.697429\pi$
$853$	−33.9511	−1.16246	−0.581232	+	0.813738i	$0.697429\pi$
$854$	−2.69464	−0.0922088
$855$	4.77575	0.163327
$856$	−9.28375	−0.317312
$857$	44.2941	1.51306	0.756529	−	0.653961i	$-0.226894\pi$
$857$	44.2941	1.51306	0.756529	+	0.653961i	$0.226894\pi$
$858$	2.00985	0.0686151
$859$	−3.13918	−0.107108	−0.0535538	−	0.998565i	$-0.517055\pi$
$859$	−3.13918	−0.107108	−0.0535538	+	0.998565i	$0.517055\pi$
$860$	3.77716	0.128800
$861$	−11.5066	−0.392143
$862$	2.35168	0.0800984
$863$	20.3331	0.692147	0.346074	−	0.938207i	$-0.387515\pi$
$863$	20.3331	0.692147	0.346074	+	0.938207i	$0.387515\pi$
$864$	−2.26916	−0.0771984
$865$	0.463096	0.0157457
$866$	−0.237428	−0.00806812
$867$	31.4749	1.06894
$868$	−5.96097	−0.202329
$869$	16.7513	0.568249
$870$	−1.13727	−0.0385571
$871$	−74.6713	−2.53014
$872$	13.7165	0.464500
$873$	13.1187	0.444001
$874$	6.59329	0.223022
$875$	−7.53690	−0.254794
$876$	−2.40246	−0.0811717
$877$	−30.1016	−1.01646	−0.508229	−	0.861222i	$-0.669700\pi$
$877$	−30.1016	−1.01646	−0.508229	+	0.861222i	$0.669700\pi$
$878$	5.48612	0.185148
$879$	25.1392	0.847924
$880$	6.56276	0.221230
$881$	20.1925	0.680303	0.340152	−	0.940371i	$-0.389522\pi$
$881$	20.1925	0.680303	0.340152	+	0.940371i	$0.389522\pi$
$882$	−0.193937	−0.00653018
$883$	4.75528	0.160028	0.0800139	−	0.996794i	$-0.474504\pi$
$883$	4.75528	0.160028	0.0800139	+	0.996794i	$0.474504\pi$
$884$	65.6648	2.20855
$885$	1.55149	0.0521529
$886$	0.369315	0.0124074
$887$	11.1754	0.375232	0.187616	−	0.982242i	$-0.439924\pi$
$887$	11.1754	0.375232	0.187616	+	0.982242i	$0.439924\pi$
$888$	7.68452	0.257876
$889$	6.61942	0.222008
$890$	1.05220	0.0352698
$891$	−2.15633	−0.0722396
$892$	16.0654	0.537908
$893$	25.5515	0.855048
$894$	2.33709	0.0781639
$895$	−11.9149	−0.398272
$896$	−5.91748	−0.197689
$897$	27.5778	0.920797
$898$	−5.03761	−0.168107
$899$	22.0987	0.737034
$900$	8.53690	0.284563
$901$	−7.22425	−0.240675
$902$	4.81194	0.160220
$903$	−2.38787	−0.0794634
$904$	9.06679	0.301557
$905$	−1.02302	−0.0340064
$906$	3.46054	0.114969
$907$	−45.3317	−1.50521	−0.752607	−	0.658470i	$-0.771204\pi$
$907$	−45.3317	−1.50521	−0.752607	+	0.658470i	$0.771204\pi$
$908$	−24.4095	−0.810057
$909$	6.18664	0.205198
$910$	−0.751309	−0.0249056
$911$	−11.0943	−0.367570	−0.183785	−	0.982966i	$-0.558835\pi$
$911$	−11.0943	−0.367570	−0.183785	+	0.982966i	$0.558835\pi$
$912$	−22.3705	−0.740760
$913$	−15.5778	−0.515551
$914$	−4.23743	−0.140162
$915$	11.1998	0.370255
$916$	10.7466	0.355077
$917$	12.6253	0.416924
$918$	1.35026	0.0445653
$919$	−24.0409	−0.793037	−0.396519	−	0.918027i	$-0.629782\pi$
$919$	−24.0409	−0.793037	−0.396519	+	0.918027i	$0.629782\pi$
$920$	−3.55432	−0.117183
$921$	0.438658	0.0144543
$922$	−0.793385	−0.0261287
$923$	52.3244	1.72228
$924$	−4.23155	−0.139208
$925$	−43.5026	−1.43036
$926$	2.13189	0.0700582
$927$	0	0
$928$	16.5082	0.541909
$929$	−51.8310	−1.70052	−0.850260	−	0.526363i	$-0.823555\pi$
$929$	−51.8310	−1.70052	−0.850260	+	0.526363i	$0.823555\pi$
$930$	−0.474855	−0.0155711
$931$	−5.92478	−0.194177
$932$	18.9624	0.621134
$933$	−10.7005	−0.350319
$934$	−6.81477	−0.222986
$935$	−12.1016	−0.395764
$936$	3.69323	0.120717
$937$	5.83780	0.190712	0.0953562	−	0.995443i	$-0.469601\pi$
$937$	5.83780	0.190712	0.0953562	+	0.995443i	$0.469601\pi$
$938$	−3.01317	−0.0983836
$939$	18.9986	0.619995
$940$	−6.82179	−0.222502
$941$	−57.9267	−1.88836	−0.944178	−	0.329436i	$-0.893142\pi$
$941$	−57.9267	−1.88836	−0.944178	+	0.329436i	$0.893142\pi$
$942$	−1.75669	−0.0572361
$943$	66.0263	2.15011
$944$	−7.26747	−0.236536
$945$	0.806063	0.0262212
$946$	0.998585	0.0324668
$947$	−31.5574	−1.02548	−0.512738	−	0.858545i	$-0.671369\pi$
$947$	−31.5574	−1.02548	−0.512738	+	0.858545i	$0.671369\pi$
$948$	15.2447	0.495126
$949$	5.88384	0.190998
$950$	−4.99859	−0.162176
$951$	8.72496	0.282926
$952$	5.35026	0.173403
$953$	−52.1524	−1.68938	−0.844690	−	0.535255i	$-0.820215\pi$
$953$	−52.1524	−1.68938	−0.844690	+	0.535255i	$0.820215\pi$
$954$	−0.201231	−0.00651509
$955$	−4.03620	−0.130608
$956$	−36.8568	−1.19204
$957$	15.6873	0.507100
$958$	−3.22425	−0.104171
$959$	−5.22425	−0.168700
$960$	5.73226	0.185008
$961$	−21.7729	−0.702352
$962$	−9.32071	−0.300512
$963$	−12.0811	−0.389308
$964$	−19.8310	−0.638713
$965$	18.3390	0.590353
$966$	1.11283	0.0358049
$967$	−54.0381	−1.73775	−0.868874	−	0.495033i	$-0.835156\pi$
$967$	−54.0381	−1.73775	−0.868874	+	0.495033i	$0.835156\pi$
$968$	−4.87987	−0.156845
$969$	41.2506	1.32516
$970$	2.05079	0.0658467
$971$	42.8872	1.37631	0.688157	−	0.725562i	$-0.258420\pi$
$971$	42.8872	1.37631	0.688157	+	0.725562i	$0.258420\pi$
$972$	−1.96239	−0.0629436
$973$	−2.26187	−0.0725121
$974$	−3.26916	−0.104751
$975$	−20.9076	−0.669580
$976$	−52.4620	−1.67927
$977$	22.5402	0.721126	0.360563	−	0.932735i	$-0.382585\pi$
$977$	22.5402	0.721126	0.360563	+	0.932735i	$0.382585\pi$
$978$	−3.95509	−0.126470
$979$	−14.5139	−0.463866
$980$	1.58181	0.0505291
$981$	17.8496	0.569892
$982$	1.15377	0.0368183
$983$	10.1114	0.322504	0.161252	−	0.986913i	$-0.448447\pi$
$983$	10.1114	0.322504	0.161252	+	0.986913i	$0.448447\pi$
$984$	8.84226	0.281881
$985$	−3.43533	−0.109459
$986$	−9.82321	−0.312835
$987$	4.31265	0.137273
$988$	55.8787	1.77774
$989$	13.7019	0.435696
$990$	−0.337088	−0.0107134
$991$	25.9854	0.825454	0.412727	−	0.910855i	$-0.364576\pi$
$991$	25.9854	0.825454	0.412727	+	0.910855i	$0.364576\pi$
$992$	6.89282	0.218847
$993$	12.1622	0.385956
$994$	2.11142	0.0669702
$995$	−16.1016	−0.510454
$996$	−14.1768	−0.449209
$997$	2.94192	0.0931716	0.0465858	−	0.998914i	$-0.485166\pi$
$997$	2.94192	0.0931716	0.0465858	+	0.998914i	$0.485166\pi$
$998$	−2.48944	−0.0788020
$999$	10.0000	0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8043.2.a.m.1.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8043.2.a.m.1.2	✓	3	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:	\( N \)	\(=\)	\( 8043 = 3 \cdot 7 \cdot 383 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8043.a (trivial)

\(f(q)\)	\(=\)	\(q-0.193937 q^{2} +1.00000 q^{3} -1.96239 q^{4} -0.806063 q^{5} -0.193937 q^{6} -1.00000 q^{7} +0.768452 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.193937 q^{2} +1.00000 q^{3} -1.96239 q^{4} -0.806063 q^{5} -0.193937 q^{6} -1.00000 q^{7} +0.768452 q^{8} +1.00000 q^{9} +0.156325 q^{10} -2.15633 q^{11} -1.96239 q^{12} +4.80606 q^{13} +0.193937 q^{14} -0.806063 q^{15} +3.77575 q^{16} -6.96239 q^{17} -0.193937 q^{18} -5.92478 q^{19} +1.58181 q^{20} -1.00000 q^{21} +0.418190 q^{22} +5.73813 q^{23} +0.768452 q^{24} -4.35026 q^{25} -0.932071 q^{26} +1.00000 q^{27} +1.96239 q^{28} -7.27504 q^{29} +0.156325 q^{30} -3.03761 q^{31} -2.26916 q^{32} -2.15633 q^{33} +1.35026 q^{34} +0.806063 q^{35} -1.96239 q^{36} +10.0000 q^{37} +1.14903 q^{38} +4.80606 q^{39} -0.619421 q^{40} +11.5066 q^{41} +0.193937 q^{42} +2.38787 q^{43} +4.23155 q^{44} -0.806063 q^{45} -1.11283 q^{46} -4.31265 q^{47} +3.77575 q^{48} +1.00000 q^{49} +0.843675 q^{50} -6.96239 q^{51} -9.43136 q^{52} +1.03761 q^{53} -0.193937 q^{54} +1.73813 q^{55} -0.768452 q^{56} -5.92478 q^{57} +1.41090 q^{58} -1.92478 q^{59} +1.58181 q^{60} -13.8945 q^{61} +0.589104 q^{62} -1.00000 q^{63} -7.11142 q^{64} -3.87399 q^{65} +0.418190 q^{66} -15.5369 q^{67} +13.6629 q^{68} +5.73813 q^{69} -0.156325 q^{70} +10.8872 q^{71} +0.768452 q^{72} +1.22425 q^{73} -1.93937 q^{74} -4.35026 q^{75} +11.6267 q^{76} +2.15633 q^{77} -0.932071 q^{78} -7.76845 q^{79} -3.04349 q^{80} +1.00000 q^{81} -2.23155 q^{82} +7.22425 q^{83} +1.96239 q^{84} +5.61213 q^{85} -0.463096 q^{86} -7.27504 q^{87} -1.65703 q^{88} +6.73084 q^{89} +0.156325 q^{90} -4.80606 q^{91} -11.2605 q^{92} -3.03761 q^{93} +0.836381 q^{94} +4.77575 q^{95} -2.26916 q^{96} +13.1187 q^{97} -0.193937 q^{98} -2.15633 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + 3 q^{3} + 5 q^{4} - 2 q^{5} - q^{6} - 3 q^{7} - 9 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q - q^2 + 3 * q^3 + 5 * q^4 - 2 * q^5 - q^6 - 3 * q^7 - 9 * q^8 + 3 * q^9	\( 3 q - q^{2} + 3 q^{3} + 5 q^{4} - 2 q^{5} - q^{6} - 3 q^{7} - 9 q^{8} + 3 q^{9} - 10 q^{10} + 4 q^{11} + 5 q^{12} + 14 q^{13} + q^{14} - 2 q^{15} + 13 q^{16} - 10 q^{17} - q^{18} + 4 q^{19} + 6 q^{20} - 3 q^{21} + 8 q^{23} - 9 q^{24} - 3 q^{25} + 6 q^{26} + 3 q^{27} - 5 q^{28} + 10 q^{29} - 10 q^{30} - 20 q^{31} - 29 q^{32} + 4 q^{33} - 6 q^{34} + 2 q^{35} + 5 q^{36} + 30 q^{37} - 20 q^{38} + 14 q^{39} - 14 q^{40} + 14 q^{41} + q^{42} + 8 q^{43} + 24 q^{44} - 2 q^{45} - 36 q^{46} + 8 q^{47} + 13 q^{48} + 3 q^{49} + 13 q^{50} - 10 q^{51} + 14 q^{52} + 14 q^{53} - q^{54} - 4 q^{55} + 9 q^{56} + 4 q^{57} - 10 q^{58} + 16 q^{59} + 6 q^{60} - 22 q^{61} + 16 q^{62} - 3 q^{63} + 13 q^{64} - 20 q^{65} - 24 q^{67} + 10 q^{68} + 8 q^{69} + 10 q^{70} - 9 q^{72} + 2 q^{73} - 10 q^{74} - 3 q^{75} + 60 q^{76} - 4 q^{77} + 6 q^{78} - 12 q^{79} + 34 q^{80} + 3 q^{81} - 18 q^{82} + 20 q^{83} - 5 q^{84} + 16 q^{85} - 24 q^{86} + 10 q^{87} - 28 q^{88} - 2 q^{89} - 10 q^{90} - 14 q^{91} + 24 q^{92} - 20 q^{93} + 16 q^{95} - 29 q^{96} + 18 q^{97} - q^{98} + 4 q^{99}+O(q^{100}) \) 3 * q - q^2 + 3 * q^3 + 5 * q^4 - 2 * q^5 - q^6 - 3 * q^7 - 9 * q^8 + 3 * q^9 - 10 * q^10 + 4 * q^11 + 5 * q^12 + 14 * q^13 + q^14 - 2 * q^15 + 13 * q^16 - 10 * q^17 - q^18 + 4 * q^19 + 6 * q^20 - 3 * q^21 + 8 * q^23 - 9 * q^24 - 3 * q^25 + 6 * q^26 + 3 * q^27 - 5 * q^28 + 10 * q^29 - 10 * q^30 - 20 * q^31 - 29 * q^32 + 4 * q^33 - 6 * q^34 + 2 * q^35 + 5 * q^36 + 30 * q^37 - 20 * q^38 + 14 * q^39 - 14 * q^40 + 14 * q^41 + q^42 + 8 * q^43 + 24 * q^44 - 2 * q^45 - 36 * q^46 + 8 * q^47 + 13 * q^48 + 3 * q^49 + 13 * q^50 - 10 * q^51 + 14 * q^52 + 14 * q^53 - q^54 - 4 * q^55 + 9 * q^56 + 4 * q^57 - 10 * q^58 + 16 * q^59 + 6 * q^60 - 22 * q^61 + 16 * q^62 - 3 * q^63 + 13 * q^64 - 20 * q^65 - 24 * q^67 + 10 * q^68 + 8 * q^69 + 10 * q^70 - 9 * q^72 + 2 * q^73 - 10 * q^74 - 3 * q^75 + 60 * q^76 - 4 * q^77 + 6 * q^78 - 12 * q^79 + 34 * q^80 + 3 * q^81 - 18 * q^82 + 20 * q^83 - 5 * q^84 + 16 * q^85 - 24 * q^86 + 10 * q^87 - 28 * q^88 - 2 * q^89 - 10 * q^90 - 14 * q^91 + 24 * q^92 - 20 * q^93 + 16 * q^95 - 29 * q^96 + 18 * q^97 - q^98 + 4 * q^99

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