LMFDB - Embedded newform 8043.2.a.t.1.27

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:	$52$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.27
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)

$f(q)$	$=$	$q+0.0385832 q^{2} -1.00000 q^{3} -1.99851 q^{4} -4.10741 q^{5} -0.0385832 q^{6} +1.00000 q^{7} -0.154275 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+0.0385832 q^{2} -1.00000 q^{3} -1.99851 q^{4} -4.10741 q^{5} -0.0385832 q^{6} +1.00000 q^{7} -0.154275 q^{8} +1.00000 q^{9} -0.158477 q^{10} -1.32401 q^{11} +1.99851 q^{12} +0.762362 q^{13} +0.0385832 q^{14} +4.10741 q^{15} +3.99107 q^{16} -2.50757 q^{17} +0.0385832 q^{18} +6.20953 q^{19} +8.20870 q^{20} -1.00000 q^{21} -0.0510847 q^{22} +6.20294 q^{23} +0.154275 q^{24} +11.8708 q^{25} +0.0294143 q^{26} -1.00000 q^{27} -1.99851 q^{28} +3.02440 q^{29} +0.158477 q^{30} +1.34664 q^{31} +0.462538 q^{32} +1.32401 q^{33} -0.0967500 q^{34} -4.10741 q^{35} -1.99851 q^{36} -8.07420 q^{37} +0.239583 q^{38} -0.762362 q^{39} +0.633671 q^{40} -10.7169 q^{41} -0.0385832 q^{42} -8.50798 q^{43} +2.64606 q^{44} -4.10741 q^{45} +0.239329 q^{46} -11.5727 q^{47} -3.99107 q^{48} +1.00000 q^{49} +0.458013 q^{50} +2.50757 q^{51} -1.52359 q^{52} +0.514076 q^{53} -0.0385832 q^{54} +5.43827 q^{55} -0.154275 q^{56} -6.20953 q^{57} +0.116691 q^{58} +6.30138 q^{59} -8.20870 q^{60} +7.43246 q^{61} +0.0519577 q^{62} +1.00000 q^{63} -7.96429 q^{64} -3.13133 q^{65} +0.0510847 q^{66} +5.58079 q^{67} +5.01141 q^{68} -6.20294 q^{69} -0.158477 q^{70} -5.06029 q^{71} -0.154275 q^{72} -1.66869 q^{73} -0.311528 q^{74} -11.8708 q^{75} -12.4098 q^{76} -1.32401 q^{77} -0.0294143 q^{78} -5.84786 q^{79} -16.3930 q^{80} +1.00000 q^{81} -0.413492 q^{82} -2.54159 q^{83} +1.99851 q^{84} +10.2996 q^{85} -0.328265 q^{86} -3.02440 q^{87} +0.204263 q^{88} +4.55157 q^{89} -0.158477 q^{90} +0.762362 q^{91} -12.3967 q^{92} -1.34664 q^{93} -0.446510 q^{94} -25.5051 q^{95} -0.462538 q^{96} -5.81183 q^{97} +0.0385832 q^{98} -1.32401 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 52 q + 3 q^{2} - 52 q^{3} + 61 q^{4} - 7 q^{5} - 3 q^{6} + 52 q^{7} + 24 q^{8} + 52 q^{9}+O(q^{10}) $ 52 * q + 3 * q^2 - 52 * q^3 + 61 * q^4 - 7 * q^5 - 3 * q^6 + 52 * q^7 + 24 * q^8 + 52 * q^9	\( 52 q + 3 q^{2} - 52 q^{3} + 61 q^{4} - 7 q^{5} - 3 q^{6} + 52 q^{7} + 24 q^{8} + 52 q^{9} - 2 q^{10} + 9 q^{11} - 61 q^{12} + 44 q^{13} + 3 q^{14} + 7 q^{15} + 95 q^{16} - 6 q^{17} + 3 q^{18} + 7 q^{19} - 21 q^{20} - 52 q^{21} + 19 q^{22} - 4 q^{23} - 24 q^{24} + 83 q^{25} - 5 q^{26} - 52 q^{27} + 61 q^{28} + 31 q^{29} + 2 q^{30} + 11 q^{31} + 71 q^{32} - 9 q^{33} + 17 q^{34} - 7 q^{35} + 61 q^{36} + 71 q^{37} - 8 q^{38} - 44 q^{39} + 20 q^{40} - 25 q^{41} - 3 q^{42} + 75 q^{43} + 14 q^{44} - 7 q^{45} + 36 q^{46} - 20 q^{47} - 95 q^{48} + 52 q^{49} + 26 q^{50} + 6 q^{51} + 88 q^{52} + 70 q^{53} - 3 q^{54} + 12 q^{55} + 24 q^{56} - 7 q^{57} + 48 q^{58} - 27 q^{59} + 21 q^{60} + 59 q^{61} - 23 q^{62} + 52 q^{63} + 138 q^{64} + 44 q^{65} - 19 q^{66} + 65 q^{67} - 8 q^{68} + 4 q^{69} - 2 q^{70} - 11 q^{71} + 24 q^{72} + 34 q^{73} + 38 q^{74} - 83 q^{75} + 31 q^{76} + 9 q^{77} + 5 q^{78} + 74 q^{79} - 5 q^{80} + 52 q^{81} + 51 q^{82} - 30 q^{83} - 61 q^{84} + 70 q^{85} + 29 q^{86} - 31 q^{87} + 90 q^{88} - q^{89} - 2 q^{90} + 44 q^{91} + 34 q^{92} - 11 q^{93} + 27 q^{94} + 9 q^{95} - 71 q^{96} + 73 q^{97} + 3 q^{98} + 9 q^{99}+O(q^{100}) \) 52 * q + 3 * q^2 - 52 * q^3 + 61 * q^4 - 7 * q^5 - 3 * q^6 + 52 * q^7 + 24 * q^8 + 52 * q^9 - 2 * q^10 + 9 * q^11 - 61 * q^12 + 44 * q^13 + 3 * q^14 + 7 * q^15 + 95 * q^16 - 6 * q^17 + 3 * q^18 + 7 * q^19 - 21 * q^20 - 52 * q^21 + 19 * q^22 - 4 * q^23 - 24 * q^24 + 83 * q^25 - 5 * q^26 - 52 * q^27 + 61 * q^28 + 31 * q^29 + 2 * q^30 + 11 * q^31 + 71 * q^32 - 9 * q^33 + 17 * q^34 - 7 * q^35 + 61 * q^36 + 71 * q^37 - 8 * q^38 - 44 * q^39 + 20 * q^40 - 25 * q^41 - 3 * q^42 + 75 * q^43 + 14 * q^44 - 7 * q^45 + 36 * q^46 - 20 * q^47 - 95 * q^48 + 52 * q^49 + 26 * q^50 + 6 * q^51 + 88 * q^52 + 70 * q^53 - 3 * q^54 + 12 * q^55 + 24 * q^56 - 7 * q^57 + 48 * q^58 - 27 * q^59 + 21 * q^60 + 59 * q^61 - 23 * q^62 + 52 * q^63 + 138 * q^64 + 44 * q^65 - 19 * q^66 + 65 * q^67 - 8 * q^68 + 4 * q^69 - 2 * q^70 - 11 * q^71 + 24 * q^72 + 34 * q^73 + 38 * q^74 - 83 * q^75 + 31 * q^76 + 9 * q^77 + 5 * q^78 + 74 * q^79 - 5 * q^80 + 52 * q^81 + 51 * q^82 - 30 * q^83 - 61 * q^84 + 70 * q^85 + 29 * q^86 - 31 * q^87 + 90 * q^88 - q^89 - 2 * q^90 + 44 * q^91 + 34 * q^92 - 11 * q^93 + 27 * q^94 + 9 * q^95 - 71 * q^96 + 73 * q^97 + 3 * q^98 + 9 * 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 $

$2$	0.0385832	0.0272824	0.0136412	−	0.999907i	$-0.495658\pi$
$2$	0.0385832	0.0272824	0.0136412	+	0.999907i	$0.495658\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.99851	−0.999256
$5$	−4.10741	−1.83689	−0.918444	−	0.395550i	$-0.870554\pi$
$5$	−4.10741	−1.83689	−0.918444	+	0.395550i	$0.870554\pi$
$6$	−0.0385832	−0.0157515
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−0.154275	−0.0545445
$9$	1.00000	0.333333
$10$	−0.158477	−0.0501148
$11$	−1.32401	−0.399205	−0.199603	−	0.979877i	$-0.563965\pi$
$11$	−1.32401	−0.399205	−0.199603	+	0.979877i	$0.563965\pi$
$12$	1.99851	0.576921
$13$	0.762362	0.211441	0.105721	−	0.994396i	$-0.466285\pi$
$13$	0.762362	0.211441	0.105721	+	0.994396i	$0.466285\pi$
$14$	0.0385832	0.0103118
$15$	4.10741	1.06053
$16$	3.99107	0.997768
$17$	−2.50757	−0.608176	−0.304088	−	0.952644i	$-0.598352\pi$
$17$	−2.50757	−0.608176	−0.304088	+	0.952644i	$0.598352\pi$
$18$	0.0385832	0.00909414
$19$	6.20953	1.42456	0.712282	−	0.701893i	$-0.247662\pi$
$19$	6.20953	1.42456	0.712282	+	0.701893i	$0.247662\pi$
$20$	8.20870	1.83552
$21$	−1.00000	−0.218218
$22$	−0.0510847	−0.0108913
$23$	6.20294	1.29340	0.646701	−	0.762743i	$-0.276148\pi$
$23$	6.20294	1.29340	0.646701	+	0.762743i	$0.276148\pi$
$24$	0.154275	0.0314913
$25$	11.8708	2.37416
$26$	0.0294143	0.00576863
$27$	−1.00000	−0.192450
$28$	−1.99851	−0.377683
$29$	3.02440	0.561617	0.280809	−	0.959764i	$-0.409397\pi$
$29$	3.02440	0.561617	0.280809	+	0.959764i	$0.409397\pi$
$30$	0.158477	0.0289338
$31$	1.34664	0.241864	0.120932	−	0.992661i	$-0.461412\pi$
$31$	1.34664	0.241864	0.120932	+	0.992661i	$0.461412\pi$
$32$	0.462538	0.0817660
$33$	1.32401	0.230481
$34$	−0.0967500	−0.0165925
$35$	−4.10741	−0.694279
$36$	−1.99851	−0.333085
$37$	−8.07420	−1.32739	−0.663696	−	0.748003i	$-0.731013\pi$
$37$	−8.07420	−1.32739	−0.663696	+	0.748003i	$0.731013\pi$
$38$	0.239583	0.0388656
$39$	−0.762362	−0.122076
$40$	0.633671	0.100192
$41$	−10.7169	−1.67370	−0.836851	−	0.547431i	$-0.815606\pi$
$41$	−10.7169	−1.67370	−0.836851	+	0.547431i	$0.815606\pi$
$42$	−0.0385832	−0.00595351
$43$	−8.50798	−1.29746	−0.648728	−	0.761021i	$-0.724699\pi$
$43$	−8.50798	−1.29746	−0.648728	+	0.761021i	$0.724699\pi$
$44$	2.64606	0.398908
$45$	−4.10741	−0.612296
$46$	0.239329	0.0352871
$47$	−11.5727	−1.68805	−0.844023	−	0.536307i	$-0.819819\pi$
$47$	−11.5727	−1.68805	−0.844023	+	0.536307i	$0.819819\pi$
$48$	−3.99107	−0.576061
$49$	1.00000	0.142857
$50$	0.458013	0.0647728
$51$	2.50757	0.351130
$52$	−1.52359	−0.211284
$53$	0.514076	0.0706137	0.0353069	−	0.999377i	$-0.488759\pi$
$53$	0.514076	0.0706137	0.0353069	+	0.999377i	$0.488759\pi$
$54$	−0.0385832	−0.00525050
$55$	5.43827	0.733296
$56$	−0.154275	−0.0206159
$57$	−6.20953	−0.822473
$58$	0.116691	0.0153223
$59$	6.30138	0.820370	0.410185	−	0.912002i	$-0.365464\pi$
$59$	6.30138	0.820370	0.410185	+	0.912002i	$0.365464\pi$
$60$	−8.20870	−1.05974
$61$	7.43246	0.951629	0.475815	−	0.879546i	$-0.342153\pi$
$61$	7.43246	0.951629	0.475815	+	0.879546i	$0.342153\pi$
$62$	0.0519577	0.00659863
$63$	1.00000	0.125988
$64$	−7.96429	−0.995537
$65$	−3.13133	−0.388394
$66$	0.0510847	0.00628809
$67$	5.58079	0.681802	0.340901	−	0.940099i	$-0.389268\pi$
$67$	5.58079	0.681802	0.340901	+	0.940099i	$0.389268\pi$
$68$	5.01141	0.607723
$69$	−6.20294	−0.746747
$70$	−0.158477	−0.0189416
$71$	−5.06029	−0.600546	−0.300273	−	0.953853i	$-0.597078\pi$
$71$	−5.06029	−0.600546	−0.300273	+	0.953853i	$0.597078\pi$
$72$	−0.154275	−0.0181815
$73$	−1.66869	−0.195305	−0.0976525	−	0.995221i	$-0.531133\pi$
$73$	−1.66869	−0.195305	−0.0976525	+	0.995221i	$0.531133\pi$
$74$	−0.311528	−0.0362144
$75$	−11.8708	−1.37072
$76$	−12.4098	−1.42350
$77$	−1.32401	−0.150885
$78$	−0.0294143	−0.00333052
$79$	−5.84786	−0.657935	−0.328968	−	0.944341i	$-0.606701\pi$
$79$	−5.84786	−0.657935	−0.328968	+	0.944341i	$0.606701\pi$
$80$	−16.3930	−1.83279
$81$	1.00000	0.111111
$82$	−0.413492	−0.0456626
$83$	−2.54159	−0.278975	−0.139488	−	0.990224i	$-0.544546\pi$
$83$	−2.54159	−0.278975	−0.139488	+	0.990224i	$0.544546\pi$
$84$	1.99851	0.218055
$85$	10.2996	1.11715
$86$	−0.328265	−0.0353977
$87$	−3.02440	−0.324250
$88$	0.204263	0.0217745
$89$	4.55157	0.482466	0.241233	−	0.970467i	$-0.422448\pi$
$89$	4.55157	0.482466	0.241233	+	0.970467i	$0.422448\pi$
$90$	−0.158477	−0.0167049
$91$	0.762362	0.0799173
$92$	−12.3967	−1.29244
$93$	−1.34664	−0.139640
$94$	−0.446510	−0.0460540
$95$	−25.5051	−2.61677
$96$	−0.462538	−0.0472076
$97$	−5.81183	−0.590102	−0.295051	−	0.955482i	$-0.595337\pi$
$97$	−5.81183	−0.590102	−0.295051	+	0.955482i	$0.595337\pi$
$98$	0.0385832	0.00389749
$99$	−1.32401	−0.133068
$100$	−23.7239	−2.37239
$101$	−10.0770	−1.00270	−0.501350	−	0.865245i	$-0.667163\pi$
$101$	−10.0770	−1.00270	−0.501350	+	0.865245i	$0.667163\pi$
$102$	0.0967500	0.00957968
$103$	10.7485	1.05908	0.529539	−	0.848285i	$-0.322365\pi$
$103$	10.7485	1.05908	0.529539	+	0.848285i	$0.322365\pi$
$104$	−0.117614	−0.0115330
$105$	4.10741	0.400842
$106$	0.0198347	0.00192651
$107$	−3.56259	−0.344408	−0.172204	−	0.985061i	$-0.555089\pi$
$107$	−3.56259	−0.344408	−0.172204	+	0.985061i	$0.555089\pi$
$108$	1.99851	0.192307
$109$	−1.03550	−0.0991825	−0.0495912	−	0.998770i	$-0.515792\pi$
$109$	−1.03550	−0.0991825	−0.0495912	+	0.998770i	$0.515792\pi$
$110$	0.209826	0.0200061
$111$	8.07420	0.766370
$112$	3.99107	0.377121
$113$	12.6265	1.18780	0.593902	−	0.804537i	$-0.297587\pi$
$113$	12.6265	1.18780	0.593902	+	0.804537i	$0.297587\pi$
$114$	−0.239583	−0.0224390
$115$	−25.4780	−2.37584
$116$	−6.04430	−0.561199
$117$	0.762362	0.0704804
$118$	0.243127	0.0223817
$119$	−2.50757	−0.229869
$120$	−0.633671	−0.0578460
$121$	−9.24698	−0.840635
$122$	0.286768	0.0259627
$123$	10.7169	0.966312
$124$	−2.69128	−0.241684
$125$	−28.2212	−2.52418
$126$	0.0385832	0.00343726
$127$	20.4253	1.81245	0.906225	−	0.422796i	$-0.138951\pi$
$127$	20.4253	1.81245	0.906225	+	0.422796i	$0.138951\pi$
$128$	−1.23236	−0.108927
$129$	8.50798	0.749086
$130$	−0.120817	−0.0105963
$131$	−5.68441	−0.496650	−0.248325	−	0.968677i	$-0.579880\pi$
$131$	−5.68441	−0.496650	−0.248325	+	0.968677i	$0.579880\pi$
$132$	−2.64606	−0.230310
$133$	6.20953	0.538435
$134$	0.215324	0.0186012
$135$	4.10741	0.353509
$136$	0.386856	0.0331726
$137$	−15.2435	−1.30234	−0.651170	−	0.758932i	$-0.725721\pi$
$137$	−15.2435	−1.30234	−0.651170	+	0.758932i	$0.725721\pi$
$138$	−0.239329	−0.0203730
$139$	−17.7333	−1.50411	−0.752057	−	0.659098i	$-0.770938\pi$
$139$	−17.7333	−1.50411	−0.752057	+	0.659098i	$0.770938\pi$
$140$	8.20870	0.693762
$141$	11.5727	0.974594
$142$	−0.195242	−0.0163843
$143$	−1.00938	−0.0844085
$144$	3.99107	0.332589
$145$	−12.4225	−1.03163
$146$	−0.0643832	−0.00532839
$147$	−1.00000	−0.0824786
$148$	16.1364	1.32640
$149$	3.03642	0.248753	0.124377	−	0.992235i	$-0.460307\pi$
$149$	3.03642	0.248753	0.124377	+	0.992235i	$0.460307\pi$
$150$	−0.458013	−0.0373966
$151$	11.5994	0.943943	0.471972	−	0.881614i	$-0.343542\pi$
$151$	11.5994	0.943943	0.471972	+	0.881614i	$0.343542\pi$
$152$	−0.957977	−0.0777022
$153$	−2.50757	−0.202725
$154$	−0.0510847	−0.00411652
$155$	−5.53121	−0.444277
$156$	1.52359	0.121985
$157$	16.5814	1.32334	0.661669	−	0.749796i	$-0.269849\pi$
$157$	16.5814	1.32334	0.661669	+	0.749796i	$0.269849\pi$
$158$	−0.225629	−0.0179501
$159$	−0.514076	−0.0407689
$160$	−1.89983	−0.150195
$161$	6.20294	0.488860
$162$	0.0385832	0.00303138
$163$	22.2013	1.73894	0.869471	−	0.493983i	$-0.164460\pi$
$163$	22.2013	1.73894	0.869471	+	0.493983i	$0.164460\pi$
$164$	21.4179	1.67246
$165$	−5.43827	−0.423369
$166$	−0.0980624	−0.00761112
$167$	−3.67903	−0.284692	−0.142346	−	0.989817i	$-0.545465\pi$
$167$	−3.67903	−0.284692	−0.142346	+	0.989817i	$0.545465\pi$
$168$	0.154275	0.0119026
$169$	−12.4188	−0.955293
$170$	0.397392	0.0304786
$171$	6.20953	0.474855
$172$	17.0033	1.29649
$173$	−8.23352	−0.625983	−0.312992	−	0.949756i	$-0.601331\pi$
$173$	−8.23352	−0.625983	−0.312992	+	0.949756i	$0.601331\pi$
$174$	−0.116691	−0.00884632
$175$	11.8708	0.897348
$176$	−5.28424	−0.398314
$177$	−6.30138	−0.473641
$178$	0.175614	0.0131628
$179$	13.2845	0.992928	0.496464	−	0.868057i	$-0.334632\pi$
$179$	13.2845	0.992928	0.496464	+	0.868057i	$0.334632\pi$
$180$	8.20870	0.611841
$181$	0.118310	0.00879393	0.00439696	−	0.999990i	$-0.498600\pi$
$181$	0.118310	0.00879393	0.00439696	+	0.999990i	$0.498600\pi$
$182$	0.0294143	0.00218034
$183$	−7.43246	−0.549424
$184$	−0.956960	−0.0705480
$185$	33.1641	2.43827
$186$	−0.0519577	−0.00380972
$187$	3.32006	0.242787
$188$	23.1281	1.68679
$189$	−1.00000	−0.0727393
$190$	−0.984067	−0.0713917
$191$	13.1045	0.948209	0.474104	−	0.880469i	$-0.342772\pi$
$191$	13.1045	0.948209	0.474104	+	0.880469i	$0.342772\pi$
$192$	7.96429	0.574773
$193$	2.00372	0.144231	0.0721155	−	0.997396i	$-0.477025\pi$
$193$	2.00372	0.144231	0.0721155	+	0.997396i	$0.477025\pi$
$194$	−0.224239	−0.0160994
$195$	3.13133	0.224239
$196$	−1.99851	−0.142751
$197$	−7.80884	−0.556357	−0.278178	−	0.960529i	$-0.589731\pi$
$197$	−7.80884	−0.556357	−0.278178	+	0.960529i	$0.589731\pi$
$198$	−0.0510847	−0.00363043
$199$	−12.6532	−0.896959	−0.448479	−	0.893793i	$-0.648034\pi$
$199$	−12.6532	−0.896959	−0.448479	+	0.893793i	$0.648034\pi$
$200$	−1.83137	−0.129497
$201$	−5.58079	−0.393638
$202$	−0.388803	−0.0273561
$203$	3.02440	0.212271
$204$	−5.01141	−0.350869
$205$	44.0187	3.07440
$206$	0.414710	0.0288942
$207$	6.20294	0.431134
$208$	3.04264	0.210969
$209$	−8.22151	−0.568694
$210$	0.158477	0.0109359
$211$	21.6356	1.48946	0.744728	−	0.667368i	$-0.232579\pi$
$211$	21.6356	1.48946	0.744728	+	0.667368i	$0.232579\pi$
$212$	−1.02739	−0.0705612
$213$	5.06029	0.346725
$214$	−0.137456	−0.00939628
$215$	34.9458	2.38328
$216$	0.154275	0.0104971
$217$	1.34664	0.0914160
$218$	−0.0399527	−0.00270594
$219$	1.66869	0.112759
$220$	−10.8684	−0.732750
$221$	−1.91168	−0.128593
$222$	0.311528	0.0209084
$223$	−11.6221	−0.778274	−0.389137	−	0.921180i	$-0.627227\pi$
$223$	−11.6221	−0.778274	−0.389137	+	0.921180i	$0.627227\pi$
$224$	0.462538	0.0309046
$225$	11.8708	0.791387
$226$	0.487172	0.0324062
$227$	2.24339	0.148899	0.0744495	−	0.997225i	$-0.476280\pi$
$227$	2.24339	0.148899	0.0744495	+	0.997225i	$0.476280\pi$
$228$	12.4098	0.821861
$229$	−19.0073	−1.25604	−0.628018	−	0.778199i	$-0.716133\pi$
$229$	−19.0073	−1.25604	−0.628018	+	0.778199i	$0.716133\pi$
$230$	−0.983022	−0.0648186
$231$	1.32401	0.0871138
$232$	−0.466590	−0.0306331
$233$	9.49141	0.621803	0.310901	−	0.950442i	$-0.399369\pi$
$233$	9.49141	0.621803	0.310901	+	0.950442i	$0.399369\pi$
$234$	0.0294143	0.00192288
$235$	47.5336	3.10075
$236$	−12.5934	−0.819759
$237$	5.84786	0.379859
$238$	−0.0967500	−0.00627137
$239$	−15.7194	−1.01680	−0.508401	−	0.861120i	$-0.669763\pi$
$239$	−15.7194	−1.01680	−0.508401	+	0.861120i	$0.669763\pi$
$240$	16.3930	1.05816
$241$	15.3920	0.991488	0.495744	−	0.868469i	$-0.334895\pi$
$241$	15.3920	0.991488	0.495744	+	0.868469i	$0.334895\pi$
$242$	−0.356778	−0.0229345
$243$	−1.00000	−0.0641500
$244$	−14.8539	−0.950921
$245$	−4.10741	−0.262413
$246$	0.413492	0.0263633
$247$	4.73391	0.301212
$248$	−0.207753	−0.0131924
$249$	2.54159	0.161067
$250$	−1.08886	−0.0688657
$251$	2.38571	0.150585	0.0752925	−	0.997161i	$-0.476011\pi$
$251$	2.38571	0.150585	0.0752925	+	0.997161i	$0.476011\pi$
$252$	−1.99851	−0.125894
$253$	−8.21279	−0.516334
$254$	0.788071	0.0494480
$255$	−10.2996	−0.644987
$256$	15.8810	0.992565
$257$	−13.3061	−0.830012	−0.415006	−	0.909819i	$-0.636220\pi$
$257$	−13.3061	−0.830012	−0.415006	+	0.909819i	$0.636220\pi$
$258$	0.328265	0.0204369
$259$	−8.07420	−0.501707
$260$	6.25800	0.388105
$261$	3.02440	0.187206
$262$	−0.219323	−0.0135498
$263$	−4.53672	−0.279746	−0.139873	−	0.990169i	$-0.544669\pi$
$263$	−4.53672	−0.279746	−0.139873	+	0.990169i	$0.544669\pi$
$264$	−0.204263	−0.0125715
$265$	−2.11152	−0.129710
$266$	0.239583	0.0146898
$267$	−4.55157	−0.278552
$268$	−11.1533	−0.681294
$269$	22.2293	1.35534	0.677672	−	0.735365i	$-0.262989\pi$
$269$	22.2293	1.35534	0.677672	+	0.735365i	$0.262989\pi$
$270$	0.158477	0.00964459
$271$	−22.2823	−1.35355	−0.676776	−	0.736189i	$-0.736624\pi$
$271$	−22.2823	−1.35355	−0.676776	+	0.736189i	$0.736624\pi$
$272$	−10.0079	−0.606818
$273$	−0.762362	−0.0461403
$274$	−0.588142	−0.0355310
$275$	−15.7171	−0.947778
$276$	12.3967	0.746191
$277$	0.726184	0.0436322	0.0218161	−	0.999762i	$-0.493055\pi$
$277$	0.726184	0.0436322	0.0218161	+	0.999762i	$0.493055\pi$
$278$	−0.684205	−0.0410359
$279$	1.34664	0.0806214
$280$	0.633671	0.0378691
$281$	−14.9929	−0.894403	−0.447202	−	0.894433i	$-0.647579\pi$
$281$	−14.9929	−0.894403	−0.447202	+	0.894433i	$0.647579\pi$
$282$	0.446510	0.0265893
$283$	24.5524	1.45949	0.729744	−	0.683721i	$-0.239639\pi$
$283$	24.5524	1.45949	0.729744	+	0.683721i	$0.239639\pi$
$284$	10.1130	0.600099
$285$	25.5051	1.51079
$286$	−0.0389450	−0.00230287
$287$	−10.7169	−0.632600
$288$	0.462538	0.0272553
$289$	−10.7121	−0.630123
$290$	−0.479297	−0.0281453
$291$	5.81183	0.340696
$292$	3.33489	0.195160
$293$	−15.1631	−0.885838	−0.442919	−	0.896562i	$-0.646057\pi$
$293$	−15.1631	−0.885838	−0.442919	+	0.896562i	$0.646057\pi$
$294$	−0.0385832	−0.00225022
$295$	−25.8823	−1.50693
$296$	1.24565	0.0724019
$297$	1.32401	0.0768271
$298$	0.117155	0.00678659
$299$	4.72889	0.273479
$300$	23.7239	1.36970
$301$	−8.50798	−0.490392
$302$	0.447540	0.0257531
$303$	10.0770	0.578909
$304$	24.7827	1.42138
$305$	−30.5282	−1.74804
$306$	−0.0967500	−0.00553083
$307$	13.4192	0.765874	0.382937	−	0.923774i	$-0.374913\pi$
$307$	13.4192	0.765874	0.382937	+	0.923774i	$0.374913\pi$
$308$	2.64606	0.150773
$309$	−10.7485	−0.611459
$310$	−0.213411	−0.0121210
$311$	−1.81894	−0.103143	−0.0515714	−	0.998669i	$-0.516423\pi$
$311$	−1.81894	−0.103143	−0.0515714	+	0.998669i	$0.516423\pi$
$312$	0.117614	0.00665856
$313$	−15.0473	−0.850522	−0.425261	−	0.905071i	$-0.639818\pi$
$313$	−15.0473	−0.850522	−0.425261	+	0.905071i	$0.639818\pi$
$314$	0.639762	0.0361038
$315$	−4.10741	−0.231426
$316$	11.6870	0.657445
$317$	13.6762	0.768134	0.384067	−	0.923305i	$-0.374523\pi$
$317$	13.6762	0.768134	0.384067	+	0.923305i	$0.374523\pi$
$318$	−0.0198347	−0.00111227
$319$	−4.00435	−0.224201
$320$	32.7126	1.82869
$321$	3.56259	0.198844
$322$	0.239329	0.0133373
$323$	−15.5709	−0.866385
$324$	−1.99851	−0.111028
$325$	9.04985	0.501995
$326$	0.856598	0.0474426
$327$	1.03550	0.0572630
$328$	1.65335	0.0912912
$329$	−11.5727	−0.638021
$330$	−0.209826	−0.0115505
$331$	−22.3294	−1.22733	−0.613667	−	0.789565i	$-0.710306\pi$
$331$	−22.3294	−1.22733	−0.613667	+	0.789565i	$0.710306\pi$
$332$	5.07939	0.278768
$333$	−8.07420	−0.442464
$334$	−0.141949	−0.00776709
$335$	−22.9226	−1.25239
$336$	−3.99107	−0.217731
$337$	−25.4576	−1.38676	−0.693381	−	0.720571i	$-0.743880\pi$
$337$	−25.4576	−1.38676	−0.693381	+	0.720571i	$0.743880\pi$
$338$	−0.479157	−0.0260627
$339$	−12.6265	−0.685779
$340$	−20.5839	−1.11632
$341$	−1.78297	−0.0965535
$342$	0.239583	0.0129552
$343$	1.00000	0.0539949
$344$	1.31257	0.0707691
$345$	25.4780	1.37169
$346$	−0.317675	−0.0170783
$347$	32.8833	1.76527	0.882633	−	0.470063i	$-0.155769\pi$
$347$	32.8833	1.76527	0.882633	+	0.470063i	$0.155769\pi$
$348$	6.04430	0.324009
$349$	−14.4463	−0.773290	−0.386645	−	0.922229i	$-0.626366\pi$
$349$	−14.4463	−0.773290	−0.386645	+	0.922229i	$0.626366\pi$
$350$	0.458013	0.0244818
$351$	−0.762362	−0.0406919
$352$	−0.612408	−0.0326414
$353$	−2.18102	−0.116084	−0.0580419	−	0.998314i	$-0.518486\pi$
$353$	−2.18102	−0.116084	−0.0580419	+	0.998314i	$0.518486\pi$
$354$	−0.243127	−0.0129221
$355$	20.7847	1.10314
$356$	−9.09637	−0.482106
$357$	2.50757	0.132715
$358$	0.512557	0.0270895
$359$	12.6693	0.668659	0.334329	−	0.942456i	$-0.391490\pi$
$359$	12.6693	0.668659	0.334329	+	0.942456i	$0.391490\pi$
$360$	0.633671	0.0333974
$361$	19.5583	1.02938
$362$	0.00456478	0.000239920 0
$363$	9.24698	0.485341
$364$	−1.52359	−0.0798578
$365$	6.85398	0.358753
$366$	−0.286768	−0.0149896
$367$	33.7606	1.76229	0.881144	−	0.472847i	$-0.156774\pi$
$367$	33.7606	1.76229	0.881144	+	0.472847i	$0.156774\pi$
$368$	24.7564	1.29052
$369$	−10.7169	−0.557900
$370$	1.27957	0.0665219
$371$	0.514076	0.0266895
$372$	2.69128	0.139536
$373$	−1.91476	−0.0991423	−0.0495711	−	0.998771i	$-0.515785\pi$
$373$	−1.91476	−0.0991423	−0.0495711	+	0.998771i	$0.515785\pi$
$374$	0.128098	0.00662381
$375$	28.2212	1.45734
$376$	1.78537	0.0920736
$377$	2.30569	0.118749
$378$	−0.0385832	−0.00198450
$379$	20.6787	1.06219	0.531096	−	0.847312i	$-0.321780\pi$
$379$	20.6787	1.06219	0.531096	+	0.847312i	$0.321780\pi$
$380$	50.9722	2.61482
$381$	−20.4253	−1.04642
$382$	0.505613	0.0258694
$383$	−1.00000	−0.0510976
$383$	−1.00000	−0.0510976
$384$	1.23236	0.0628888
$385$	5.43827	0.277160
$386$	0.0773098	0.00393497
$387$	−8.50798	−0.432485
$388$	11.6150	0.589663
$389$	−4.58510	−0.232474	−0.116237	−	0.993222i	$-0.537083\pi$
$389$	−4.58510	−0.232474	−0.116237	+	0.993222i	$0.537083\pi$
$390$	0.120817	0.00611779
$391$	−15.5543	−0.786616
$392$	−0.154275	−0.00779207
$393$	5.68441	0.286741
$394$	−0.301290	−0.0151788
$395$	24.0195	1.20855
$396$	2.64606	0.132969
$397$	0.992991	0.0498368	0.0249184	−	0.999689i	$-0.492067\pi$
$397$	0.992991	0.0498368	0.0249184	+	0.999689i	$0.492067\pi$
$398$	−0.488199	−0.0244712
$399$	−6.20953	−0.310865
$400$	47.3772	2.36886
$401$	−35.0723	−1.75142	−0.875712	−	0.482833i	$-0.839608\pi$
$401$	−35.0723	−1.75142	−0.875712	+	0.482833i	$0.839608\pi$
$402$	−0.215324	−0.0107394
$403$	1.02663	0.0511400
$404$	20.1390	1.00195
$405$	−4.10741	−0.204099
$406$	0.116691	0.00579127
$407$	10.6904	0.529902
$408$	−0.386856	−0.0191522
$409$	−13.6634	−0.675614	−0.337807	−	0.941215i	$-0.609685\pi$
$409$	−13.6634	−0.675614	−0.337807	+	0.941215i	$0.609685\pi$
$410$	1.69838	0.0838771
$411$	15.2435	0.751906
$412$	−21.4809	−1.05829
$413$	6.30138	0.310071
$414$	0.239329	0.0117624
$415$	10.4393	0.512447
$416$	0.352622	0.0172887
$417$	17.7333	0.868401
$418$	−0.317212	−0.0155153
$419$	−26.4744	−1.29336	−0.646679	−	0.762762i	$-0.723843\pi$
$419$	−26.4744	−1.29336	−0.646679	+	0.762762i	$0.723843\pi$
$420$	−8.20870	−0.400544
$421$	17.5757	0.856589	0.428294	−	0.903639i	$-0.359115\pi$
$421$	17.5757	0.856589	0.428294	+	0.903639i	$0.359115\pi$
$422$	0.834769	0.0406359
$423$	−11.5727	−0.562682
$424$	−0.0793091	−0.00385159
$425$	−29.7669	−1.44391
$426$	0.195242	0.00945950
$427$	7.43246	0.359682
$428$	7.11987	0.344152
$429$	1.00938	0.0487333
$430$	1.34832	0.0650216
$431$	−11.0983	−0.534586	−0.267293	−	0.963615i	$-0.586129\pi$
$431$	−11.0983	−0.534586	−0.267293	+	0.963615i	$0.586129\pi$
$432$	−3.99107	−0.192020
$433$	22.6188	1.08699	0.543496	−	0.839412i	$-0.317100\pi$
$433$	22.6188	1.08699	0.543496	+	0.839412i	$0.317100\pi$
$434$	0.0519577	0.00249405
$435$	12.4225	0.595611
$436$	2.06945	0.0991086
$437$	38.5174	1.84254
$438$	0.0643832	0.00307635
$439$	−8.28369	−0.395359	−0.197679	−	0.980267i	$-0.563340\pi$
$439$	−8.28369	−0.395359	−0.197679	+	0.980267i	$0.563340\pi$
$440$	−0.838990	−0.0399973
$441$	1.00000	0.0476190
$442$	−0.0737586	−0.00350834
$443$	31.4591	1.49467	0.747334	−	0.664448i	$-0.231333\pi$
$443$	31.4591	1.49467	0.747334	+	0.664448i	$0.231333\pi$
$444$	−16.1364	−0.765799
$445$	−18.6952	−0.886236
$446$	−0.448418	−0.0212332
$447$	−3.03642	−0.143618
$448$	−7.96429	−0.376278
$449$	19.7017	0.929779	0.464890	−	0.885369i	$-0.346094\pi$
$449$	19.7017	0.929779	0.464890	+	0.885369i	$0.346094\pi$
$450$	0.458013	0.0215909
$451$	14.1894	0.668151
$452$	−25.2343	−1.18692
$453$	−11.5994	−0.544986
$454$	0.0865571	0.00406232
$455$	−3.13133	−0.146799
$456$	0.957977	0.0448614
$457$	1.49243	0.0698130	0.0349065	−	0.999391i	$-0.488887\pi$
$457$	1.49243	0.0698130	0.0349065	+	0.999391i	$0.488887\pi$
$458$	−0.733360	−0.0342677
$459$	2.50757	0.117043
$460$	50.9181	2.37407
$461$	12.7998	0.596144	0.298072	−	0.954543i	$-0.403656\pi$
$461$	12.7998	0.596144	0.298072	+	0.954543i	$0.403656\pi$
$462$	0.0510847	0.00237667
$463$	−20.3369	−0.945138	−0.472569	−	0.881294i	$-0.656673\pi$
$463$	−20.3369	−0.945138	−0.472569	+	0.881294i	$0.656673\pi$
$464$	12.0706	0.560364
$465$	5.53121	0.256504
$466$	0.366208	0.0169643
$467$	29.2282	1.35252	0.676260	−	0.736663i	$-0.263599\pi$
$467$	29.2282	1.35252	0.676260	+	0.736663i	$0.263599\pi$
$468$	−1.52359	−0.0704279
$469$	5.58079	0.257697
$470$	1.83400	0.0845960
$471$	−16.5814	−0.764029
$472$	−0.972146	−0.0447467
$473$	11.2647	0.517951
$474$	0.225629	0.0103635
$475$	73.7122	3.38215
$476$	5.01141	0.229698
$477$	0.514076	0.0235379
$478$	−0.606504	−0.0277408
$479$	5.10385	0.233201	0.116600	−	0.993179i	$-0.462800\pi$
$479$	5.10385	0.233201	0.116600	+	0.993179i	$0.462800\pi$
$480$	1.89983	0.0867152
$481$	−6.15547	−0.280665
$482$	0.593873	0.0270502
$483$	−6.20294	−0.282244
$484$	18.4802	0.840009
$485$	23.8716	1.08395
$486$	−0.0385832	−0.00175017
$487$	−20.5911	−0.933073	−0.466537	−	0.884502i	$-0.654498\pi$
$487$	−20.5911	−0.933073	−0.466537	+	0.884502i	$0.654498\pi$
$488$	−1.14664	−0.0519062
$489$	−22.2013	−1.00398
$490$	−0.158477	−0.00715925
$491$	−2.23724	−0.100965	−0.0504826	−	0.998725i	$-0.516076\pi$
$491$	−2.23724	−0.100965	−0.0504826	+	0.998725i	$0.516076\pi$
$492$	−21.4179	−0.965592
$493$	−7.58390	−0.341562
$494$	0.182649	0.00821778
$495$	5.43827	0.244432
$496$	5.37454	0.241324
$497$	−5.06029	−0.226985
$498$	0.0980624	0.00439428
$499$	32.0424	1.43441	0.717207	−	0.696860i	$-0.245420\pi$
$499$	32.0424	1.43441	0.717207	+	0.696860i	$0.245420\pi$
$500$	56.4004	2.52230
$501$	3.67903	0.164367
$502$	0.0920484	0.00410832
$503$	−30.4922	−1.35958	−0.679789	−	0.733408i	$-0.737929\pi$
$503$	−30.4922	−1.35958	−0.679789	+	0.733408i	$0.737929\pi$
$504$	−0.154275	−0.00687196
$505$	41.3904	1.84185
$506$	−0.316875	−0.0140868
$507$	12.4188	0.551538
$508$	−40.8201	−1.81110
$509$	−22.5170	−0.998048	−0.499024	−	0.866588i	$-0.666308\pi$
$509$	−22.5170	−0.998048	−0.499024	+	0.866588i	$0.666308\pi$
$510$	−0.397392	−0.0175968
$511$	−1.66869	−0.0738183
$512$	3.07747	0.136006
$513$	−6.20953	−0.274158
$514$	−0.513391	−0.0226447
$515$	−44.1484	−1.94541
$516$	−17.0033	−0.748528
$517$	15.3224	0.673877
$518$	−0.311528	−0.0136878
$519$	8.23352	0.361412
$520$	0.483087	0.0211848
$521$	6.61512	0.289814	0.144907	−	0.989445i	$-0.453712\pi$
$521$	6.61512	0.289814	0.144907	+	0.989445i	$0.453712\pi$
$522$	0.116691	0.00510742
$523$	16.8741	0.737851	0.368925	−	0.929459i	$-0.379726\pi$
$523$	16.8741	0.737851	0.368925	+	0.929459i	$0.379726\pi$
$524$	11.3604	0.496280
$525$	−11.8708	−0.518084
$526$	−0.175041	−0.00763215
$527$	−3.37680	−0.147096
$528$	5.28424	0.229967
$529$	15.4765	0.672891
$530$	−0.0814691	−0.00353879
$531$	6.30138	0.273457
$532$	−12.4098	−0.538034
$533$	−8.17017	−0.353889
$534$	−0.175614	−0.00759956
$535$	14.6330	0.632639
$536$	−0.860977	−0.0371885
$537$	−13.2845	−0.573267
$538$	0.857676	0.0369770
$539$	−1.32401	−0.0570294
$540$	−8.20870	−0.353246
$541$	−19.0145	−0.817497	−0.408748	−	0.912647i	$-0.634035\pi$
$541$	−19.0145	−0.817497	−0.408748	+	0.912647i	$0.634035\pi$
$542$	−0.859721	−0.0369282
$543$	−0.118310	−0.00507718
$544$	−1.15985	−0.0497281
$545$	4.25320	0.182187
$546$	−0.0294143	−0.00125882
$547$	2.60183	0.111246	0.0556231	−	0.998452i	$-0.482285\pi$
$547$	2.60183	0.111246	0.0556231	+	0.998452i	$0.482285\pi$
$548$	30.4643	1.30137
$549$	7.43246	0.317210
$550$	−0.606416	−0.0258577
$551$	18.7801	0.800060
$552$	0.956960	0.0407309
$553$	−5.84786	−0.248676
$554$	0.0280185	0.00119039
$555$	−33.1641	−1.40774
$556$	35.4401	1.50300
$557$	28.0428	1.18821	0.594105	−	0.804387i	$-0.297506\pi$
$557$	28.0428	1.18821	0.594105	+	0.804387i	$0.297506\pi$
$558$	0.0519577	0.00219954
$559$	−6.48616	−0.274335
$560$	−16.3930	−0.692729
$561$	−3.32006	−0.140173
$562$	−0.578475	−0.0244015
$563$	−14.2217	−0.599372	−0.299686	−	0.954038i	$-0.596882\pi$
$563$	−14.2217	−0.599372	−0.299686	+	0.954038i	$0.596882\pi$
$564$	−23.1281	−0.973868
$565$	−51.8624	−2.18187
$566$	0.947308	0.0398183
$567$	1.00000	0.0419961
$568$	0.780677	0.0327565
$569$	−22.5070	−0.943542	−0.471771	−	0.881721i	$-0.656385\pi$
$569$	−22.5070	−0.943542	−0.471771	+	0.881721i	$0.656385\pi$
$570$	0.984067	0.0412180
$571$	−9.69509	−0.405727	−0.202863	−	0.979207i	$-0.565025\pi$
$571$	−9.69509	−0.405727	−0.202863	+	0.979207i	$0.565025\pi$
$572$	2.01725	0.0843457
$573$	−13.1045	−0.547449
$574$	−0.413492	−0.0172588
$575$	73.6339	3.07075
$576$	−7.96429	−0.331846
$577$	−7.89730	−0.328769	−0.164385	−	0.986396i	$-0.552564\pi$
$577$	−7.89730	−0.328769	−0.164385	+	0.986396i	$0.552564\pi$
$578$	−0.413306	−0.0171913
$579$	−2.00372	−0.0832718
$580$	24.8264	1.03086
$581$	−2.54159	−0.105443
$582$	0.224239	0.00929499
$583$	−0.680644	−0.0281894
$584$	0.257437	0.0106528
$585$	−3.13133	−0.129465
$586$	−0.585040	−0.0241678
$587$	−23.4228	−0.966763	−0.483382	−	0.875410i	$-0.660592\pi$
$587$	−23.4228	−0.966763	−0.483382	+	0.875410i	$0.660592\pi$
$588$	1.99851	0.0824172
$589$	8.36202	0.344551
$590$	−0.998622	−0.0411126
$591$	7.80884	0.321213
$592$	−32.2247	−1.32443
$593$	10.9479	0.449574	0.224787	−	0.974408i	$-0.427831\pi$
$593$	10.9479	0.449574	0.224787	+	0.974408i	$0.427831\pi$
$594$	0.0510847	0.00209603
$595$	10.2996	0.422243
$596$	−6.06832	−0.248568
$597$	12.6532	0.517859
$598$	0.182455	0.00746116
$599$	24.7606	1.01169	0.505846	−	0.862624i	$-0.331180\pi$
$599$	24.7606	1.01169	0.505846	+	0.862624i	$0.331180\pi$
$600$	1.83137	0.0747654
$601$	7.37245	0.300729	0.150364	−	0.988631i	$-0.451955\pi$
$601$	7.37245	0.300729	0.150364	+	0.988631i	$0.451955\pi$
$602$	−0.328265	−0.0133791
$603$	5.58079	0.227267
$604$	−23.1815	−0.943241
$605$	37.9811	1.54415
$606$	0.388803	0.0157940
$607$	21.7013	0.880828	0.440414	−	0.897795i	$-0.354832\pi$
$607$	21.7013	0.880828	0.440414	+	0.897795i	$0.354832\pi$
$608$	2.87215	0.116481
$609$	−3.02440	−0.122555
$610$	−1.17787	−0.0476907
$611$	−8.82256	−0.356922
$612$	5.01141	0.202574
$613$	26.4242	1.06726	0.533631	−	0.845718i	$-0.320827\pi$
$613$	26.4242	1.06726	0.533631	+	0.845718i	$0.320827\pi$
$614$	0.517755	0.0208949
$615$	−44.0187	−1.77501
$616$	0.204263	0.00822997
$617$	10.8136	0.435341	0.217670	−	0.976022i	$-0.430154\pi$
$617$	10.8136	0.435341	0.217670	+	0.976022i	$0.430154\pi$
$618$	−0.414710	−0.0166821
$619$	12.3691	0.497157	0.248578	−	0.968612i	$-0.420037\pi$
$619$	12.3691	0.497157	0.248578	+	0.968612i	$0.420037\pi$
$620$	11.0542	0.443947
$621$	−6.20294	−0.248916
$622$	−0.0701806	−0.00281398
$623$	4.55157	0.182355
$624$	−3.04264	−0.121803
$625$	56.5620	2.26248
$626$	−0.580571	−0.0232043
$627$	8.22151	0.328336
$628$	−33.1381	−1.32235
$629$	20.2466	0.807287
$630$	−0.158477	−0.00631387
$631$	5.27630	0.210046	0.105023	−	0.994470i	$-0.466508\pi$
$631$	5.27630	0.210046	0.105023	+	0.994470i	$0.466508\pi$
$632$	0.902179	0.0358868
$633$	−21.6356	−0.859938
$634$	0.527672	0.0209566
$635$	−83.8949	−3.32927
$636$	1.02739	0.0407385
$637$	0.762362	0.0302059
$638$	−0.154501	−0.00611673
$639$	−5.06029	−0.200182
$640$	5.06182	0.200086
$641$	−38.6773	−1.52766	−0.763832	−	0.645416i	$-0.776684\pi$
$641$	−38.6773	−1.52766	−0.763832	+	0.645416i	$0.776684\pi$
$642$	0.137456	0.00542495
$643$	−24.0759	−0.949461	−0.474730	−	0.880131i	$-0.657454\pi$
$643$	−24.0759	−0.949461	−0.474730	+	0.880131i	$0.657454\pi$
$644$	−12.3967	−0.488496
$645$	−34.9458	−1.37599
$646$	−0.600773	−0.0236371
$647$	30.8641	1.21339	0.606696	−	0.794934i	$-0.292495\pi$
$647$	30.8641	1.21339	0.606696	+	0.794934i	$0.292495\pi$
$648$	−0.154275	−0.00606050
$649$	−8.34312	−0.327496
$650$	0.349172	0.0136956
$651$	−1.34664	−0.0527791
$652$	−44.3696	−1.73765
$653$	31.1739	1.21993	0.609965	−	0.792428i	$-0.291183\pi$
$653$	31.1739	1.21993	0.609965	+	0.792428i	$0.291183\pi$
$654$	0.0399527	0.00156227
$655$	23.3482	0.912290
$656$	−42.7720	−1.66996
$657$	−1.66869	−0.0651016
$658$	−0.446510	−0.0174068
$659$	21.4614	0.836016	0.418008	−	0.908443i	$-0.362728\pi$
$659$	21.4614	0.836016	0.418008	+	0.908443i	$0.362728\pi$
$660$	10.8684	0.423054
$661$	−12.7689	−0.496653	−0.248326	−	0.968676i	$-0.579881\pi$
$661$	−12.7689	−0.496653	−0.248326	+	0.968676i	$0.579881\pi$
$662$	−0.861538	−0.0334846
$663$	1.91168	0.0742434
$664$	0.392104	0.0152166
$665$	−25.5051	−0.989045
$666$	−0.311528	−0.0120715
$667$	18.7602	0.726397
$668$	7.35259	0.284480
$669$	11.6221	0.449337
$670$	−0.884425	−0.0341683
$671$	−9.84069	−0.379896
$672$	−0.462538	−0.0178428
$673$	43.7072	1.68479	0.842394	−	0.538862i	$-0.181146\pi$
$673$	43.7072	1.68479	0.842394	+	0.538862i	$0.181146\pi$
$674$	−0.982233	−0.0378342
$675$	−11.8708	−0.456907
$676$	24.8191	0.954582
$677$	24.8332	0.954416	0.477208	−	0.878790i	$-0.341649\pi$
$677$	24.8332	0.954416	0.477208	+	0.878790i	$0.341649\pi$
$678$	−0.487172	−0.0187097
$679$	−5.81183	−0.223038
$680$	−1.58898	−0.0609344
$681$	−2.24339	−0.0859669
$682$	−0.0687928	−0.00263421
$683$	−9.61906	−0.368063	−0.184032	−	0.982920i	$-0.558915\pi$
$683$	−9.61906	−0.368063	−0.184032	+	0.982920i	$0.558915\pi$
$684$	−12.4098	−0.474501
$685$	62.6113	2.39225
$686$	0.0385832	0.00147311
$687$	19.0073	0.725172
$688$	−33.9560	−1.29456
$689$	0.391912	0.0149307
$690$	0.983022	0.0374230
$691$	−14.5964	−0.555272	−0.277636	−	0.960686i	$-0.589551\pi$
$691$	−14.5964	−0.555272	−0.277636	+	0.960686i	$0.589551\pi$
$692$	16.4548	0.625517
$693$	−1.32401	−0.0502952
$694$	1.26874	0.0481607
$695$	72.8377	2.76289
$696$	0.466590	0.0176861
$697$	26.8734	1.01790
$698$	−0.557382	−0.0210972
$699$	−9.49141	−0.358998
$700$	−23.7239	−0.896680
$701$	12.6660	0.478387	0.239194	−	0.970972i	$-0.423117\pi$
$701$	12.6660	0.478387	0.239194	+	0.970972i	$0.423117\pi$
$702$	−0.0294143	−0.00111017
$703$	−50.1370	−1.89095
$704$	10.5448	0.397424
$705$	−47.5336	−1.79022
$706$	−0.0841505	−0.00316704
$707$	−10.0770	−0.378985
$708$	12.5934	0.473288
$709$	35.6653	1.33944	0.669719	−	0.742615i	$-0.266415\pi$
$709$	35.6653	1.33944	0.669719	+	0.742615i	$0.266415\pi$
$710$	0.801938	0.0300962
$711$	−5.84786	−0.219312
$712$	−0.702194	−0.0263159
$713$	8.35314	0.312828
$714$	0.0967500	0.00362078
$715$	4.14593	0.155049
$716$	−26.5492	−0.992189
$717$	15.7194	0.587051
$718$	0.488821	0.0182426
$719$	17.2411	0.642985	0.321493	−	0.946912i	$-0.395815\pi$
$719$	17.2411	0.642985	0.321493	+	0.946912i	$0.395815\pi$
$720$	−16.3930	−0.610929
$721$	10.7485	0.400294
$722$	0.754621	0.0280841
$723$	−15.3920	−0.572436
$724$	−0.236444	−0.00878738
$725$	35.9021	1.33337
$726$	0.356778	0.0132413
$727$	−5.64497	−0.209360	−0.104680	−	0.994506i	$-0.533382\pi$
$727$	−5.64497	−0.209360	−0.104680	+	0.994506i	$0.533382\pi$
$728$	−0.117614	−0.00435905
$729$	1.00000	0.0370370
$730$	0.264448	0.00978766
$731$	21.3344	0.789080
$732$	14.8539	0.549015
$733$	−19.6845	−0.727063	−0.363531	−	0.931582i	$-0.618429\pi$
$733$	−19.6845	−0.727063	−0.363531	+	0.931582i	$0.618429\pi$
$734$	1.30259	0.0480795
$735$	4.10741	0.151504
$736$	2.86910	0.105756
$737$	−7.38905	−0.272179
$738$	−0.413492	−0.0152209
$739$	44.5524	1.63889	0.819444	−	0.573159i	$-0.194282\pi$
$739$	44.5524	1.63889	0.819444	+	0.573159i	$0.194282\pi$
$740$	−66.2787	−2.43645
$741$	−4.73391	−0.173905
$742$	0.0198347	0.000728153 0
$743$	8.68393	0.318582	0.159291	−	0.987232i	$-0.449079\pi$
$743$	8.68393	0.318582	0.159291	+	0.987232i	$0.449079\pi$
$744$	0.207753	0.00761661
$745$	−12.4718	−0.456932
$746$	−0.0738773	−0.00270484
$747$	−2.54159	−0.0929918
$748$	−6.63518	−0.242606
$749$	−3.56259	−0.130174
$750$	1.08886	0.0397596
$751$	39.2287	1.43148	0.715738	−	0.698369i	$-0.246091\pi$
$751$	39.2287	1.43148	0.715738	+	0.698369i	$0.246091\pi$
$752$	−46.1873	−1.68428
$753$	−2.38571	−0.0869403
$754$	0.0889608	0.00323976
$755$	−47.6434	−1.73392
$756$	1.99851	0.0726852
$757$	−41.6642	−1.51431	−0.757156	−	0.653234i	$-0.773412\pi$
$757$	−41.6642	−1.51431	−0.757156	+	0.653234i	$0.773412\pi$
$758$	0.797848	0.0289791
$759$	8.21279	0.298105
$760$	3.93480	0.142730
$761$	42.0484	1.52425	0.762126	−	0.647428i	$-0.224156\pi$
$761$	42.0484	1.52425	0.762126	+	0.647428i	$0.224156\pi$
$762$	−0.788071	−0.0285488
$763$	−1.03550	−0.0374874
$764$	−26.1895	−0.947503
$765$	10.2996	0.372384
$766$	−0.0385832	−0.00139407
$767$	4.80393	0.173460
$768$	−15.8810	−0.573058
$769$	24.2143	0.873191	0.436595	−	0.899658i	$-0.356184\pi$
$769$	24.2143	0.873191	0.436595	+	0.899658i	$0.356184\pi$
$770$	0.209826	0.00756159
$771$	13.3061	0.479208
$772$	−4.00446	−0.144124
$773$	41.5176	1.49328	0.746642	−	0.665226i	$-0.231665\pi$
$773$	41.5176	1.49328	0.746642	+	0.665226i	$0.231665\pi$
$774$	−0.328265	−0.0117992
$775$	15.9857	0.574224
$776$	0.896621	0.0321868
$777$	8.07420	0.289660
$778$	−0.176908	−0.00634244
$779$	−66.5470	−2.38430
$780$	−6.25800	−0.224072
$781$	6.69990	0.239741
$782$	−0.600135	−0.0214608
$783$	−3.02440	−0.108083
$784$	3.99107	0.142538
$785$	−68.1065	−2.43082
$786$	0.219323	0.00782298
$787$	−11.7115	−0.417469	−0.208735	−	0.977972i	$-0.566934\pi$
$787$	−11.7115	−0.417469	−0.208735	+	0.977972i	$0.566934\pi$
$788$	15.6061	0.555943
$789$	4.53672	0.161512
$790$	0.926749	0.0329723
$791$	12.6265	0.448948
$792$	0.204263	0.00725816
$793$	5.66623	0.201214
$794$	0.0383127	0.00135967
$795$	2.11152	0.0748879
$796$	25.2875	0.896291
$797$	−7.15130	−0.253312	−0.126656	−	0.991947i	$-0.540424\pi$
$797$	−7.15130	−0.253312	−0.126656	+	0.991947i	$0.540424\pi$
$798$	−0.239583	−0.00848116
$799$	29.0193	1.02663
$800$	5.49070	0.194126
$801$	4.55157	0.160822
$802$	−1.35320	−0.0477831
$803$	2.20936	0.0779668
$804$	11.1533	0.393345
$805$	−25.4780	−0.897982
$806$	0.0396106	0.00139522
$807$	−22.2293	−0.782508
$808$	1.55463	0.0546918
$809$	−24.0927	−0.847053	−0.423526	−	0.905884i	$-0.639208\pi$
$809$	−24.0927	−0.847053	−0.423526	+	0.905884i	$0.639208\pi$
$810$	−0.158477	−0.00556831
$811$	−3.97925	−0.139730	−0.0698652	−	0.997556i	$-0.522257\pi$
$811$	−3.97925	−0.139730	−0.0698652	+	0.997556i	$0.522257\pi$
$812$	−6.04430	−0.212113
$813$	22.2823	0.781474
$814$	0.412468	0.0144570
$815$	−91.1900	−3.19424
$816$	10.0079	0.350346
$817$	−52.8306	−1.84831
$818$	−0.527179	−0.0184324
$819$	0.762362	0.0266391
$820$	−87.9720	−3.07211
$821$	−7.79264	−0.271965	−0.135983	−	0.990711i	$-0.543419\pi$
$821$	−7.79264	−0.271965	−0.135983	+	0.990711i	$0.543419\pi$
$822$	0.588142	0.0205138
$823$	39.4895	1.37652	0.688258	−	0.725466i	$-0.258376\pi$
$823$	39.4895	1.37652	0.688258	+	0.725466i	$0.258376\pi$
$824$	−1.65822	−0.0577669
$825$	15.7171	0.547200
$826$	0.243127	0.00845947
$827$	−1.09654	−0.0381304	−0.0190652	−	0.999818i	$-0.506069\pi$
$827$	−1.09654	−0.0381304	−0.0190652	+	0.999818i	$0.506069\pi$
$828$	−12.3967	−0.430813
$829$	15.3429	0.532880	0.266440	−	0.963852i	$-0.414153\pi$
$829$	15.3429	0.532880	0.266440	+	0.963852i	$0.414153\pi$
$830$	0.402782	0.0139808
$831$	−0.726184	−0.0251910
$832$	−6.07168	−0.210498
$833$	−2.50757	−0.0868822
$834$	0.684205	0.0236921
$835$	15.1113	0.522948
$836$	16.4308	0.568271
$837$	−1.34664	−0.0465468
$838$	−1.02146	−0.0352859
$839$	20.2962	0.700704	0.350352	−	0.936618i	$-0.386062\pi$
$839$	20.2962	0.700704	0.350352	+	0.936618i	$0.386062\pi$
$840$	−0.633671	−0.0218637
$841$	−19.8530	−0.684586
$842$	0.678127	0.0233698
$843$	14.9929	0.516384
$844$	−43.2390	−1.48835
$845$	51.0091	1.75477
$846$	−0.446510	−0.0153513
$847$	−9.24698	−0.317730
$848$	2.05171	0.0704561
$849$	−24.5524	−0.842635
$850$	−1.14850	−0.0393932
$851$	−50.0838	−1.71685
$852$	−10.1130	−0.346467
$853$	−39.0396	−1.33669	−0.668346	−	0.743851i	$-0.732997\pi$
$853$	−39.0396	−1.33669	−0.668346	+	0.743851i	$0.732997\pi$
$854$	0.286768	0.00981299
$855$	−25.5051	−0.872256
$856$	0.549618	0.0187856
$857$	−9.18755	−0.313841	−0.156920	−	0.987611i	$-0.550157\pi$
$857$	−9.18755	−0.313841	−0.156920	+	0.987611i	$0.550157\pi$
$858$	0.0389450	0.00132956
$859$	23.7012	0.808676	0.404338	−	0.914610i	$-0.367502\pi$
$859$	23.7012	0.808676	0.404338	+	0.914610i	$0.367502\pi$
$860$	−69.8395	−2.38151
$861$	10.7169	0.365232
$862$	−0.428207	−0.0145848
$863$	−52.4809	−1.78647	−0.893236	−	0.449589i	$-0.851570\pi$
$863$	−52.4809	−1.78647	−0.893236	+	0.449589i	$0.851570\pi$
$864$	−0.462538	−0.0157359
$865$	33.8184	1.14986
$866$	0.872705	0.0296557
$867$	10.7121	0.363801
$868$	−2.69128	−0.0913480
$869$	7.74265	0.262651
$870$	0.479297	0.0162497
$871$	4.25458	0.144161
$872$	0.159751	0.00540986
$873$	−5.81183	−0.196701
$874$	1.48612	0.0502688
$875$	−28.2212	−0.954051
$876$	−3.33489	−0.112675
$877$	35.3503	1.19369	0.596847	−	0.802355i	$-0.296420\pi$
$877$	35.3503	1.19369	0.596847	+	0.802355i	$0.296420\pi$
$878$	−0.319611	−0.0107863
$879$	15.1631	0.511439
$880$	21.7045	0.731659
$881$	14.1212	0.475757	0.237878	−	0.971295i	$-0.423548\pi$
$881$	14.1212	0.475757	0.237878	+	0.971295i	$0.423548\pi$
$882$	0.0385832	0.00129916
$883$	47.1077	1.58530	0.792651	−	0.609676i	$-0.208700\pi$
$883$	47.1077	1.58530	0.792651	+	0.609676i	$0.208700\pi$
$884$	3.82051	0.128498
$885$	25.8823	0.870025
$886$	1.21379	0.0407782
$887$	−13.3441	−0.448053	−0.224026	−	0.974583i	$-0.571920\pi$
$887$	−13.3441	−0.448053	−0.224026	+	0.974583i	$0.571920\pi$
$888$	−1.24565	−0.0418013
$889$	20.4253	0.685041
$890$	−0.721318	−0.0241786
$891$	−1.32401	−0.0443562
$892$	23.2269	0.777695
$893$	−71.8608	−2.40473
$894$	−0.117155	−0.00391824
$895$	−54.5647	−1.82390
$896$	−1.23236	−0.0411704
$897$	−4.72889	−0.157893
$898$	0.760152	0.0253666
$899$	4.07279	0.135835
$900$	−23.7239	−0.790798
$901$	−1.28908	−0.0429455
$902$	0.547470	0.0182288
$903$	8.50798	0.283128
$904$	−1.94796	−0.0647882
$905$	−0.485948	−0.0161535
$906$	−0.447540	−0.0148685
$907$	−48.3800	−1.60643	−0.803216	−	0.595687i	$-0.796880\pi$
$907$	−48.3800	−1.60643	−0.803216	+	0.595687i	$0.796880\pi$
$908$	−4.48344	−0.148788
$909$	−10.0770	−0.334233
$910$	−0.120817	−0.00400503
$911$	17.0726	0.565640	0.282820	−	0.959173i	$-0.408730\pi$
$911$	17.0726	0.565640	0.282820	+	0.959173i	$0.408730\pi$
$912$	−24.7827	−0.820637
$913$	3.36510	0.111369
$914$	0.0575827	0.00190467
$915$	30.5282	1.00923
$916$	37.9862	1.25510
$917$	−5.68441	−0.187716
$918$	0.0967500	0.00319323
$919$	36.8476	1.21549	0.607745	−	0.794132i	$-0.292074\pi$
$919$	36.8476	1.21549	0.607745	+	0.794132i	$0.292074\pi$
$920$	3.93063	0.129589
$921$	−13.4192	−0.442178
$922$	0.493855	0.0162643
$923$	−3.85777	−0.126980
$924$	−2.64606	−0.0870489
$925$	−95.8473	−3.15144
$926$	−0.784663	−0.0257856
$927$	10.7485	0.353026
$928$	1.39890	0.0459212
$929$	11.0087	0.361184	0.180592	−	0.983558i	$-0.442199\pi$
$929$	11.0087	0.361184	0.180592	+	0.983558i	$0.442199\pi$
$930$	0.213411	0.00699804
$931$	6.20953	0.203509
$932$	−18.9687	−0.621340
$933$	1.81894	0.0595495
$934$	1.12772	0.0369000
$935$	−13.6369	−0.445973
$936$	−0.117614	−0.00384432
$937$	35.7309	1.16728	0.583638	−	0.812014i	$-0.301629\pi$
$937$	35.7309	1.16728	0.583638	+	0.812014i	$0.301629\pi$
$938$	0.215324	0.00703059
$939$	15.0473	0.491049
$940$	−94.9965	−3.09844
$941$	49.7977	1.62336	0.811680	−	0.584103i	$-0.198553\pi$
$941$	49.7977	1.62336	0.811680	+	0.584103i	$0.198553\pi$
$942$	−0.639762	−0.0208446
$943$	−66.4764	−2.16477
$944$	25.1492	0.818538
$945$	4.10741	0.133614
$946$	0.434627	0.0141310
$947$	50.9569	1.65588	0.827939	−	0.560819i	$-0.189514\pi$
$947$	50.9569	1.65588	0.827939	+	0.560819i	$0.189514\pi$
$948$	−11.6870	−0.379576
$949$	−1.27214	−0.0412955
$950$	2.84405	0.0922731
$951$	−13.6762	−0.443482
$952$	0.386856	0.0125381
$953$	16.4784	0.533789	0.266895	−	0.963726i	$-0.414002\pi$
$953$	16.4784	0.533789	0.266895	+	0.963726i	$0.414002\pi$
$954$	0.0198347	0.000642171 0
$955$	−53.8256	−1.74175
$956$	31.4154	1.01605
$957$	4.00435	0.129442
$958$	0.196923	0.00636228
$959$	−15.2435	−0.492238
$960$	−32.7126	−1.05579
$961$	−29.1866	−0.941502
$962$	−0.237497	−0.00765722
$963$	−3.56259	−0.114803
$964$	−30.7612	−0.990750
$965$	−8.23010	−0.264936
$966$	−0.239329	−0.00770029
$967$	30.0682	0.966928	0.483464	−	0.875364i	$-0.339378\pi$
$967$	30.0682	0.966928	0.483464	+	0.875364i	$0.339378\pi$
$968$	1.42658	0.0458520
$969$	15.5709	0.500208
$970$	0.921040	0.0295728
$971$	1.29098	0.0414296	0.0207148	−	0.999785i	$-0.493406\pi$
$971$	1.29098	0.0414296	0.0207148	+	0.999785i	$0.493406\pi$
$972$	1.99851	0.0641023
$973$	−17.7333	−0.568502
$974$	−0.794471	−0.0254565
$975$	−9.04985	−0.289827
$976$	29.6635	0.949505
$977$	39.4498	1.26211	0.631055	−	0.775738i	$-0.282622\pi$
$977$	39.4498	1.26211	0.631055	+	0.775738i	$0.282622\pi$
$978$	−0.856598	−0.0273910
$979$	−6.02635	−0.192603
$980$	8.20870	0.262217
$981$	−1.03550	−0.0330608
$982$	−0.0863198	−0.00275458
$983$	3.51642	0.112157	0.0560783	−	0.998426i	$-0.482140\pi$
$983$	3.51642	0.112157	0.0560783	+	0.998426i	$0.482140\pi$
$984$	−1.65335	−0.0527070
$985$	32.0741	1.02197
$986$	−0.292611	−0.00931863
$987$	11.5727	0.368362
$988$	−9.46078	−0.300987
$989$	−52.7745	−1.67813
$990$	0.209826	0.00666869
$991$	29.1067	0.924603	0.462302	−	0.886723i	$-0.347024\pi$
$991$	29.1067	0.924603	0.462302	+	0.886723i	$0.347024\pi$
$992$	0.622874	0.0197763
$993$	22.3294	0.708601
$994$	−0.195242	−0.00619270
$995$	51.9717	1.64761
$996$	−5.07939	−0.160947
$997$	−18.2877	−0.579176	−0.289588	−	0.957151i	$-0.593518\pi$
$997$	−18.2877	−0.579176	−0.289588	+	0.957151i	$0.593518\pi$
$998$	1.23630	0.0391343
$999$	8.07420	0.255457

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8043.2.a.t.1.27	✓	52

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8043.2.a.t.1.27	✓	52	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)

\(f(q)\)	\(=\)	\(q+0.0385832 q^{2} -1.00000 q^{3} -1.99851 q^{4} -4.10741 q^{5} -0.0385832 q^{6} +1.00000 q^{7} -0.154275 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+0.0385832 q^{2} -1.00000 q^{3} -1.99851 q^{4} -4.10741 q^{5} -0.0385832 q^{6} +1.00000 q^{7} -0.154275 q^{8} +1.00000 q^{9} -0.158477 q^{10} -1.32401 q^{11} +1.99851 q^{12} +0.762362 q^{13} +0.0385832 q^{14} +4.10741 q^{15} +3.99107 q^{16} -2.50757 q^{17} +0.0385832 q^{18} +6.20953 q^{19} +8.20870 q^{20} -1.00000 q^{21} -0.0510847 q^{22} +6.20294 q^{23} +0.154275 q^{24} +11.8708 q^{25} +0.0294143 q^{26} -1.00000 q^{27} -1.99851 q^{28} +3.02440 q^{29} +0.158477 q^{30} +1.34664 q^{31} +0.462538 q^{32} +1.32401 q^{33} -0.0967500 q^{34} -4.10741 q^{35} -1.99851 q^{36} -8.07420 q^{37} +0.239583 q^{38} -0.762362 q^{39} +0.633671 q^{40} -10.7169 q^{41} -0.0385832 q^{42} -8.50798 q^{43} +2.64606 q^{44} -4.10741 q^{45} +0.239329 q^{46} -11.5727 q^{47} -3.99107 q^{48} +1.00000 q^{49} +0.458013 q^{50} +2.50757 q^{51} -1.52359 q^{52} +0.514076 q^{53} -0.0385832 q^{54} +5.43827 q^{55} -0.154275 q^{56} -6.20953 q^{57} +0.116691 q^{58} +6.30138 q^{59} -8.20870 q^{60} +7.43246 q^{61} +0.0519577 q^{62} +1.00000 q^{63} -7.96429 q^{64} -3.13133 q^{65} +0.0510847 q^{66} +5.58079 q^{67} +5.01141 q^{68} -6.20294 q^{69} -0.158477 q^{70} -5.06029 q^{71} -0.154275 q^{72} -1.66869 q^{73} -0.311528 q^{74} -11.8708 q^{75} -12.4098 q^{76} -1.32401 q^{77} -0.0294143 q^{78} -5.84786 q^{79} -16.3930 q^{80} +1.00000 q^{81} -0.413492 q^{82} -2.54159 q^{83} +1.99851 q^{84} +10.2996 q^{85} -0.328265 q^{86} -3.02440 q^{87} +0.204263 q^{88} +4.55157 q^{89} -0.158477 q^{90} +0.762362 q^{91} -12.3967 q^{92} -1.34664 q^{93} -0.446510 q^{94} -25.5051 q^{95} -0.462538 q^{96} -5.81183 q^{97} +0.0385832 q^{98} -1.32401 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 52 q + 3 q^{2} - 52 q^{3} + 61 q^{4} - 7 q^{5} - 3 q^{6} + 52 q^{7} + 24 q^{8} + 52 q^{9}+O(q^{10}) \) 52 * q + 3 * q^2 - 52 * q^3 + 61 * q^4 - 7 * q^5 - 3 * q^6 + 52 * q^7 + 24 * q^8 + 52 * q^9	\( 52 q + 3 q^{2} - 52 q^{3} + 61 q^{4} - 7 q^{5} - 3 q^{6} + 52 q^{7} + 24 q^{8} + 52 q^{9} - 2 q^{10} + 9 q^{11} - 61 q^{12} + 44 q^{13} + 3 q^{14} + 7 q^{15} + 95 q^{16} - 6 q^{17} + 3 q^{18} + 7 q^{19} - 21 q^{20} - 52 q^{21} + 19 q^{22} - 4 q^{23} - 24 q^{24} + 83 q^{25} - 5 q^{26} - 52 q^{27} + 61 q^{28} + 31 q^{29} + 2 q^{30} + 11 q^{31} + 71 q^{32} - 9 q^{33} + 17 q^{34} - 7 q^{35} + 61 q^{36} + 71 q^{37} - 8 q^{38} - 44 q^{39} + 20 q^{40} - 25 q^{41} - 3 q^{42} + 75 q^{43} + 14 q^{44} - 7 q^{45} + 36 q^{46} - 20 q^{47} - 95 q^{48} + 52 q^{49} + 26 q^{50} + 6 q^{51} + 88 q^{52} + 70 q^{53} - 3 q^{54} + 12 q^{55} + 24 q^{56} - 7 q^{57} + 48 q^{58} - 27 q^{59} + 21 q^{60} + 59 q^{61} - 23 q^{62} + 52 q^{63} + 138 q^{64} + 44 q^{65} - 19 q^{66} + 65 q^{67} - 8 q^{68} + 4 q^{69} - 2 q^{70} - 11 q^{71} + 24 q^{72} + 34 q^{73} + 38 q^{74} - 83 q^{75} + 31 q^{76} + 9 q^{77} + 5 q^{78} + 74 q^{79} - 5 q^{80} + 52 q^{81} + 51 q^{82} - 30 q^{83} - 61 q^{84} + 70 q^{85} + 29 q^{86} - 31 q^{87} + 90 q^{88} - q^{89} - 2 q^{90} + 44 q^{91} + 34 q^{92} - 11 q^{93} + 27 q^{94} + 9 q^{95} - 71 q^{96} + 73 q^{97} + 3 q^{98} + 9 q^{99}+O(q^{100}) \) 52 * q + 3 * q^2 - 52 * q^3 + 61 * q^4 - 7 * q^5 - 3 * q^6 + 52 * q^7 + 24 * q^8 + 52 * q^9 - 2 * q^10 + 9 * q^11 - 61 * q^12 + 44 * q^13 + 3 * q^14 + 7 * q^15 + 95 * q^16 - 6 * q^17 + 3 * q^18 + 7 * q^19 - 21 * q^20 - 52 * q^21 + 19 * q^22 - 4 * q^23 - 24 * q^24 + 83 * q^25 - 5 * q^26 - 52 * q^27 + 61 * q^28 + 31 * q^29 + 2 * q^30 + 11 * q^31 + 71 * q^32 - 9 * q^33 + 17 * q^34 - 7 * q^35 + 61 * q^36 + 71 * q^37 - 8 * q^38 - 44 * q^39 + 20 * q^40 - 25 * q^41 - 3 * q^42 + 75 * q^43 + 14 * q^44 - 7 * q^45 + 36 * q^46 - 20 * q^47 - 95 * q^48 + 52 * q^49 + 26 * q^50 + 6 * q^51 + 88 * q^52 + 70 * q^53 - 3 * q^54 + 12 * q^55 + 24 * q^56 - 7 * q^57 + 48 * q^58 - 27 * q^59 + 21 * q^60 + 59 * q^61 - 23 * q^62 + 52 * q^63 + 138 * q^64 + 44 * q^65 - 19 * q^66 + 65 * q^67 - 8 * q^68 + 4 * q^69 - 2 * q^70 - 11 * q^71 + 24 * q^72 + 34 * q^73 + 38 * q^74 - 83 * q^75 + 31 * q^76 + 9 * q^77 + 5 * q^78 + 74 * q^79 - 5 * q^80 + 52 * q^81 + 51 * q^82 - 30 * q^83 - 61 * q^84 + 70 * q^85 + 29 * q^86 - 31 * q^87 + 90 * q^88 - q^89 - 2 * q^90 + 44 * q^91 + 34 * q^92 - 11 * q^93 + 27 * q^94 + 9 * q^95 - 71 * q^96 + 73 * q^97 + 3 * q^98 + 9 * q^99

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