LMFDB - Embedded newform 4031.2.a.c.1.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4031,2,Mod(1,4031)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4031, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4031.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4031 = 29 \cdot 139 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4031.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:	$32.1876970548$
Analytic rank:	$1$
Dimension:	$61$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Character	$\chi$	$=$	4031.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.13471 q^{2} +1.15618 q^{3} -0.712442 q^{4} +3.10962 q^{5} -1.31192 q^{6} -1.02298 q^{7} +3.07782 q^{8} -1.66325 q^{9} +O(q^{10})$	\(q-1.13471 q^{2} +1.15618 q^{3} -0.712442 q^{4} +3.10962 q^{5} -1.31192 q^{6} -1.02298 q^{7} +3.07782 q^{8} -1.66325 q^{9} -3.52850 q^{10} -1.83622 q^{11} -0.823711 q^{12} +3.48520 q^{13} +1.16078 q^{14} +3.59528 q^{15} -2.06754 q^{16} -1.09492 q^{17} +1.88730 q^{18} -2.20806 q^{19} -2.21542 q^{20} -1.18275 q^{21} +2.08357 q^{22} -1.23653 q^{23} +3.55852 q^{24} +4.66973 q^{25} -3.95467 q^{26} -5.39155 q^{27} +0.728812 q^{28} -1.00000 q^{29} -4.07958 q^{30} -4.93412 q^{31} -3.80960 q^{32} -2.12300 q^{33} +1.24242 q^{34} -3.18107 q^{35} +1.18497 q^{36} -10.7505 q^{37} +2.50550 q^{38} +4.02951 q^{39} +9.57086 q^{40} -3.31015 q^{41} +1.34207 q^{42} +7.70702 q^{43} +1.30820 q^{44} -5.17207 q^{45} +1.40309 q^{46} +7.43359 q^{47} -2.39045 q^{48} -5.95352 q^{49} -5.29877 q^{50} -1.26593 q^{51} -2.48300 q^{52} -2.52651 q^{53} +6.11783 q^{54} -5.70995 q^{55} -3.14854 q^{56} -2.55292 q^{57} +1.13471 q^{58} +4.84374 q^{59} -2.56143 q^{60} -5.33508 q^{61} +5.59878 q^{62} +1.70147 q^{63} +8.45786 q^{64} +10.8376 q^{65} +2.40899 q^{66} -10.1714 q^{67} +0.780070 q^{68} -1.42965 q^{69} +3.60958 q^{70} -8.54404 q^{71} -5.11919 q^{72} +6.94608 q^{73} +12.1986 q^{74} +5.39905 q^{75} +1.57312 q^{76} +1.87841 q^{77} -4.57231 q^{78} +6.60000 q^{79} -6.42927 q^{80} -1.24386 q^{81} +3.75605 q^{82} -9.84384 q^{83} +0.842638 q^{84} -3.40480 q^{85} -8.74520 q^{86} -1.15618 q^{87} -5.65157 q^{88} -8.43394 q^{89} +5.86878 q^{90} -3.56528 q^{91} +0.880954 q^{92} -5.70473 q^{93} -8.43494 q^{94} -6.86623 q^{95} -4.40458 q^{96} -1.62517 q^{97} +6.75549 q^{98} +3.05409 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 61 q - q^{2} - 4 q^{3} + 43 q^{4} - 7 q^{5} - 13 q^{6} - 10 q^{7} - 6 q^{8} + 23 q^{9}+O(q^{10}) $ 61 * q - q^2 - 4 * q^3 + 43 * q^4 - 7 * q^5 - 13 * q^6 - 10 * q^7 - 6 * q^8 + 23 * q^9	\( 61 q - q^{2} - 4 q^{3} + 43 q^{4} - 7 q^{5} - 13 q^{6} - 10 q^{7} - 6 q^{8} + 23 q^{9} - 16 q^{10} - 3 q^{11} - 18 q^{12} - 28 q^{13} - 14 q^{14} - 12 q^{15} + 11 q^{16} - 21 q^{17} - 17 q^{18} - 36 q^{19} - 16 q^{20} - 12 q^{21} - 42 q^{22} - 15 q^{23} - 28 q^{24} - 16 q^{25} - 13 q^{26} - 10 q^{27} - 25 q^{28} - 61 q^{29} - 12 q^{30} - 18 q^{31} - 3 q^{32} - 42 q^{33} - 22 q^{34} - 29 q^{35} - 38 q^{36} - 30 q^{37} - 27 q^{38} - 31 q^{39} - 22 q^{40} - 28 q^{41} - 9 q^{42} - 58 q^{43} - 2 q^{44} - 31 q^{45} - 40 q^{46} - 6 q^{47} - 37 q^{48} - 37 q^{49} - 15 q^{50} - 44 q^{51} - 43 q^{52} - 27 q^{53} - 18 q^{54} - 38 q^{55} - 22 q^{56} - 50 q^{57} + q^{58} - 24 q^{59} + 6 q^{60} - 76 q^{61} - 17 q^{62} - 6 q^{63} - 60 q^{64} - 65 q^{65} - 7 q^{66} - 45 q^{67} - 31 q^{68} - 16 q^{69} - 48 q^{70} - 28 q^{71} - 40 q^{72} - 50 q^{73} - 17 q^{74} - 35 q^{75} - 100 q^{76} + q^{77} - 6 q^{78} - 66 q^{79} - 10 q^{80} - 63 q^{81} - 5 q^{82} - 9 q^{83} - 24 q^{84} - 77 q^{85} + 29 q^{86} + 4 q^{87} - 62 q^{88} - 30 q^{89} + 50 q^{90} - 52 q^{91} - 53 q^{92} - 42 q^{93} - 92 q^{94} - 20 q^{95} - 47 q^{96} - 34 q^{97} + 36 q^{98} - 29 q^{99}+O(q^{100}) \) 61 * q - q^2 - 4 * q^3 + 43 * q^4 - 7 * q^5 - 13 * q^6 - 10 * q^7 - 6 * q^8 + 23 * q^9 - 16 * q^10 - 3 * q^11 - 18 * q^12 - 28 * q^13 - 14 * q^14 - 12 * q^15 + 11 * q^16 - 21 * q^17 - 17 * q^18 - 36 * q^19 - 16 * q^20 - 12 * q^21 - 42 * q^22 - 15 * q^23 - 28 * q^24 - 16 * q^25 - 13 * q^26 - 10 * q^27 - 25 * q^28 - 61 * q^29 - 12 * q^30 - 18 * q^31 - 3 * q^32 - 42 * q^33 - 22 * q^34 - 29 * q^35 - 38 * q^36 - 30 * q^37 - 27 * q^38 - 31 * q^39 - 22 * q^40 - 28 * q^41 - 9 * q^42 - 58 * q^43 - 2 * q^44 - 31 * q^45 - 40 * q^46 - 6 * q^47 - 37 * q^48 - 37 * q^49 - 15 * q^50 - 44 * q^51 - 43 * q^52 - 27 * q^53 - 18 * q^54 - 38 * q^55 - 22 * q^56 - 50 * q^57 + q^58 - 24 * q^59 + 6 * q^60 - 76 * q^61 - 17 * q^62 - 6 * q^63 - 60 * q^64 - 65 * q^65 - 7 * q^66 - 45 * q^67 - 31 * q^68 - 16 * q^69 - 48 * q^70 - 28 * q^71 - 40 * q^72 - 50 * q^73 - 17 * q^74 - 35 * q^75 - 100 * q^76 + q^77 - 6 * q^78 - 66 * q^79 - 10 * q^80 - 63 * q^81 - 5 * q^82 - 9 * q^83 - 24 * q^84 - 77 * q^85 + 29 * q^86 + 4 * q^87 - 62 * q^88 - 30 * q^89 + 50 * q^90 - 52 * q^91 - 53 * q^92 - 42 * q^93 - 92 * q^94 - 20 * q^95 - 47 * q^96 - 34 * q^97 + 36 * q^98 - 29 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.13471	−0.802358	−0.401179	−	0.916000i	$-0.631400\pi$
$2$	−1.13471	−0.802358	−0.401179	+	0.916000i	$0.631400\pi$
$3$	1.15618	0.667521	0.333760	−	0.942658i	$-0.391682\pi$
$3$	1.15618	0.667521	0.333760	+	0.942658i	$0.391682\pi$
$4$	−0.712442	−0.356221
$5$	3.10962	1.39066	0.695332	−	0.718689i	$-0.255257\pi$
$5$	3.10962	1.39066	0.695332	+	0.718689i	$0.255257\pi$
$6$	−1.31192	−0.535591
$7$	−1.02298	−0.386649	−0.193325	−	0.981135i	$-0.561927\pi$
$7$	−1.02298	−0.386649	−0.193325	+	0.981135i	$0.561927\pi$
$8$	3.07782	1.08818
$9$	−1.66325	−0.554416
$10$	−3.52850	−1.11581
$11$	−1.83622	−0.553642	−0.276821	−	0.960922i	$-0.589281\pi$
$11$	−1.83622	−0.553642	−0.276821	+	0.960922i	$0.589281\pi$
$12$	−0.823711	−0.237785
$13$	3.48520	0.966620	0.483310	−	0.875449i	$-0.339434\pi$
$13$	3.48520	0.966620	0.483310	+	0.875449i	$0.339434\pi$
$14$	1.16078	0.310231
$15$	3.59528	0.928297
$16$	−2.06754	−0.516885
$17$	−1.09492	−0.265558	−0.132779	−	0.991146i	$-0.542390\pi$
$17$	−1.09492	−0.265558	−0.132779	+	0.991146i	$0.542390\pi$
$18$	1.88730	0.444840
$19$	−2.20806	−0.506564	−0.253282	−	0.967392i	$-0.581510\pi$
$19$	−2.20806	−0.506564	−0.253282	+	0.967392i	$0.581510\pi$
$20$	−2.21542	−0.495384
$21$	−1.18275	−0.258096
$22$	2.08357	0.444219
$23$	−1.23653	−0.257834	−0.128917	−	0.991655i	$-0.541150\pi$
$23$	−1.23653	−0.257834	−0.128917	+	0.991655i	$0.541150\pi$
$24$	3.55852	0.726380
$25$	4.66973	0.933946
$26$	−3.95467	−0.775575
$27$	−5.39155	−1.03760
$28$	0.728812	0.137733
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	−4.07958	−0.744827
$31$	−4.93412	−0.886194	−0.443097	−	0.896474i	$-0.646120\pi$
$31$	−4.93412	−0.886194	−0.443097	+	0.896474i	$0.646120\pi$
$32$	−3.80960	−0.673448
$33$	−2.12300	−0.369568
$34$	1.24242	0.213073
$35$	−3.18107	−0.537699
$36$	1.18497	0.197495
$37$	−10.7505	−1.76737	−0.883684	−	0.468085i	$-0.844944\pi$
$37$	−10.7505	−1.76737	−0.883684	+	0.468085i	$0.844944\pi$
$38$	2.50550	0.406446
$39$	4.02951	0.645239
$40$	9.57086	1.51329
$41$	−3.31015	−0.516959	−0.258479	−	0.966017i	$-0.583221\pi$
$41$	−3.31015	−0.516959	−0.258479	+	0.966017i	$0.583221\pi$
$42$	1.34207	0.207086
$43$	7.70702	1.17531	0.587655	−	0.809112i	$-0.300051\pi$
$43$	7.70702	1.17531	0.587655	+	0.809112i	$0.300051\pi$
$44$	1.30820	0.197219
$45$	−5.17207	−0.771006
$46$	1.40309	0.206875
$47$	7.43359	1.08430	0.542150	−	0.840281i	$-0.317610\pi$
$47$	7.43359	1.08430	0.542150	+	0.840281i	$0.317610\pi$
$48$	−2.39045	−0.345032
$49$	−5.95352	−0.850502
$50$	−5.29877	−0.749359
$51$	−1.26593	−0.177266
$52$	−2.48300	−0.344330
$53$	−2.52651	−0.347043	−0.173521	−	0.984830i	$-0.555515\pi$
$53$	−2.52651	−0.347043	−0.173521	+	0.984830i	$0.555515\pi$
$54$	6.11783	0.832531
$55$	−5.70995	−0.769930
$56$	−3.14854	−0.420742
$57$	−2.55292	−0.338142
$58$	1.13471	0.148994
$59$	4.84374	0.630601	0.315300	−	0.948992i	$-0.397895\pi$
$59$	4.84374	0.630601	0.315300	+	0.948992i	$0.397895\pi$
$60$	−2.56143	−0.330679
$61$	−5.33508	−0.683087	−0.341544	−	0.939866i	$-0.610950\pi$
$61$	−5.33508	−0.683087	−0.341544	+	0.939866i	$0.610950\pi$
$62$	5.59878	0.711045
$63$	1.70147	0.214364
$64$	8.45786	1.05723
$65$	10.8376	1.34424
$66$	2.40899	0.296526
$67$	−10.1714	−1.24264	−0.621319	−	0.783557i	$-0.713403\pi$
$67$	−10.1714	−1.24264	−0.621319	+	0.783557i	$0.713403\pi$
$68$	0.780070	0.0945974
$69$	−1.42965	−0.172109
$70$	3.60958	0.431427
$71$	−8.54404	−1.01399	−0.506995	−	0.861949i	$-0.669244\pi$
$71$	−8.54404	−1.01399	−0.506995	+	0.861949i	$0.669244\pi$
$72$	−5.11919	−0.603302
$73$	6.94608	0.812977	0.406488	−	0.913656i	$-0.366753\pi$
$73$	6.94608	0.812977	0.406488	+	0.913656i	$0.366753\pi$
$74$	12.1986	1.41806
$75$	5.39905	0.623428
$76$	1.57312	0.180449
$77$	1.87841	0.214065
$78$	−4.57231	−0.517713
$79$	6.60000	0.742558	0.371279	−	0.928521i	$-0.378919\pi$
$79$	6.60000	0.742558	0.371279	+	0.928521i	$0.378919\pi$
$80$	−6.42927	−0.718814
$81$	−1.24386	−0.138207
$82$	3.75605	0.414786
$83$	−9.84384	−1.08050	−0.540251	−	0.841504i	$-0.681671\pi$
$83$	−9.84384	−1.08050	−0.540251	+	0.841504i	$0.681671\pi$
$84$	0.842638	0.0919394
$85$	−3.40480	−0.369302
$86$	−8.74520	−0.943020
$87$	−1.15618	−0.123955
$88$	−5.65157	−0.602460
$89$	−8.43394	−0.893995	−0.446998	−	0.894535i	$-0.647507\pi$
$89$	−8.43394	−0.893995	−0.446998	+	0.894535i	$0.647507\pi$
$90$	5.86878	0.618623
$91$	−3.56528	−0.373743
$92$	0.880954	0.0918458
$93$	−5.70473	−0.591553
$94$	−8.43494	−0.869998
$95$	−6.86623	−0.704460
$96$	−4.40458	−0.449540
$97$	−1.62517	−0.165011	−0.0825055	−	0.996591i	$-0.526292\pi$
$97$	−1.62517	−0.165011	−0.0825055	+	0.996591i	$0.526292\pi$
$98$	6.75549	0.682408
$99$	3.05409	0.306948
$100$	−3.32691	−0.332691
$101$	4.03793	0.401789	0.200895	−	0.979613i	$-0.435615\pi$
$101$	4.03793	0.401789	0.200895	+	0.979613i	$0.435615\pi$
$102$	1.43646	0.142231
$103$	3.28005	0.323193	0.161597	−	0.986857i	$-0.448336\pi$
$103$	3.28005	0.323193	0.161597	+	0.986857i	$0.448336\pi$
$104$	10.7268	1.05185
$105$	−3.67789	−0.358925
$106$	2.86684	0.278453
$107$	−2.42399	−0.234336	−0.117168	−	0.993112i	$-0.537382\pi$
$107$	−2.42399	−0.234336	−0.117168	+	0.993112i	$0.537382\pi$
$108$	3.84117	0.369617
$109$	−2.19586	−0.210326	−0.105163	−	0.994455i	$-0.533536\pi$
$109$	−2.19586	−0.210326	−0.105163	+	0.994455i	$0.533536\pi$
$110$	6.47912	0.617760
$111$	−12.4295	−1.17975
$112$	2.11505	0.199853
$113$	−6.09089	−0.572983	−0.286491	−	0.958083i	$-0.592489\pi$
$113$	−6.09089	−0.572983	−0.286491	+	0.958083i	$0.592489\pi$
$114$	2.89681	0.271311
$115$	−3.84513	−0.358560
$116$	0.712442	0.0661486
$117$	−5.79675	−0.535909
$118$	−5.49622	−0.505968
$119$	1.12008	0.102678
$120$	11.0656	1.01015
$121$	−7.62829	−0.693481
$122$	6.05375	0.548081
$123$	−3.82713	−0.345081
$124$	3.51528	0.315681
$125$	−1.02702	−0.0918593
$126$	−1.93066	−0.171997
$127$	−6.01420	−0.533674	−0.266837	−	0.963742i	$-0.585978\pi$
$127$	−6.01420	−0.533674	−0.266837	+	0.963742i	$0.585978\pi$
$128$	−1.97799	−0.174831
$129$	8.91070	0.784544
$130$	−12.2975	−1.07856
$131$	−6.54336	−0.571696	−0.285848	−	0.958275i	$-0.592275\pi$
$131$	−6.54336	−0.571696	−0.285848	+	0.958275i	$0.592275\pi$
$132$	1.51252	0.131648
$133$	2.25880	0.195863
$134$	11.5416	0.997042
$135$	−16.7657	−1.44296
$136$	−3.36998	−0.288974
$137$	−8.80948	−0.752645	−0.376322	−	0.926489i	$-0.622811\pi$
$137$	−8.80948	−0.752645	−0.376322	+	0.926489i	$0.622811\pi$
$138$	1.62223	0.138093
$139$	−1.00000	−0.0848189
$139$	−1.00000	−0.0848189
$140$	2.26633	0.191540
$141$	8.59457	0.723793
$142$	9.69497	0.813584
$143$	−6.39960	−0.535161
$144$	3.43883	0.286570
$145$	−3.10962	−0.258240
$146$	−7.88175	−0.652299
$147$	−6.88334	−0.567728
$148$	7.65909	0.629573
$149$	12.2218	1.00125	0.500624	−	0.865665i	$-0.333104\pi$
$149$	12.2218	1.00125	0.500624	+	0.865665i	$0.333104\pi$
$150$	−6.12633	−0.500213
$151$	18.2053	1.48153	0.740765	−	0.671765i	$-0.234463\pi$
$151$	18.2053	1.48153	0.740765	+	0.671765i	$0.234463\pi$
$152$	−6.79603	−0.551230
$153$	1.82113	0.147230
$154$	−2.13145	−0.171757
$155$	−15.3432	−1.23240
$156$	−2.87080	−0.229848
$157$	20.8300	1.66241	0.831207	−	0.555963i	$-0.187650\pi$
$157$	20.8300	1.66241	0.831207	+	0.555963i	$0.187650\pi$
$158$	−7.48906	−0.595797
$159$	−2.92110	−0.231658
$160$	−11.8464	−0.936540
$161$	1.26494	0.0996911
$162$	1.41142	0.110891
$163$	10.4895	0.821603	0.410802	−	0.911725i	$-0.365249\pi$
$163$	10.4895	0.821603	0.410802	+	0.911725i	$0.365249\pi$
$164$	2.35829	0.184152
$165$	−6.60173	−0.513944
$166$	11.1699	0.866950
$167$	−12.0413	−0.931782	−0.465891	−	0.884842i	$-0.654266\pi$
$167$	−12.0413	−0.931782	−0.465891	+	0.884842i	$0.654266\pi$
$168$	−3.64028	−0.280854
$169$	−0.853407	−0.0656467
$170$	3.86344	0.296313
$171$	3.67255	0.280847
$172$	−5.49081	−0.418670
$173$	−10.1502	−0.771707	−0.385853	−	0.922560i	$-0.626093\pi$
$173$	−10.1502	−0.771707	−0.385853	+	0.922560i	$0.626093\pi$
$174$	1.31192	0.0994567
$175$	−4.77703	−0.361109
$176$	3.79647	0.286169
$177$	5.60023	0.420939
$178$	9.57004	0.717305
$179$	4.37340	0.326884	0.163442	−	0.986553i	$-0.447740\pi$
$179$	4.37340	0.326884	0.163442	+	0.986553i	$0.447740\pi$
$180$	3.68480	0.274649
$181$	−15.9108	−1.18264	−0.591321	−	0.806436i	$-0.701393\pi$
$181$	−15.9108	−1.18264	−0.591321	+	0.806436i	$0.701393\pi$
$182$	4.04554	0.299875
$183$	−6.16832	−0.455975
$184$	−3.80581	−0.280568
$185$	−33.4299	−2.45781
$186$	6.47319	0.474637
$187$	2.01052	0.147024
$188$	−5.29601	−0.386251
$189$	5.51544	0.401189
$190$	7.79115	0.565229
$191$	5.42754	0.392723	0.196362	−	0.980532i	$-0.437087\pi$
$191$	5.42754	0.392723	0.196362	+	0.980532i	$0.437087\pi$
$192$	9.77880	0.705724
$193$	9.78327	0.704215	0.352108	−	0.935960i	$-0.385465\pi$
$193$	9.78327	0.704215	0.352108	+	0.935960i	$0.385465\pi$
$194$	1.84409	0.132398
$195$	12.5303	0.897310
$196$	4.24154	0.302967
$197$	1.27664	0.0909570	0.0454785	−	0.998965i	$-0.485519\pi$
$197$	1.27664	0.0909570	0.0454785	+	0.998965i	$0.485519\pi$
$198$	−3.46550	−0.246282
$199$	11.3715	0.806105	0.403052	−	0.915177i	$-0.367949\pi$
$199$	11.3715	0.806105	0.403052	+	0.915177i	$0.367949\pi$
$200$	14.3726	1.01630
$201$	−11.7600	−0.829487
$202$	−4.58186	−0.322379
$203$	1.02298	0.0717989
$204$	0.901902	0.0631457
$205$	−10.2933	−0.718916
$206$	−3.72189	−0.259317
$207$	2.05665	0.142947
$208$	−7.20579	−0.499632
$209$	4.05449	0.280455
$210$	4.17332	0.287987
$211$	8.33933	0.574103	0.287052	−	0.957915i	$-0.407325\pi$
$211$	8.33933	0.574103	0.287052	+	0.957915i	$0.407325\pi$
$212$	1.79999	0.123624
$213$	−9.87845	−0.676860
$214$	2.75051	0.188021
$215$	23.9659	1.63446
$216$	−16.5943	−1.12910
$217$	5.04749	0.342646
$218$	2.49166	0.168757
$219$	8.03091	0.542679
$220$	4.06801	0.274265
$221$	−3.81603	−0.256694
$222$	14.1038	0.946586
$223$	−3.60700	−0.241543	−0.120771	−	0.992680i	$-0.538537\pi$
$223$	−3.60700	−0.241543	−0.120771	+	0.992680i	$0.538537\pi$
$224$	3.89713	0.260388
$225$	−7.76692	−0.517795
$226$	6.91137	0.459738
$227$	27.9665	1.85620	0.928101	−	0.372327i	$-0.121440\pi$
$227$	27.9665	1.85620	0.928101	+	0.372327i	$0.121440\pi$
$228$	1.81881	0.120453
$229$	22.9302	1.51527	0.757634	−	0.652679i	$-0.226355\pi$
$229$	22.9302	1.51527	0.757634	+	0.652679i	$0.226355\pi$
$230$	4.36309	0.287693
$231$	2.17178	0.142893
$232$	−3.07782	−0.202069
$233$	−15.2211	−0.997165	−0.498582	−	0.866842i	$-0.666146\pi$
$233$	−15.2211	−0.997165	−0.498582	+	0.866842i	$0.666146\pi$
$234$	6.57760	0.429991
$235$	23.1156	1.50790
$236$	−3.45088	−0.224633
$237$	7.63078	0.495673
$238$	−1.27096	−0.0823844
$239$	−11.0498	−0.714751	−0.357376	−	0.933961i	$-0.616328\pi$
$239$	−11.0498	−0.714751	−0.357376	+	0.933961i	$0.616328\pi$
$240$	−7.43339	−0.479823
$241$	4.97450	0.320436	0.160218	−	0.987082i	$-0.448780\pi$
$241$	4.97450	0.320436	0.160218	+	0.987082i	$0.448780\pi$
$242$	8.65586	0.556420
$243$	14.7365	0.945349
$244$	3.80094	0.243330
$245$	−18.5132	−1.18276
$246$	4.34267	0.276878
$247$	−7.69553	−0.489655
$248$	−15.1864	−0.964335
$249$	−11.3813	−0.721258
$250$	1.16536	0.0737041
$251$	−1.91328	−0.120765	−0.0603825	−	0.998175i	$-0.519232\pi$
$251$	−1.91328	−0.120765	−0.0603825	+	0.998175i	$0.519232\pi$
$252$	−1.21220	−0.0763612
$253$	2.27054	0.142747
$254$	6.82434	0.428197
$255$	−3.93656	−0.246517
$256$	−14.6713	−0.916955
$257$	−25.8728	−1.61390	−0.806952	−	0.590617i	$-0.798884\pi$
$257$	−25.8728	−1.61390	−0.806952	+	0.590617i	$0.798884\pi$
$258$	−10.1110	−0.629485
$259$	10.9975	0.683351
$260$	−7.72119	−0.478848
$261$	1.66325	0.102952
$262$	7.42479	0.458705
$263$	−1.31821	−0.0812845	−0.0406422	−	0.999174i	$-0.512940\pi$
$263$	−1.31821	−0.0812845	−0.0406422	+	0.999174i	$0.512940\pi$
$264$	−6.53423	−0.402154
$265$	−7.85648	−0.482620
$266$	−2.56307	−0.157152
$267$	−9.75115	−0.596760
$268$	7.24656	0.442654
$269$	−13.4812	−0.821965	−0.410983	−	0.911643i	$-0.634814\pi$
$269$	−13.4812	−0.821965	−0.410983	+	0.911643i	$0.634814\pi$
$270$	19.0241	1.15777
$271$	−26.3532	−1.60085	−0.800423	−	0.599436i	$-0.795392\pi$
$271$	−26.3532	−1.60085	−0.800423	+	0.599436i	$0.795392\pi$
$272$	2.26380	0.137263
$273$	−4.12210	−0.249481
$274$	9.99617	0.603891
$275$	−8.57466	−0.517072
$276$	1.01854	0.0613089
$277$	−27.9624	−1.68010	−0.840048	−	0.542512i	$-0.817473\pi$
$277$	−27.9624	−1.68010	−0.840048	+	0.542512i	$0.817473\pi$
$278$	1.13471	0.0680551
$279$	8.20667	0.491320
$280$	−9.79077	−0.585111
$281$	15.8224	0.943887	0.471944	−	0.881629i	$-0.343553\pi$
$281$	15.8224	0.943887	0.471944	+	0.881629i	$0.343553\pi$
$282$	−9.75231	−0.580742
$283$	−3.79565	−0.225628	−0.112814	−	0.993616i	$-0.535986\pi$
$283$	−3.79565	−0.225628	−0.112814	+	0.993616i	$0.535986\pi$
$284$	6.08713	0.361205
$285$	−7.93859	−0.470242
$286$	7.26166	0.429391
$287$	3.38621	0.199882
$288$	6.33631	0.373370
$289$	−15.8011	−0.929479
$290$	3.52850	0.207201
$291$	−1.87899	−0.110148
$292$	−4.94868	−0.289599
$293$	−13.9562	−0.815330	−0.407665	−	0.913132i	$-0.633657\pi$
$293$	−13.9562	−0.815330	−0.407665	+	0.913132i	$0.633657\pi$
$294$	7.81056	0.455521
$295$	15.0622	0.876954
$296$	−33.0881	−1.92321
$297$	9.90009	0.574462
$298$	−13.8681	−0.803360
$299$	−4.30954	−0.249227
$300$	−3.84651	−0.222078
$301$	−7.88411	−0.454432
$302$	−20.6577	−1.18872
$303$	4.66857	0.268203
$304$	4.56526	0.261836
$305$	−16.5901	−0.949945
$306$	−2.06645	−0.118131
$307$	−11.9453	−0.681752	−0.340876	−	0.940108i	$-0.610724\pi$
$307$	−11.9453	−0.681752	−0.340876	+	0.940108i	$0.610724\pi$
$308$	−1.33826	−0.0762545
$309$	3.79233	0.215738
$310$	17.4101	0.988825
$311$	25.1875	1.42825	0.714125	−	0.700018i	$-0.246825\pi$
$311$	25.1875	1.42825	0.714125	+	0.700018i	$0.246825\pi$
$312$	12.4021	0.702133
$313$	−34.6271	−1.95724	−0.978620	−	0.205674i	$-0.934061\pi$
$313$	−34.6271	−1.95724	−0.978620	+	0.205674i	$0.934061\pi$
$314$	−23.6359	−1.33385
$315$	5.29091	0.298109
$316$	−4.70212	−0.264515
$317$	−23.2512	−1.30592	−0.652959	−	0.757393i	$-0.726473\pi$
$317$	−23.2512	−1.30592	−0.652959	+	0.757393i	$0.726473\pi$
$318$	3.31459	0.185873
$319$	1.83622	0.102809
$320$	26.3007	1.47025
$321$	−2.80256	−0.156424
$322$	−1.43533	−0.0799880
$323$	2.41766	0.134522
$324$	0.886179	0.0492321
$325$	16.2749	0.902770
$326$	−11.9025	−0.659220
$327$	−2.53881	−0.140397
$328$	−10.1881	−0.562542
$329$	−7.60440	−0.419244
$330$	7.49103	0.412367
$331$	15.4242	0.847790	0.423895	−	0.905711i	$-0.360662\pi$
$331$	15.4242	0.847790	0.423895	+	0.905711i	$0.360662\pi$
$332$	7.01317	0.384898
$333$	17.8807	0.979857
$334$	13.6633	0.747623
$335$	−31.6293	−1.72809
$336$	2.44538	0.133406
$337$	13.4221	0.731149	0.365575	−	0.930782i	$-0.380873\pi$
$337$	13.4221	0.731149	0.365575	+	0.930782i	$0.380873\pi$
$338$	0.968366	0.0526722
$339$	−7.04217	−0.382478
$340$	2.42572	0.131553
$341$	9.06014	0.490634
$342$	−4.16727	−0.225340
$343$	13.2512	0.715495
$344$	23.7209	1.27894
$345$	−4.44566	−0.239346
$346$	11.5175	0.619186
$347$	26.5231	1.42383	0.711916	−	0.702264i	$-0.247828\pi$
$347$	26.5231	1.42383	0.711916	+	0.702264i	$0.247828\pi$
$348$	0.823711	0.0441556
$349$	0.159546	0.00854032	0.00427016	−	0.999991i	$-0.498641\pi$
$349$	0.159546	0.00854032	0.00427016	+	0.999991i	$0.498641\pi$
$350$	5.42052	0.289739
$351$	−18.7906	−1.00297
$352$	6.99527	0.372849
$353$	8.91712	0.474610	0.237305	−	0.971435i	$-0.423736\pi$
$353$	8.91712	0.474610	0.237305	+	0.971435i	$0.423736\pi$
$354$	−6.35461	−0.337744
$355$	−26.5687	−1.41012
$356$	6.00869	0.318460
$357$	1.29502	0.0685396
$358$	−4.96253	−0.262278
$359$	−17.2674	−0.911337	−0.455669	−	0.890149i	$-0.650600\pi$
$359$	−17.2674	−0.911337	−0.455669	+	0.890149i	$0.650600\pi$
$360$	−15.9187	−0.838990
$361$	−14.1245	−0.743393
$362$	18.0541	0.948903
$363$	−8.81967	−0.462913
$364$	2.54005	0.133135
$365$	21.5996	1.13058
$366$	6.99923	0.365855
$367$	−9.05603	−0.472721	−0.236360	−	0.971665i	$-0.575955\pi$
$367$	−9.05603	−0.472721	−0.236360	+	0.971665i	$0.575955\pi$
$368$	2.55657	0.133270
$369$	5.50560	0.286610
$370$	37.9331	1.97205
$371$	2.58456	0.134184
$372$	4.06429	0.210724
$373$	−34.0146	−1.76121	−0.880606	−	0.473850i	$-0.842864\pi$
$373$	−34.0146	−1.76121	−0.880606	+	0.473850i	$0.842864\pi$
$374$	−2.28135	−0.117966
$375$	−1.18742	−0.0613180
$376$	22.8793	1.17991
$377$	−3.48520	−0.179497
$378$	−6.25840	−0.321897
$379$	7.26268	0.373059	0.186529	−	0.982449i	$-0.440276\pi$
$379$	7.26268	0.373059	0.186529	+	0.982449i	$0.440276\pi$
$380$	4.89179	0.250944
$381$	−6.95349	−0.356238
$382$	−6.15867	−0.315105
$383$	5.41538	0.276713	0.138356	−	0.990383i	$-0.455818\pi$
$383$	5.41538	0.276713	0.138356	+	0.990383i	$0.455818\pi$
$384$	−2.28691	−0.116703
$385$	5.84115	0.297693
$386$	−11.1011	−0.565033
$387$	−12.8187	−0.651611
$388$	1.15784	0.0587804
$389$	−24.2020	−1.22709	−0.613545	−	0.789660i	$-0.710257\pi$
$389$	−24.2020	−1.22709	−0.613545	+	0.789660i	$0.710257\pi$
$390$	−14.2182	−0.719964
$391$	1.35390	0.0684698
$392$	−18.3239	−0.925496
$393$	−7.56530	−0.381619
$394$	−1.44861	−0.0729801
$395$	20.5235	1.03265
$396$	−2.17587	−0.109341
$397$	1.02440	0.0514133	0.0257067	−	0.999670i	$-0.491816\pi$
$397$	1.02440	0.0514133	0.0257067	+	0.999670i	$0.491816\pi$
$398$	−12.9033	−0.646785
$399$	2.61158	0.130742
$400$	−9.65486	−0.482743
$401$	−18.6452	−0.931096	−0.465548	−	0.885023i	$-0.654143\pi$
$401$	−18.6452	−0.931096	−0.465548	+	0.885023i	$0.654143\pi$
$402$	13.3442	0.665546
$403$	−17.1964	−0.856613
$404$	−2.87679	−0.143126
$405$	−3.86793	−0.192199
$406$	−1.16078	−0.0576085
$407$	19.7403	0.978489
$408$	−3.89631	−0.192896
$409$	24.4204	1.20751	0.603756	−	0.797169i	$-0.293670\pi$
$409$	24.4204	1.20751	0.603756	+	0.797169i	$0.293670\pi$
$410$	11.6799	0.576828
$411$	−10.1853	−0.502406
$412$	−2.33685	−0.115128
$413$	−4.95503	−0.243821
$414$	−2.33369	−0.114695
$415$	−30.6106	−1.50262
$416$	−13.2772	−0.650968
$417$	−1.15618	−0.0566184
$418$	−4.60066	−0.225025
$419$	13.3773	0.653523	0.326761	−	0.945107i	$-0.394043\pi$
$419$	13.3773	0.653523	0.326761	+	0.945107i	$0.394043\pi$
$420$	2.62028	0.127857
$421$	32.8198	1.59954	0.799769	−	0.600308i	$-0.204955\pi$
$421$	32.8198	1.59954	0.799769	+	0.600308i	$0.204955\pi$
$422$	−9.46269	−0.460636
$423$	−12.3639	−0.601154
$424$	−7.77615	−0.377643
$425$	−5.11300	−0.248017
$426$	11.2091	0.543084
$427$	5.45767	0.264115
$428$	1.72695	0.0834753
$429$	−7.39908	−0.357231
$430$	−27.1942	−1.31142
$431$	2.14292	0.103221	0.0516103	−	0.998667i	$-0.483565\pi$
$431$	2.14292	0.103221	0.0516103	+	0.998667i	$0.483565\pi$
$432$	11.1473	0.536323
$433$	13.9674	0.671231	0.335616	−	0.941999i	$-0.391056\pi$
$433$	13.9674	0.671231	0.335616	+	0.941999i	$0.391056\pi$
$434$	−5.72742	−0.274925
$435$	−3.59528	−0.172380
$436$	1.56443	0.0749225
$437$	2.73033	0.130609
$438$	−9.11273	−0.435423
$439$	9.98146	0.476389	0.238195	−	0.971217i	$-0.423444\pi$
$439$	9.98146	0.476389	0.238195	+	0.971217i	$0.423444\pi$
$440$	−17.5742	−0.837819
$441$	9.90218	0.471532
$442$	4.33007	0.205960
$443$	−7.71677	−0.366635	−0.183317	−	0.983054i	$-0.558684\pi$
$443$	−7.71677	−0.366635	−0.183317	+	0.983054i	$0.558684\pi$
$444$	8.85529	0.420253
$445$	−26.2263	−1.24325
$446$	4.09289	0.193804
$447$	14.1306	0.668354
$448$	−8.65220	−0.408778
$449$	6.39685	0.301886	0.150943	−	0.988542i	$-0.451769\pi$
$449$	6.39685	0.301886	0.150943	+	0.988542i	$0.451769\pi$
$450$	8.81317	0.415457
$451$	6.07817	0.286210
$452$	4.33941	0.204109
$453$	21.0486	0.988952
$454$	−31.7338	−1.48934
$455$	−11.0867	−0.519750
$456$	−7.85743	−0.367958
$457$	−23.1608	−1.08342	−0.541709	−	0.840566i	$-0.682223\pi$
$457$	−23.1608	−1.08342	−0.541709	+	0.840566i	$0.682223\pi$
$458$	−26.0190	−1.21579
$459$	5.90334	0.275544
$460$	2.73943	0.127727
$461$	−8.80772	−0.410216	−0.205108	−	0.978739i	$-0.565755\pi$
$461$	−8.80772	−0.410216	−0.205108	+	0.978739i	$0.565755\pi$
$462$	−2.46434	−0.114651
$463$	−25.5369	−1.18680	−0.593401	−	0.804907i	$-0.702215\pi$
$463$	−25.5369	−1.18680	−0.593401	+	0.804907i	$0.702215\pi$
$464$	2.06754	0.0959832
$465$	−17.7395	−0.822651
$466$	17.2714	0.800083
$467$	−10.5970	−0.490373	−0.245186	−	0.969476i	$-0.578849\pi$
$467$	−10.5970	−0.490373	−0.245186	+	0.969476i	$0.578849\pi$
$468$	4.12985	0.190902
$469$	10.4052	0.480465
$470$	−26.2295	−1.20987
$471$	24.0832	1.10970
$472$	14.9082	0.686204
$473$	−14.1518	−0.650701
$474$	−8.65870	−0.397707
$475$	−10.3110	−0.473103
$476$	−0.797994	−0.0365760
$477$	4.20221	0.192406
$478$	12.5383	0.573486
$479$	−0.483332	−0.0220840	−0.0110420	−	0.999939i	$-0.503515\pi$
$479$	−0.483332	−0.0220840	−0.0110420	+	0.999939i	$0.503515\pi$
$480$	−13.6966	−0.625160
$481$	−37.4675	−1.70837
$482$	−5.64460	−0.257104
$483$	1.46250	0.0665459
$484$	5.43471	0.247032
$485$	−5.05366	−0.229475
$486$	−16.7216	−0.758509
$487$	−13.3441	−0.604678	−0.302339	−	0.953200i	$-0.597768\pi$
$487$	−13.3441	−0.604678	−0.302339	+	0.953200i	$0.597768\pi$
$488$	−16.4205	−0.743319
$489$	12.1278	0.548437
$490$	21.0070	0.949000
$491$	−42.2360	−1.90609	−0.953043	−	0.302836i	$-0.902067\pi$
$491$	−42.2360	−1.90609	−0.953043	+	0.302836i	$0.902067\pi$
$492$	2.72661	0.122925
$493$	1.09492	0.0493129
$494$	8.73216	0.392878
$495$	9.49707	0.426862
$496$	10.2015	0.458061
$497$	8.74036	0.392059
$498$	12.9144	0.578707
$499$	−12.1737	−0.544968	−0.272484	−	0.962160i	$-0.587845\pi$
$499$	−12.1737	−0.544968	−0.272484	+	0.962160i	$0.587845\pi$
$500$	0.731691	0.0327222
$501$	−13.9219	−0.621984
$502$	2.17101	0.0968969
$503$	−13.4299	−0.598808	−0.299404	−	0.954126i	$-0.596788\pi$
$503$	−13.4299	−0.598808	−0.299404	+	0.954126i	$0.596788\pi$
$504$	5.23681	0.233266
$505$	12.5564	0.558754
$506$	−2.57639	−0.114535
$507$	−0.986692	−0.0438205
$508$	4.28477	0.190106
$509$	−17.2396	−0.764132	−0.382066	−	0.924135i	$-0.624787\pi$
$509$	−17.2396	−0.764132	−0.382066	+	0.924135i	$0.624787\pi$
$510$	4.46684	0.197795
$511$	−7.10568	−0.314337
$512$	20.6036	0.910557
$513$	11.9049	0.525613
$514$	29.3581	1.29493
$515$	10.1997	0.449453
$516$	−6.34836	−0.279471
$517$	−13.6497	−0.600315
$518$	−12.4789	−0.548292
$519$	−11.7355	−0.515130
$520$	33.3563	1.46277
$521$	19.3190	0.846382	0.423191	−	0.906040i	$-0.360910\pi$
$521$	19.3190	0.846382	0.423191	+	0.906040i	$0.360910\pi$
$522$	−1.88730	−0.0826048
$523$	−1.34569	−0.0588429	−0.0294214	−	0.999567i	$-0.509366\pi$
$523$	−1.34569	−0.0588429	−0.0294214	+	0.999567i	$0.509366\pi$
$524$	4.66176	0.203650
$525$	−5.52310	−0.241048
$526$	1.49578	0.0652193
$527$	5.40249	0.235336
$528$	4.38940	0.191024
$529$	−21.4710	−0.933522
$530$	8.91479	0.387234
$531$	−8.05634	−0.349615
$532$	−1.60926	−0.0697704
$533$	−11.5365	−0.499702
$534$	11.0647	0.478816
$535$	−7.53768	−0.325882
$536$	−31.3059	−1.35221
$537$	5.05644	0.218202
$538$	15.2972	0.659511
$539$	10.9320	0.470874
$540$	11.9446	0.514013
$541$	−25.3566	−1.09016	−0.545082	−	0.838382i	$-0.683502\pi$
$541$	−25.3566	−1.09016	−0.545082	+	0.838382i	$0.683502\pi$
$542$	29.9032	1.28445
$543$	−18.3958	−0.789438
$544$	4.17122	0.178840
$545$	−6.82830	−0.292492
$546$	4.67737	0.200173
$547$	−34.5100	−1.47554	−0.737770	−	0.675052i	$-0.764121\pi$
$547$	−34.5100	−1.47554	−0.737770	+	0.675052i	$0.764121\pi$
$548$	6.27624	0.268108
$549$	8.87357	0.378715
$550$	9.72972	0.414877
$551$	2.20806	0.0940666
$552$	−4.40020	−0.187285
$553$	−6.75165	−0.287109
$554$	31.7291	1.34804
$555$	−38.6509	−1.64064
$556$	0.712442	0.0302143
$557$	18.1572	0.769345	0.384672	−	0.923053i	$-0.374314\pi$
$557$	18.1572	0.769345	0.384672	+	0.923053i	$0.374314\pi$
$558$	−9.31215	−0.394215
$559$	26.8605	1.13608
$560$	6.57699	0.277929
$561$	2.32453	0.0981417
$562$	−17.9538	−0.757336
$563$	33.2195	1.40004	0.700018	−	0.714125i	$-0.253175\pi$
$563$	33.2195	1.40004	0.700018	+	0.714125i	$0.253175\pi$
$564$	−6.12313	−0.257830
$565$	−18.9404	−0.796827
$566$	4.30695	0.181034
$567$	1.27244	0.0534375
$568$	−26.2971	−1.10340
$569$	31.0751	1.30274	0.651368	−	0.758762i	$-0.274196\pi$
$569$	31.0751	1.30274	0.651368	+	0.758762i	$0.274196\pi$
$570$	9.00797	0.377302
$571$	19.2105	0.803934	0.401967	−	0.915654i	$-0.368327\pi$
$571$	19.2105	0.803934	0.401967	+	0.915654i	$0.368327\pi$
$572$	4.55934	0.190636
$573$	6.27522	0.262151
$574$	−3.84235	−0.160377
$575$	−5.77424	−0.240803
$576$	−14.0675	−0.586146
$577$	−46.2815	−1.92673	−0.963363	−	0.268201i	$-0.913571\pi$
$577$	−46.2815	−1.92673	−0.963363	+	0.268201i	$0.913571\pi$
$578$	17.9296	0.745775
$579$	11.3112	0.470078
$580$	2.21542	0.0919905
$581$	10.0700	0.417775
$582$	2.13210	0.0883784
$583$	4.63923	0.192137
$584$	21.3788	0.884661
$585$	−18.0257	−0.745270
$586$	15.8362	0.654187
$587$	14.4568	0.596695	0.298347	−	0.954457i	$-0.403565\pi$
$587$	14.4568	0.596695	0.298347	+	0.954457i	$0.403565\pi$
$588$	4.90398	0.202237
$589$	10.8948	0.448914
$590$	−17.0911	−0.703631
$591$	1.47603	0.0607157
$592$	22.2271	0.913526
$593$	33.9034	1.39225	0.696124	−	0.717922i	$-0.254907\pi$
$593$	33.9034	1.39225	0.696124	+	0.717922i	$0.254907\pi$
$594$	−11.2337	−0.460924
$595$	3.48303	0.142790
$596$	−8.70732	−0.356666
$597$	13.1475	0.538091
$598$	4.89006	0.199969
$599$	18.1098	0.739948	0.369974	−	0.929042i	$-0.379367\pi$
$599$	18.1098	0.739948	0.369974	+	0.929042i	$0.379367\pi$
$600$	16.6173	0.678399
$601$	25.0280	1.02091	0.510457	−	0.859903i	$-0.329476\pi$
$601$	25.0280	1.02091	0.510457	+	0.859903i	$0.329476\pi$
$602$	8.94615	0.364618
$603$	16.9176	0.688939
$604$	−12.9703	−0.527752
$605$	−23.7211	−0.964398
$606$	−5.29746	−0.215195
$607$	7.62872	0.309640	0.154820	−	0.987943i	$-0.450520\pi$
$607$	7.62872	0.309640	0.154820	+	0.987943i	$0.450520\pi$
$608$	8.41182	0.341145
$609$	1.18275	0.0479273
$610$	18.8249	0.762196
$611$	25.9075	1.04811
$612$	−1.29745	−0.0524463
$613$	46.5036	1.87826	0.939131	−	0.343558i	$-0.111632\pi$
$613$	46.5036	1.87826	0.939131	+	0.343558i	$0.111632\pi$
$614$	13.5544	0.547010
$615$	−11.9009	−0.479891
$616$	5.78143	0.232940
$617$	6.20042	0.249619	0.124810	−	0.992181i	$-0.460168\pi$
$617$	6.20042	0.249619	0.124810	+	0.992181i	$0.460168\pi$
$618$	−4.30318	−0.173099
$619$	5.86974	0.235925	0.117962	−	0.993018i	$-0.462364\pi$
$619$	5.86974	0.235925	0.117962	+	0.993018i	$0.462364\pi$
$620$	10.9312	0.439006
$621$	6.66680	0.267529
$622$	−28.5804	−1.14597
$623$	8.62773	0.345663
$624$	−8.33119	−0.333514
$625$	−26.5423	−1.06169
$626$	39.2916	1.57041
$627$	4.68772	0.187210
$628$	−14.8402	−0.592187
$629$	11.7710	0.469339
$630$	−6.00363	−0.239190
$631$	20.3125	0.808627	0.404314	−	0.914620i	$-0.367510\pi$
$631$	20.3125	0.808627	0.404314	+	0.914620i	$0.367510\pi$
$632$	20.3136	0.808033
$633$	9.64177	0.383226
$634$	26.3833	1.04781
$635$	−18.7019	−0.742160
$636$	2.08111	0.0825215
$637$	−20.7492	−0.822112
$638$	−2.08357	−0.0824894
$639$	14.2109	0.562173
$640$	−6.15078	−0.243131
$641$	40.2965	1.59162	0.795808	−	0.605549i	$-0.207047\pi$
$641$	40.2965	1.59162	0.795808	+	0.605549i	$0.207047\pi$
$642$	3.18009	0.125508
$643$	13.1288	0.517750	0.258875	−	0.965911i	$-0.416648\pi$
$643$	13.1288	0.517750	0.258875	+	0.965911i	$0.416648\pi$
$644$	−0.901196	−0.0355121
$645$	27.7089	1.09104
$646$	−2.74333	−0.107935
$647$	20.7344	0.815155	0.407577	−	0.913171i	$-0.366374\pi$
$647$	20.7344	0.815155	0.407577	+	0.913171i	$0.366374\pi$
$648$	−3.82838	−0.150393
$649$	−8.89418	−0.349127
$650$	−18.4672	−0.724345
$651$	5.83581	0.228723
$652$	−7.47318	−0.292672
$653$	−19.0052	−0.743730	−0.371865	−	0.928287i	$-0.621281\pi$
$653$	−19.0052	−0.743730	−0.371865	+	0.928287i	$0.621281\pi$
$654$	2.88081	0.112649
$655$	−20.3473	−0.795036
$656$	6.84388	0.267208
$657$	−11.5530	−0.450727
$658$	8.62876	0.336384
$659$	27.7789	1.08211	0.541055	−	0.840987i	$-0.318025\pi$
$659$	27.7789	1.08211	0.541055	+	0.840987i	$0.318025\pi$
$660$	4.70335	0.183078
$661$	16.2944	0.633777	0.316889	−	0.948463i	$-0.397362\pi$
$661$	16.2944	0.633777	0.316889	+	0.948463i	$0.397362\pi$
$662$	−17.5019	−0.680232
$663$	−4.41201	−0.171348
$664$	−30.2976	−1.17578
$665$	7.02400	0.272379
$666$	−20.2893	−0.786196
$667$	1.23653	0.0478785
$668$	8.57871	0.331920
$669$	−4.17034	−0.161235
$670$	35.8899	1.38655
$671$	9.79640	0.378186
$672$	4.50579	0.173814
$673$	−34.6053	−1.33394	−0.666968	−	0.745087i	$-0.732408\pi$
$673$	−34.6053	−1.33394	−0.666968	+	0.745087i	$0.732408\pi$
$674$	−15.2302	−0.586644
$675$	−25.1771	−0.969067
$676$	0.608003	0.0233847
$677$	26.9314	1.03506	0.517529	−	0.855666i	$-0.326852\pi$
$677$	26.9314	1.03506	0.517529	+	0.855666i	$0.326852\pi$
$678$	7.99079	0.306884
$679$	1.66251	0.0638014
$680$	−10.4794	−0.401865
$681$	32.3343	1.23905
$682$	−10.2806	−0.393665
$683$	17.8228	0.681972	0.340986	−	0.940068i	$-0.389239\pi$
$683$	17.8228	0.681972	0.340986	+	0.940068i	$0.389239\pi$
$684$	−2.61648	−0.100044
$685$	−27.3941	−1.04668
$686$	−15.0362	−0.574083
$687$	26.5114	1.01147
$688$	−15.9346	−0.607500
$689$	−8.80538	−0.335458
$690$	5.04451	0.192041
$691$	8.42965	0.320679	0.160339	−	0.987062i	$-0.448741\pi$
$691$	8.42965	0.320679	0.160339	+	0.987062i	$0.448741\pi$
$692$	7.23145	0.274898
$693$	−3.12427	−0.118681
$694$	−30.0959	−1.14242
$695$	−3.10962	−0.117955
$696$	−3.55852	−0.134885
$697$	3.62436	0.137283
$698$	−0.181038	−0.00685240
$699$	−17.5983	−0.665628
$700$	3.40336	0.128635
$701$	−20.5531	−0.776281	−0.388141	−	0.921600i	$-0.626882\pi$
$701$	−20.5531	−0.776281	−0.388141	+	0.921600i	$0.626882\pi$
$702$	21.3218	0.804741
$703$	23.7377	0.895285
$704$	−15.5305	−0.585328
$705$	26.7258	1.00655
$706$	−10.1183	−0.380808
$707$	−4.13071	−0.155351
$708$	−3.98984	−0.149947
$709$	−23.5312	−0.883733	−0.441866	−	0.897081i	$-0.645683\pi$
$709$	−23.5312	−0.883733	−0.441866	+	0.897081i	$0.645683\pi$
$710$	30.1477	1.13142
$711$	−10.9774	−0.411686
$712$	−25.9582	−0.972824
$713$	6.10117	0.228491
$714$	−1.46946	−0.0549933
$715$	−19.9003	−0.744229
$716$	−3.11580	−0.116443
$717$	−12.7755	−0.477111
$718$	19.5934	0.731219
$719$	23.8108	0.887993	0.443996	−	0.896029i	$-0.353560\pi$
$719$	23.8108	0.887993	0.443996	+	0.896029i	$0.353560\pi$
$720$	10.6935	0.398522
$721$	−3.35542	−0.124962
$722$	16.0271	0.596467
$723$	5.75142	0.213898
$724$	11.3355	0.421282
$725$	−4.66973	−0.173429
$726$	10.0077	0.371422
$727$	−33.7258	−1.25082	−0.625411	−	0.780296i	$-0.715069\pi$
$727$	−33.7258	−1.25082	−0.625411	+	0.780296i	$0.715069\pi$
$728$	−10.9733	−0.406697
$729$	20.7697	0.769247
$730$	−24.5093	−0.907128
$731$	−8.43861	−0.312113
$732$	4.39457	0.162428
$733$	−34.7441	−1.28330	−0.641650	−	0.766997i	$-0.721750\pi$
$733$	−34.7441	−1.28330	−0.641650	+	0.766997i	$0.721750\pi$
$734$	10.2759	0.379291
$735$	−21.4046	−0.789519
$736$	4.71067	0.173637
$737$	18.6770	0.687977
$738$	−6.24724	−0.229964
$739$	−48.4882	−1.78367	−0.891833	−	0.452365i	$-0.850580\pi$
$739$	−48.4882	−1.78367	−0.891833	+	0.452365i	$0.850580\pi$
$740$	23.8169	0.875525
$741$	−8.89741	−0.326855
$742$	−2.93272	−0.107663
$743$	−35.6302	−1.30715	−0.653573	−	0.756863i	$-0.726731\pi$
$743$	−35.6302	−1.30715	−0.653573	+	0.756863i	$0.726731\pi$
$744$	−17.5582	−0.643713
$745$	38.0051	1.39240
$746$	38.5966	1.41312
$747$	16.3728	0.599048
$748$	−1.43238	−0.0523731
$749$	2.47968	0.0906057
$750$	1.34737	0.0491990
$751$	−19.1689	−0.699483	−0.349742	−	0.936846i	$-0.613731\pi$
$751$	−19.1689	−0.699483	−0.349742	+	0.936846i	$0.613731\pi$
$752$	−15.3693	−0.560459
$753$	−2.21209	−0.0806132
$754$	3.95467	0.144021
$755$	56.6117	2.06031
$756$	−3.92943	−0.142912
$757$	20.8207	0.756740	0.378370	−	0.925654i	$-0.376485\pi$
$757$	20.8207	0.756740	0.378370	+	0.925654i	$0.376485\pi$
$758$	−8.24101	−0.299327
$759$	2.62515	0.0952869
$760$	−21.1330	−0.766576
$761$	27.8987	1.01133	0.505664	−	0.862730i	$-0.331247\pi$
$761$	27.8987	1.01133	0.505664	+	0.862730i	$0.331247\pi$
$762$	7.89017	0.285831
$763$	2.24632	0.0813222
$764$	−3.86681	−0.139896
$765$	5.66302	0.204747
$766$	−6.14486	−0.222023
$767$	16.8814	0.609551
$768$	−16.9626	−0.612086
$769$	−5.28632	−0.190630	−0.0953148	−	0.995447i	$-0.530386\pi$
$769$	−5.28632	−0.190630	−0.0953148	+	0.995447i	$0.530386\pi$
$770$	−6.62799	−0.238856
$771$	−29.9137	−1.07731
$772$	−6.97001	−0.250856
$773$	−45.2651	−1.62807	−0.814037	−	0.580813i	$-0.802735\pi$
$773$	−45.2651	−1.62807	−0.814037	+	0.580813i	$0.802735\pi$
$774$	14.5454	0.522825
$775$	−23.0410	−0.827657
$776$	−5.00199	−0.179561
$777$	12.7151	0.456151
$778$	27.4621	0.984565
$779$	7.30902	0.261873
$780$	−8.92708	−0.319641
$781$	15.6888	0.561388
$782$	−1.53628	−0.0549373
$783$	5.39155	0.192678
$784$	12.3091	0.439612
$785$	64.7733	2.31186
$786$	8.58439	0.306195
$787$	−9.12880	−0.325407	−0.162703	−	0.986675i	$-0.552021\pi$
$787$	−9.12880	−0.325407	−0.162703	+	0.986675i	$0.552021\pi$
$788$	−0.909534	−0.0324008
$789$	−1.52409	−0.0542591
$790$	−23.2881	−0.828554
$791$	6.23084	0.221543
$792$	9.39997	0.334013
$793$	−18.5938	−0.660286
$794$	−1.16240	−0.0412519
$795$	−9.08350	−0.322159
$796$	−8.10154	−0.287151
$797$	−3.03946	−0.107663	−0.0538316	−	0.998550i	$-0.517143\pi$
$797$	−3.03946	−0.107663	−0.0538316	+	0.998550i	$0.517143\pi$
$798$	−2.96337	−0.104902
$799$	−8.13922	−0.287945
$800$	−17.7898	−0.628964
$801$	14.0277	0.495645
$802$	21.1568	0.747073
$803$	−12.7545	−0.450098
$804$	8.37833	0.295481
$805$	3.93348	0.138637
$806$	19.5128	0.687310
$807$	−15.5867	−0.548679
$808$	12.4280	0.437217
$809$	13.4793	0.473909	0.236954	−	0.971521i	$-0.423851\pi$
$809$	13.4793	0.473909	0.236954	+	0.971521i	$0.423851\pi$
$810$	4.38897	0.154213
$811$	−8.73276	−0.306649	−0.153324	−	0.988176i	$-0.548998\pi$
$811$	−8.73276	−0.306649	−0.153324	+	0.988176i	$0.548998\pi$
$812$	−0.728812	−0.0255763
$813$	−30.4691	−1.06860
$814$	−22.3994	−0.785099
$815$	32.6184	1.14257
$816$	2.61736	0.0916260
$817$	−17.0176	−0.595370
$818$	−27.7100	−0.968857
$819$	5.92994	0.207209
$820$	7.33339	0.256093
$821$	−32.7753	−1.14387	−0.571933	−	0.820300i	$-0.693806\pi$
$821$	−32.7753	−1.14387	−0.571933	+	0.820300i	$0.693806\pi$
$822$	11.5574	0.403109
$823$	44.4981	1.55111	0.775553	−	0.631282i	$-0.217471\pi$
$823$	44.4981	1.55111	0.775553	+	0.631282i	$0.217471\pi$
$824$	10.0954	0.351691
$825$	−9.91385	−0.345156
$826$	5.62251	0.195632
$827$	3.49530	0.121543	0.0607717	−	0.998152i	$-0.480644\pi$
$827$	3.49530	0.121543	0.0607717	+	0.998152i	$0.480644\pi$
$828$	−1.46524	−0.0509208
$829$	32.0909	1.11456	0.557281	−	0.830324i	$-0.311845\pi$
$829$	32.0909	1.11456	0.557281	+	0.830324i	$0.311845\pi$
$830$	34.7340	1.20564
$831$	−32.3295	−1.12150
$832$	29.4773	1.02194
$833$	6.51865	0.225858
$834$	1.31192	0.0454282
$835$	−37.4438	−1.29579
$836$	−2.88859	−0.0999040
$837$	26.6026	0.919519
$838$	−15.1793	−0.524359
$839$	−25.2239	−0.870827	−0.435414	−	0.900231i	$-0.643398\pi$
$839$	−25.2239	−0.870827	−0.435414	+	0.900231i	$0.643398\pi$
$840$	−11.3199	−0.390573
$841$	1.00000	0.0344828
$842$	−37.2408	−1.28340
$843$	18.2936	0.630064
$844$	−5.94129	−0.204508
$845$	−2.65377	−0.0912925
$846$	14.0294	0.482341
$847$	7.80356	0.268134
$848$	5.22366	0.179381
$849$	−4.38845	−0.150611
$850$	5.80175	0.198998
$851$	13.2932	0.455687
$852$	7.03782	0.241112
$853$	0.785023	0.0268787	0.0134393	−	0.999910i	$-0.495722\pi$
$853$	0.785023	0.0268787	0.0134393	+	0.999910i	$0.495722\pi$
$854$	−6.19285	−0.211915
$855$	11.4202	0.390564
$856$	−7.46061	−0.254998
$857$	18.6587	0.637368	0.318684	−	0.947861i	$-0.396759\pi$
$857$	18.6587	0.637368	0.318684	+	0.947861i	$0.396759\pi$
$858$	8.39579	0.286627
$859$	44.1153	1.50519	0.752597	−	0.658481i	$-0.228801\pi$
$859$	44.1153	1.50519	0.752597	+	0.658481i	$0.228801\pi$
$860$	−17.0743	−0.582229
$861$	3.91507	0.133425
$862$	−2.43158	−0.0828199
$863$	45.7919	1.55878	0.779388	−	0.626542i	$-0.215530\pi$
$863$	45.7919	1.55878	0.779388	+	0.626542i	$0.215530\pi$
$864$	20.5396	0.698773
$865$	−31.5633	−1.07318
$866$	−15.8489	−0.538568
$867$	−18.2690	−0.620446
$868$	−3.59605	−0.122058
$869$	−12.1191	−0.411111
$870$	4.07958	0.138311
$871$	−35.4495	−1.20116
$872$	−6.75849	−0.228871
$873$	2.70306	0.0914848
$874$	−3.09812	−0.104795
$875$	1.05062	0.0355173
$876$	−5.72156	−0.193314
$877$	35.4618	1.19746	0.598730	−	0.800951i	$-0.295672\pi$
$877$	35.4618	1.19746	0.598730	+	0.800951i	$0.295672\pi$
$878$	−11.3260	−0.382235
$879$	−16.1359	−0.544250
$880$	11.8056	0.397966
$881$	20.6969	0.697297	0.348649	−	0.937253i	$-0.386641\pi$
$881$	20.6969	0.697297	0.348649	+	0.937253i	$0.386641\pi$
$882$	−11.2361	−0.378338
$883$	26.0174	0.875555	0.437778	−	0.899083i	$-0.355766\pi$
$883$	26.0174	0.875555	0.437778	+	0.899083i	$0.355766\pi$
$884$	2.71870	0.0914397
$885$	17.4146	0.585385
$886$	8.75627	0.294173
$887$	−20.1889	−0.677877	−0.338939	−	0.940809i	$-0.610068\pi$
$887$	−20.1889	−0.677877	−0.338939	+	0.940809i	$0.610068\pi$
$888$	−38.2558	−1.28378
$889$	6.15239	0.206344
$890$	29.7592	0.997530
$891$	2.28400	0.0765170
$892$	2.56978	0.0860426
$893$	−16.4138	−0.549268
$894$	−16.0341	−0.536259
$895$	13.5996	0.454585
$896$	2.02343	0.0675982
$897$	−4.98260	−0.166364
$898$	−7.25854	−0.242221
$899$	4.93412	0.164562
$900$	5.53348	0.184449
$901$	2.76634	0.0921600
$902$	−6.89694	−0.229643
$903$	−9.11545	−0.303343
$904$	−18.7467	−0.623506
$905$	−49.4766	−1.64466
$906$	−23.8840	−0.793494
$907$	8.27034	0.274612	0.137306	−	0.990529i	$-0.456156\pi$
$907$	8.27034	0.274612	0.137306	+	0.990529i	$0.456156\pi$
$908$	−19.9245	−0.661219
$909$	−6.71608	−0.222758
$910$	12.5801	0.417026
$911$	−20.6807	−0.685182	−0.342591	−	0.939485i	$-0.611304\pi$
$911$	−20.6807	−0.685182	−0.342591	+	0.939485i	$0.611304\pi$
$912$	5.27826	0.174781
$913$	18.0755	0.598211
$914$	26.2808	0.869290
$915$	−19.1811	−0.634108
$916$	−16.3364	−0.539771
$917$	6.69371	0.221046
$918$	−6.69856	−0.221085
$919$	17.0450	0.562264	0.281132	−	0.959669i	$-0.409290\pi$
$919$	17.0450	0.562264	0.281132	+	0.959669i	$0.409290\pi$
$920$	−11.8346	−0.390176
$921$	−13.8109	−0.455084
$922$	9.99417	0.329141
$923$	−29.7777	−0.980143
$924$	−1.54727	−0.0509015
$925$	−50.2018	−1.65062
$926$	28.9769	0.952241
$927$	−5.45554	−0.179183
$928$	3.80960	0.125056
$929$	22.3370	0.732852	0.366426	−	0.930447i	$-0.380581\pi$
$929$	22.3370	0.732852	0.366426	+	0.930447i	$0.380581\pi$
$930$	20.1292	0.660061
$931$	13.1457	0.430834
$932$	10.8441	0.355211
$933$	29.1213	0.953387
$934$	12.0245	0.393455
$935$	6.25197	0.204461
$936$	−17.8414	−0.583163
$937$	15.3988	0.503058	0.251529	−	0.967850i	$-0.419067\pi$
$937$	15.3988	0.503058	0.251529	+	0.967850i	$0.419067\pi$
$938$	−11.8068	−0.385505
$939$	−40.0352	−1.30650
$940$	−16.4686	−0.537145
$941$	−24.1663	−0.787797	−0.393899	−	0.919154i	$-0.628874\pi$
$941$	−24.1663	−0.787797	−0.393899	+	0.919154i	$0.628874\pi$
$942$	−27.3274	−0.890374
$943$	4.09309	0.133289
$944$	−10.0146	−0.325948
$945$	17.1509	0.557919
$946$	16.0581	0.522095
$947$	−21.4752	−0.697851	−0.348925	−	0.937150i	$-0.613453\pi$
$947$	−21.4752	−0.697851	−0.348925	+	0.937150i	$0.613453\pi$
$948$	−5.43649	−0.176569
$949$	24.2084	0.785839
$950$	11.7000	0.379598
$951$	−26.8826	−0.871728
$952$	3.44742	0.111731
$953$	42.8180	1.38701	0.693506	−	0.720451i	$-0.256065\pi$
$953$	42.8180	1.38701	0.693506	+	0.720451i	$0.256065\pi$
$954$	−4.76827	−0.154379
$955$	16.8776	0.546146
$956$	7.87233	0.254609
$957$	2.12300	0.0686270
$958$	0.548440	0.0177193
$959$	9.01190	0.291009
$960$	30.4083	0.981425
$961$	−6.65446	−0.214660
$962$	42.5146	1.37073
$963$	4.03169	0.129919
$964$	−3.54405	−0.114146
$965$	30.4222	0.979326
$966$	−1.65950	−0.0533936
$967$	−8.68920	−0.279426	−0.139713	−	0.990192i	$-0.544618\pi$
$967$	−8.68920	−0.279426	−0.139713	+	0.990192i	$0.544618\pi$
$968$	−23.4785	−0.754628
$969$	2.79525	0.0897964
$970$	5.73442	0.184121
$971$	21.4172	0.687312	0.343656	−	0.939096i	$-0.388335\pi$
$971$	21.4172	0.687312	0.343656	+	0.939096i	$0.388335\pi$
$972$	−10.4989	−0.336753
$973$	1.02298	0.0327951
$974$	15.1416	0.485169
$975$	18.8167	0.602618
$976$	11.0305	0.353078
$977$	54.5753	1.74602	0.873009	−	0.487704i	$-0.162166\pi$
$977$	54.5753	1.74602	0.873009	+	0.487704i	$0.162166\pi$
$978$	−13.7615	−0.440043
$979$	15.4866	0.494953
$980$	13.1896	0.421325
$981$	3.65227	0.116608
$982$	47.9255	1.52936
$983$	−54.5397	−1.73955	−0.869773	−	0.493452i	$-0.835735\pi$
$983$	−54.5397	−1.73955	−0.869773	+	0.493452i	$0.835735\pi$
$984$	−11.7792	−0.375508
$985$	3.96987	0.126491
$986$	−1.24242	−0.0395666
$987$	−8.79205	−0.279854
$988$	5.48262	0.174425
$989$	−9.52993	−0.303034
$990$	−10.7764	−0.342496
$991$	−8.84470	−0.280961	−0.140481	−	0.990083i	$-0.544865\pi$
$991$	−8.84470	−0.280961	−0.140481	+	0.990083i	$0.544865\pi$
$992$	18.7970	0.596806
$993$	17.8331	0.565918
$994$	−9.91774	−0.314572
$995$	35.3610	1.12102
$996$	8.10849	0.256927
$997$	−14.4225	−0.456765	−0.228382	−	0.973571i	$-0.573344\pi$
$997$	−14.4225	−0.456765	−0.228382	+	0.973571i	$0.573344\pi$
$998$	13.8135	0.437260
$999$	57.9618	1.83383

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4031.2.a.c.1.20	✓	61

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4031.2.a.c.1.20	✓	61	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:	\( N \)	\(=\)	\( 4031 = 29 \cdot 139 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4031.a (trivial)

\(f(q)\)	\(=\)	\(q-1.13471 q^{2} +1.15618 q^{3} -0.712442 q^{4} +3.10962 q^{5} -1.31192 q^{6} -1.02298 q^{7} +3.07782 q^{8} -1.66325 q^{9} +O(q^{10})\)	\(q-1.13471 q^{2} +1.15618 q^{3} -0.712442 q^{4} +3.10962 q^{5} -1.31192 q^{6} -1.02298 q^{7} +3.07782 q^{8} -1.66325 q^{9} -3.52850 q^{10} -1.83622 q^{11} -0.823711 q^{12} +3.48520 q^{13} +1.16078 q^{14} +3.59528 q^{15} -2.06754 q^{16} -1.09492 q^{17} +1.88730 q^{18} -2.20806 q^{19} -2.21542 q^{20} -1.18275 q^{21} +2.08357 q^{22} -1.23653 q^{23} +3.55852 q^{24} +4.66973 q^{25} -3.95467 q^{26} -5.39155 q^{27} +0.728812 q^{28} -1.00000 q^{29} -4.07958 q^{30} -4.93412 q^{31} -3.80960 q^{32} -2.12300 q^{33} +1.24242 q^{34} -3.18107 q^{35} +1.18497 q^{36} -10.7505 q^{37} +2.50550 q^{38} +4.02951 q^{39} +9.57086 q^{40} -3.31015 q^{41} +1.34207 q^{42} +7.70702 q^{43} +1.30820 q^{44} -5.17207 q^{45} +1.40309 q^{46} +7.43359 q^{47} -2.39045 q^{48} -5.95352 q^{49} -5.29877 q^{50} -1.26593 q^{51} -2.48300 q^{52} -2.52651 q^{53} +6.11783 q^{54} -5.70995 q^{55} -3.14854 q^{56} -2.55292 q^{57} +1.13471 q^{58} +4.84374 q^{59} -2.56143 q^{60} -5.33508 q^{61} +5.59878 q^{62} +1.70147 q^{63} +8.45786 q^{64} +10.8376 q^{65} +2.40899 q^{66} -10.1714 q^{67} +0.780070 q^{68} -1.42965 q^{69} +3.60958 q^{70} -8.54404 q^{71} -5.11919 q^{72} +6.94608 q^{73} +12.1986 q^{74} +5.39905 q^{75} +1.57312 q^{76} +1.87841 q^{77} -4.57231 q^{78} +6.60000 q^{79} -6.42927 q^{80} -1.24386 q^{81} +3.75605 q^{82} -9.84384 q^{83} +0.842638 q^{84} -3.40480 q^{85} -8.74520 q^{86} -1.15618 q^{87} -5.65157 q^{88} -8.43394 q^{89} +5.86878 q^{90} -3.56528 q^{91} +0.880954 q^{92} -5.70473 q^{93} -8.43494 q^{94} -6.86623 q^{95} -4.40458 q^{96} -1.62517 q^{97} +6.75549 q^{98} +3.05409 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 61 q - q^{2} - 4 q^{3} + 43 q^{4} - 7 q^{5} - 13 q^{6} - 10 q^{7} - 6 q^{8} + 23 q^{9}+O(q^{10}) \) 61 * q - q^2 - 4 * q^3 + 43 * q^4 - 7 * q^5 - 13 * q^6 - 10 * q^7 - 6 * q^8 + 23 * q^9	\( 61 q - q^{2} - 4 q^{3} + 43 q^{4} - 7 q^{5} - 13 q^{6} - 10 q^{7} - 6 q^{8} + 23 q^{9} - 16 q^{10} - 3 q^{11} - 18 q^{12} - 28 q^{13} - 14 q^{14} - 12 q^{15} + 11 q^{16} - 21 q^{17} - 17 q^{18} - 36 q^{19} - 16 q^{20} - 12 q^{21} - 42 q^{22} - 15 q^{23} - 28 q^{24} - 16 q^{25} - 13 q^{26} - 10 q^{27} - 25 q^{28} - 61 q^{29} - 12 q^{30} - 18 q^{31} - 3 q^{32} - 42 q^{33} - 22 q^{34} - 29 q^{35} - 38 q^{36} - 30 q^{37} - 27 q^{38} - 31 q^{39} - 22 q^{40} - 28 q^{41} - 9 q^{42} - 58 q^{43} - 2 q^{44} - 31 q^{45} - 40 q^{46} - 6 q^{47} - 37 q^{48} - 37 q^{49} - 15 q^{50} - 44 q^{51} - 43 q^{52} - 27 q^{53} - 18 q^{54} - 38 q^{55} - 22 q^{56} - 50 q^{57} + q^{58} - 24 q^{59} + 6 q^{60} - 76 q^{61} - 17 q^{62} - 6 q^{63} - 60 q^{64} - 65 q^{65} - 7 q^{66} - 45 q^{67} - 31 q^{68} - 16 q^{69} - 48 q^{70} - 28 q^{71} - 40 q^{72} - 50 q^{73} - 17 q^{74} - 35 q^{75} - 100 q^{76} + q^{77} - 6 q^{78} - 66 q^{79} - 10 q^{80} - 63 q^{81} - 5 q^{82} - 9 q^{83} - 24 q^{84} - 77 q^{85} + 29 q^{86} + 4 q^{87} - 62 q^{88} - 30 q^{89} + 50 q^{90} - 52 q^{91} - 53 q^{92} - 42 q^{93} - 92 q^{94} - 20 q^{95} - 47 q^{96} - 34 q^{97} + 36 q^{98} - 29 q^{99}+O(q^{100}) \) 61 * q - q^2 - 4 * q^3 + 43 * q^4 - 7 * q^5 - 13 * q^6 - 10 * q^7 - 6 * q^8 + 23 * q^9 - 16 * q^10 - 3 * q^11 - 18 * q^12 - 28 * q^13 - 14 * q^14 - 12 * q^15 + 11 * q^16 - 21 * q^17 - 17 * q^18 - 36 * q^19 - 16 * q^20 - 12 * q^21 - 42 * q^22 - 15 * q^23 - 28 * q^24 - 16 * q^25 - 13 * q^26 - 10 * q^27 - 25 * q^28 - 61 * q^29 - 12 * q^30 - 18 * q^31 - 3 * q^32 - 42 * q^33 - 22 * q^34 - 29 * q^35 - 38 * q^36 - 30 * q^37 - 27 * q^38 - 31 * q^39 - 22 * q^40 - 28 * q^41 - 9 * q^42 - 58 * q^43 - 2 * q^44 - 31 * q^45 - 40 * q^46 - 6 * q^47 - 37 * q^48 - 37 * q^49 - 15 * q^50 - 44 * q^51 - 43 * q^52 - 27 * q^53 - 18 * q^54 - 38 * q^55 - 22 * q^56 - 50 * q^57 + q^58 - 24 * q^59 + 6 * q^60 - 76 * q^61 - 17 * q^62 - 6 * q^63 - 60 * q^64 - 65 * q^65 - 7 * q^66 - 45 * q^67 - 31 * q^68 - 16 * q^69 - 48 * q^70 - 28 * q^71 - 40 * q^72 - 50 * q^73 - 17 * q^74 - 35 * q^75 - 100 * q^76 + q^77 - 6 * q^78 - 66 * q^79 - 10 * q^80 - 63 * q^81 - 5 * q^82 - 9 * q^83 - 24 * q^84 - 77 * q^85 + 29 * q^86 + 4 * q^87 - 62 * q^88 - 30 * q^89 + 50 * q^90 - 52 * q^91 - 53 * q^92 - 42 * q^93 - 92 * q^94 - 20 * q^95 - 47 * q^96 - 34 * q^97 + 36 * q^98 - 29 * q^99

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