LMFDB - Embedded newform 6046.2.a.f.1.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6046,2,Mod(1,6046)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6046, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6046.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6046 = 2 \cdot 3023 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6046.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:	$48.2775530621$
Analytic rank:	$0$
Dimension:	$67$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Character	$\chi$	$=$	6046.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.34907 q^{3} +1.00000 q^{4} +0.876974 q^{5} -1.34907 q^{6} -1.47730 q^{7} +1.00000 q^{8} -1.18002 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.34907 q^{3} +1.00000 q^{4} +0.876974 q^{5} -1.34907 q^{6} -1.47730 q^{7} +1.00000 q^{8} -1.18002 q^{9} +0.876974 q^{10} -0.0292965 q^{11} -1.34907 q^{12} +6.18563 q^{13} -1.47730 q^{14} -1.18310 q^{15} +1.00000 q^{16} -6.72151 q^{17} -1.18002 q^{18} -8.02564 q^{19} +0.876974 q^{20} +1.99298 q^{21} -0.0292965 q^{22} +4.37794 q^{23} -1.34907 q^{24} -4.23092 q^{25} +6.18563 q^{26} +5.63912 q^{27} -1.47730 q^{28} +2.99748 q^{29} -1.18310 q^{30} +2.80401 q^{31} +1.00000 q^{32} +0.0395229 q^{33} -6.72151 q^{34} -1.29555 q^{35} -1.18002 q^{36} +4.76212 q^{37} -8.02564 q^{38} -8.34483 q^{39} +0.876974 q^{40} -2.08348 q^{41} +1.99298 q^{42} +0.722839 q^{43} -0.0292965 q^{44} -1.03484 q^{45} +4.37794 q^{46} +8.26996 q^{47} -1.34907 q^{48} -4.81759 q^{49} -4.23092 q^{50} +9.06777 q^{51} +6.18563 q^{52} +11.7237 q^{53} +5.63912 q^{54} -0.0256923 q^{55} -1.47730 q^{56} +10.8271 q^{57} +2.99748 q^{58} +0.169809 q^{59} -1.18310 q^{60} -4.45587 q^{61} +2.80401 q^{62} +1.74324 q^{63} +1.00000 q^{64} +5.42464 q^{65} +0.0395229 q^{66} +12.9003 q^{67} -6.72151 q^{68} -5.90614 q^{69} -1.29555 q^{70} -8.68095 q^{71} -1.18002 q^{72} +6.27841 q^{73} +4.76212 q^{74} +5.70779 q^{75} -8.02564 q^{76} +0.0432797 q^{77} -8.34483 q^{78} +16.4909 q^{79} +0.876974 q^{80} -4.06751 q^{81} -2.08348 q^{82} +2.43411 q^{83} +1.99298 q^{84} -5.89459 q^{85} +0.722839 q^{86} -4.04381 q^{87} -0.0292965 q^{88} +0.584767 q^{89} -1.03484 q^{90} -9.13802 q^{91} +4.37794 q^{92} -3.78280 q^{93} +8.26996 q^{94} -7.03828 q^{95} -1.34907 q^{96} +1.46694 q^{97} -4.81759 q^{98} +0.0345704 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 67 q + 67 q^{2} + 21 q^{3} + 67 q^{4} + 21 q^{5} + 21 q^{6} + 38 q^{7} + 67 q^{8} + 90 q^{9}+O(q^{10}) $ 67 * q + 67 * q^2 + 21 * q^3 + 67 * q^4 + 21 * q^5 + 21 * q^6 + 38 * q^7 + 67 * q^8 + 90 * q^9	\( 67 q + 67 q^{2} + 21 q^{3} + 67 q^{4} + 21 q^{5} + 21 q^{6} + 38 q^{7} + 67 q^{8} + 90 q^{9} + 21 q^{10} + 56 q^{11} + 21 q^{12} + 33 q^{13} + 38 q^{14} + 25 q^{15} + 67 q^{16} + 30 q^{17} + 90 q^{18} + 36 q^{19} + 21 q^{20} + 20 q^{21} + 56 q^{22} + 65 q^{23} + 21 q^{24} + 72 q^{25} + 33 q^{26} + 57 q^{27} + 38 q^{28} + 84 q^{29} + 25 q^{30} + 52 q^{31} + 67 q^{32} - 9 q^{33} + 30 q^{34} + 30 q^{35} + 90 q^{36} + 52 q^{37} + 36 q^{38} + 41 q^{39} + 21 q^{40} + 46 q^{41} + 20 q^{42} + 61 q^{43} + 56 q^{44} + 23 q^{45} + 65 q^{46} + 51 q^{47} + 21 q^{48} + 81 q^{49} + 72 q^{50} + 33 q^{51} + 33 q^{52} + 72 q^{53} + 57 q^{54} + 14 q^{55} + 38 q^{56} - 26 q^{57} + 84 q^{58} + 71 q^{59} + 25 q^{60} + 42 q^{61} + 52 q^{62} + 63 q^{63} + 67 q^{64} - 2 q^{65} - 9 q^{66} + 70 q^{67} + 30 q^{68} + 21 q^{69} + 30 q^{70} + 104 q^{71} + 90 q^{72} - 31 q^{73} + 52 q^{74} + 69 q^{75} + 36 q^{76} + 48 q^{77} + 41 q^{78} + 79 q^{79} + 21 q^{80} + 123 q^{81} + 46 q^{82} + 41 q^{83} + 20 q^{84} + 6 q^{85} + 61 q^{86} + 19 q^{87} + 56 q^{88} + 58 q^{89} + 23 q^{90} + 31 q^{91} + 65 q^{92} + 13 q^{93} + 51 q^{94} + 77 q^{95} + 21 q^{96} - 8 q^{97} + 81 q^{98} + 129 q^{99}+O(q^{100}) \) 67 * q + 67 * q^2 + 21 * q^3 + 67 * q^4 + 21 * q^5 + 21 * q^6 + 38 * q^7 + 67 * q^8 + 90 * q^9 + 21 * q^10 + 56 * q^11 + 21 * q^12 + 33 * q^13 + 38 * q^14 + 25 * q^15 + 67 * q^16 + 30 * q^17 + 90 * q^18 + 36 * q^19 + 21 * q^20 + 20 * q^21 + 56 * q^22 + 65 * q^23 + 21 * q^24 + 72 * q^25 + 33 * q^26 + 57 * q^27 + 38 * q^28 + 84 * q^29 + 25 * q^30 + 52 * q^31 + 67 * q^32 - 9 * q^33 + 30 * q^34 + 30 * q^35 + 90 * q^36 + 52 * q^37 + 36 * q^38 + 41 * q^39 + 21 * q^40 + 46 * q^41 + 20 * q^42 + 61 * q^43 + 56 * q^44 + 23 * q^45 + 65 * q^46 + 51 * q^47 + 21 * q^48 + 81 * q^49 + 72 * q^50 + 33 * q^51 + 33 * q^52 + 72 * q^53 + 57 * q^54 + 14 * q^55 + 38 * q^56 - 26 * q^57 + 84 * q^58 + 71 * q^59 + 25 * q^60 + 42 * q^61 + 52 * q^62 + 63 * q^63 + 67 * q^64 - 2 * q^65 - 9 * q^66 + 70 * q^67 + 30 * q^68 + 21 * q^69 + 30 * q^70 + 104 * q^71 + 90 * q^72 - 31 * q^73 + 52 * q^74 + 69 * q^75 + 36 * q^76 + 48 * q^77 + 41 * q^78 + 79 * q^79 + 21 * q^80 + 123 * q^81 + 46 * q^82 + 41 * q^83 + 20 * q^84 + 6 * q^85 + 61 * q^86 + 19 * q^87 + 56 * q^88 + 58 * q^89 + 23 * q^90 + 31 * q^91 + 65 * q^92 + 13 * q^93 + 51 * q^94 + 77 * q^95 + 21 * q^96 - 8 * q^97 + 81 * q^98 + 129 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.34907	−0.778884	−0.389442	−	0.921051i	$-0.627332\pi$
$3$	−1.34907	−0.778884	−0.389442	+	0.921051i	$0.627332\pi$
$4$	1.00000	0.500000
$5$	0.876974	0.392195	0.196097	−	0.980584i	$-0.437173\pi$
$5$	0.876974	0.392195	0.196097	+	0.980584i	$0.437173\pi$
$6$	−1.34907	−0.550754
$7$	−1.47730	−0.558367	−0.279183	−	0.960238i	$-0.590064\pi$
$7$	−1.47730	−0.558367	−0.279183	+	0.960238i	$0.590064\pi$
$8$	1.00000	0.353553
$9$	−1.18002	−0.393339
$10$	0.876974	0.277324
$11$	−0.0292965	−0.00883322	−0.00441661	−	0.999990i	$-0.501406\pi$
$11$	−0.0292965	−0.00883322	−0.00441661	+	0.999990i	$0.501406\pi$
$12$	−1.34907	−0.389442
$13$	6.18563	1.71558	0.857792	−	0.513997i	$-0.171836\pi$
$13$	6.18563	1.71558	0.857792	+	0.513997i	$0.171836\pi$
$14$	−1.47730	−0.394825
$15$	−1.18310	−0.305474
$16$	1.00000	0.250000
$17$	−6.72151	−1.63021	−0.815103	−	0.579316i	$-0.803320\pi$
$17$	−6.72151	−1.63021	−0.815103	+	0.579316i	$0.803320\pi$
$18$	−1.18002	−0.278133
$19$	−8.02564	−1.84121	−0.920604	−	0.390497i	$-0.872303\pi$
$19$	−8.02564	−1.84121	−0.920604	+	0.390497i	$0.872303\pi$
$20$	0.876974	0.196097
$21$	1.99298	0.434903
$22$	−0.0292965	−0.00624603
$23$	4.37794	0.912864	0.456432	−	0.889758i	$-0.349127\pi$
$23$	4.37794	0.912864	0.456432	+	0.889758i	$0.349127\pi$
$24$	−1.34907	−0.275377
$25$	−4.23092	−0.846183
$26$	6.18563	1.21310
$27$	5.63912	1.08525
$28$	−1.47730	−0.279183
$29$	2.99748	0.556619	0.278309	−	0.960492i	$-0.410226\pi$
$29$	2.99748	0.556619	0.278309	+	0.960492i	$0.410226\pi$
$30$	−1.18310	−0.216003
$31$	2.80401	0.503616	0.251808	−	0.967777i	$-0.418975\pi$
$31$	2.80401	0.503616	0.251808	+	0.967777i	$0.418975\pi$
$32$	1.00000	0.176777
$33$	0.0395229	0.00688006
$34$	−6.72151	−1.15273
$35$	−1.29555	−0.218988
$36$	−1.18002	−0.196670
$37$	4.76212	0.782887	0.391444	−	0.920202i	$-0.371976\pi$
$37$	4.76212	0.782887	0.391444	+	0.920202i	$0.371976\pi$
$38$	−8.02564	−1.30193
$39$	−8.34483	−1.33624
$40$	0.876974	0.138662
$41$	−2.08348	−0.325385	−0.162693	−	0.986677i	$-0.552018\pi$
$41$	−2.08348	−0.325385	−0.162693	+	0.986677i	$0.552018\pi$
$42$	1.99298	0.307523
$43$	0.722839	0.110232	0.0551160	−	0.998480i	$-0.482447\pi$
$43$	0.722839	0.110232	0.0551160	+	0.998480i	$0.482447\pi$
$44$	−0.0292965	−0.00441661
$45$	−1.03484	−0.154266
$46$	4.37794	0.645492
$47$	8.26996	1.20630	0.603149	−	0.797629i	$-0.293913\pi$
$47$	8.26996	1.20630	0.603149	+	0.797629i	$0.293913\pi$
$48$	−1.34907	−0.194721
$49$	−4.81759	−0.688227
$50$	−4.23092	−0.598342
$51$	9.06777	1.26974
$52$	6.18563	0.857792
$53$	11.7237	1.61037	0.805186	−	0.593022i	$-0.202065\pi$
$53$	11.7237	1.61037	0.805186	+	0.593022i	$0.202065\pi$
$54$	5.63912	0.767388
$55$	−0.0256923	−0.00346434
$56$	−1.47730	−0.197412
$57$	10.8271	1.43409
$58$	2.99748	0.393589
$59$	0.169809	0.0221072	0.0110536	−	0.999939i	$-0.496481\pi$
$59$	0.169809	0.0221072	0.0110536	+	0.999939i	$0.496481\pi$
$60$	−1.18310	−0.152737
$61$	−4.45587	−0.570515	−0.285258	−	0.958451i	$-0.592079\pi$
$61$	−4.45587	−0.570515	−0.285258	+	0.958451i	$0.592079\pi$
$62$	2.80401	0.356110
$63$	1.74324	0.219627
$64$	1.00000	0.125000
$65$	5.42464	0.672843
$66$	0.0395229	0.00486494
$67$	12.9003	1.57602	0.788010	−	0.615662i	$-0.211111\pi$
$67$	12.9003	1.57602	0.788010	+	0.615662i	$0.211111\pi$
$68$	−6.72151	−0.815103
$69$	−5.90614	−0.711016
$70$	−1.29555	−0.154848
$71$	−8.68095	−1.03024	−0.515119	−	0.857118i	$-0.672252\pi$
$71$	−8.68095	−1.03024	−0.515119	+	0.857118i	$0.672252\pi$
$72$	−1.18002	−0.139066
$73$	6.27841	0.734832	0.367416	−	0.930057i	$-0.380243\pi$
$73$	6.27841	0.734832	0.367416	+	0.930057i	$0.380243\pi$
$74$	4.76212	0.553585
$75$	5.70779	0.659079
$76$	−8.02564	−0.920604
$77$	0.0432797	0.00493218
$78$	−8.34483	−0.944866
$79$	16.4909	1.85537	0.927687	−	0.373358i	$-0.121794\pi$
$79$	16.4909	1.85537	0.927687	+	0.373358i	$0.121794\pi$
$80$	0.876974	0.0980487
$81$	−4.06751	−0.451945
$82$	−2.08348	−0.230082
$83$	2.43411	0.267178	0.133589	−	0.991037i	$-0.457350\pi$
$83$	2.43411	0.267178	0.133589	+	0.991037i	$0.457350\pi$
$84$	1.99298	0.217451
$85$	−5.89459	−0.639358
$86$	0.722839	0.0779458
$87$	−4.04381	−0.433542
$88$	−0.0292965	−0.00312302
$89$	0.584767	0.0619852	0.0309926	−	0.999520i	$-0.490133\pi$
$89$	0.584767	0.0619852	0.0309926	+	0.999520i	$0.490133\pi$
$90$	−1.03484	−0.109082
$91$	−9.13802	−0.957925
$92$	4.37794	0.456432
$93$	−3.78280	−0.392258
$94$	8.26996	0.852981
$95$	−7.03828	−0.722112
$96$	−1.34907	−0.137689
$97$	1.46694	0.148945	0.0744727	−	0.997223i	$-0.476273\pi$
$97$	1.46694	0.148945	0.0744727	+	0.997223i	$0.476273\pi$
$98$	−4.81759	−0.486650
$99$	0.0345704	0.00347445
$100$	−4.23092	−0.423092
$101$	−13.6139	−1.35463	−0.677316	−	0.735693i	$-0.736857\pi$
$101$	−13.6139	−1.35463	−0.677316	+	0.735693i	$0.736857\pi$
$102$	9.06777	0.897843
$103$	9.35314	0.921592	0.460796	−	0.887506i	$-0.347564\pi$
$103$	9.35314	0.921592	0.460796	+	0.887506i	$0.347564\pi$
$104$	6.18563	0.606551
$105$	1.74779	0.170567
$106$	11.7237	1.13870
$107$	2.14312	0.207183	0.103591	−	0.994620i	$-0.466967\pi$
$107$	2.14312	0.207183	0.103591	+	0.994620i	$0.466967\pi$
$108$	5.63912	0.542625
$109$	−8.79687	−0.842588	−0.421294	−	0.906924i	$-0.638424\pi$
$109$	−8.79687	−0.842588	−0.421294	+	0.906924i	$0.638424\pi$
$110$	−0.0256923	−0.00244966
$111$	−6.42442	−0.609779
$112$	−1.47730	−0.139592
$113$	−2.47002	−0.232360	−0.116180	−	0.993228i	$-0.537065\pi$
$113$	−2.47002	−0.232360	−0.116180	+	0.993228i	$0.537065\pi$
$114$	10.8271	1.01405
$115$	3.83934	0.358021
$116$	2.99748	0.278309
$117$	−7.29915	−0.674806
$118$	0.169809	0.0156322
$119$	9.92968	0.910252
$120$	−1.18310	−0.108002
$121$	−10.9991	−0.999922
$122$	−4.45587	−0.403415
$123$	2.81076	0.253437
$124$	2.80401	0.251808
$125$	−8.09527	−0.724063
$126$	1.74324	0.155300
$127$	12.9094	1.14552	0.572762	−	0.819722i	$-0.305872\pi$
$127$	12.9094	1.14552	0.572762	+	0.819722i	$0.305872\pi$
$128$	1.00000	0.0883883
$129$	−0.975159	−0.0858579
$130$	5.42464	0.475772
$131$	−3.55452	−0.310560	−0.155280	−	0.987871i	$-0.549628\pi$
$131$	−3.55452	−0.310560	−0.155280	+	0.987871i	$0.549628\pi$
$132$	0.0395229	0.00344003
$133$	11.8563	1.02807
$134$	12.9003	1.11441
$135$	4.94537	0.425629
$136$	−6.72151	−0.576365
$137$	12.4968	1.06767	0.533836	−	0.845588i	$-0.320750\pi$
$137$	12.4968	1.06767	0.533836	+	0.845588i	$0.320750\pi$
$138$	−5.90614	−0.502764
$139$	−4.50141	−0.381804	−0.190902	−	0.981609i	$-0.561141\pi$
$139$	−4.50141	−0.381804	−0.190902	+	0.981609i	$0.561141\pi$
$140$	−1.29555	−0.109494
$141$	−11.1567	−0.939566
$142$	−8.68095	−0.728489
$143$	−0.181217	−0.0151541
$144$	−1.18002	−0.0983348
$145$	2.62871	0.218303
$146$	6.27841	0.519605
$147$	6.49925	0.536049
$148$	4.76212	0.391444
$149$	16.4971	1.35149	0.675746	−	0.737134i	$-0.263822\pi$
$149$	16.4971	1.35149	0.675746	+	0.737134i	$0.263822\pi$
$150$	5.70779	0.466039
$151$	−1.80492	−0.146883	−0.0734414	−	0.997300i	$-0.523398\pi$
$151$	−1.80492	−0.146883	−0.0734414	+	0.997300i	$0.523398\pi$
$152$	−8.02564	−0.650966
$153$	7.93150	0.641224
$154$	0.0432797	0.00348757
$155$	2.45905	0.197515
$156$	−8.34483	−0.668121
$157$	−6.84260	−0.546099	−0.273049	−	0.962000i	$-0.588032\pi$
$157$	−6.84260	−0.546099	−0.273049	+	0.962000i	$0.588032\pi$
$158$	16.4909	1.31195
$159$	−15.8160	−1.25429
$160$	0.876974	0.0693309
$161$	−6.46753	−0.509713
$162$	−4.06751	−0.319574
$163$	15.6027	1.22210	0.611048	−	0.791594i	$-0.290748\pi$
$163$	15.6027	1.22210	0.611048	+	0.791594i	$0.290748\pi$
$164$	−2.08348	−0.162693
$165$	0.0346606	0.00269832
$166$	2.43411	0.188923
$167$	25.2705	1.95549	0.977747	−	0.209788i	$-0.0672775\pi$
$167$	25.2705	1.95549	0.977747	+	0.209788i	$0.0672775\pi$
$168$	1.99298	0.153761
$169$	25.2620	1.94323
$170$	−5.89459	−0.452095
$171$	9.47040	0.724219
$172$	0.722839	0.0551160
$173$	−4.94218	−0.375747	−0.187873	−	0.982193i	$-0.560159\pi$
$173$	−4.94218	−0.375747	−0.187873	+	0.982193i	$0.560159\pi$
$174$	−4.04381	−0.306560
$175$	6.25033	0.472480
$176$	−0.0292965	−0.00220831
$177$	−0.229083	−0.0172190
$178$	0.584767	0.0438301
$179$	−2.63896	−0.197245	−0.0986225	−	0.995125i	$-0.531444\pi$
$179$	−2.63896	−0.197245	−0.0986225	+	0.995125i	$0.531444\pi$
$180$	−1.03484	−0.0771328
$181$	10.0058	0.743725	0.371863	−	0.928288i	$-0.378719\pi$
$181$	10.0058	0.743725	0.371863	+	0.928288i	$0.378719\pi$
$182$	−9.13802	−0.677355
$183$	6.01126	0.444365
$184$	4.37794	0.322746
$185$	4.17625	0.307044
$186$	−3.78280	−0.277369
$187$	0.196917	0.0144000
$188$	8.26996	0.603149
$189$	−8.33067	−0.605967
$190$	−7.03828	−0.510611
$191$	0.461403	0.0333860	0.0166930	−	0.999861i	$-0.494686\pi$
$191$	0.461403	0.0333860	0.0166930	+	0.999861i	$0.494686\pi$
$192$	−1.34907	−0.0973606
$193$	−0.516765	−0.0371976	−0.0185988	−	0.999827i	$-0.505921\pi$
$193$	−0.516765	−0.0371976	−0.0185988	+	0.999827i	$0.505921\pi$
$194$	1.46694	0.105320
$195$	−7.31820	−0.524067
$196$	−4.81759	−0.344113
$197$	−6.80637	−0.484934	−0.242467	−	0.970160i	$-0.577957\pi$
$197$	−6.80637	−0.484934	−0.242467	+	0.970160i	$0.577957\pi$
$198$	0.0345704	0.00245681
$199$	−3.66385	−0.259723	−0.129862	−	0.991532i	$-0.541453\pi$
$199$	−3.66385	−0.259723	−0.129862	+	0.991532i	$0.541453\pi$
$200$	−4.23092	−0.299171
$201$	−17.4034	−1.22754
$202$	−13.6139	−0.957869
$203$	−4.42818	−0.310797
$204$	9.06777	0.634871
$205$	−1.82716	−0.127614
$206$	9.35314	0.651664
$207$	−5.16605	−0.359065
$208$	6.18563	0.428896
$209$	0.235123	0.0162638
$210$	1.74779	0.120609
$211$	−9.53355	−0.656316	−0.328158	−	0.944623i	$-0.606428\pi$
$211$	−9.53355	−0.656316	−0.328158	+	0.944623i	$0.606428\pi$
$212$	11.7237	0.805186
$213$	11.7112	0.802437
$214$	2.14312	0.146500
$215$	0.633911	0.0432324
$216$	5.63912	0.383694
$217$	−4.14237	−0.281202
$218$	−8.79687	−0.595800
$219$	−8.47000	−0.572349
$220$	−0.0256923	−0.00173217
$221$	−41.5768	−2.79676
$222$	−6.42442	−0.431179
$223$	18.7881	1.25814	0.629072	−	0.777347i	$-0.283435\pi$
$223$	18.7881	1.25814	0.629072	+	0.777347i	$0.283435\pi$
$224$	−1.47730	−0.0987062
$225$	4.99255	0.332837
$226$	−2.47002	−0.164303
$227$	2.43014	0.161294	0.0806471	−	0.996743i	$-0.474301\pi$
$227$	2.43014	0.161294	0.0806471	+	0.996743i	$0.474301\pi$
$228$	10.8271	0.717044
$229$	−5.70478	−0.376983	−0.188491	−	0.982075i	$-0.560360\pi$
$229$	−5.70478	−0.376983	−0.188491	+	0.982075i	$0.560360\pi$
$230$	3.83934	0.253159
$231$	−0.0583872	−0.00384159
$232$	2.99748	0.196794
$233$	19.3581	1.26819	0.634097	−	0.773254i	$-0.281372\pi$
$233$	19.3581	1.26819	0.634097	+	0.773254i	$0.281372\pi$
$234$	−7.29915	−0.477160
$235$	7.25254	0.473103
$236$	0.169809	0.0110536
$237$	−22.2474	−1.44512
$238$	9.92968	0.643646
$239$	14.6166	0.945468	0.472734	−	0.881205i	$-0.343267\pi$
$239$	14.6166	0.945468	0.472734	+	0.881205i	$0.343267\pi$
$240$	−1.18310	−0.0763686
$241$	17.9327	1.15515	0.577575	−	0.816338i	$-0.303999\pi$
$241$	17.9327	1.15515	0.577575	+	0.816338i	$0.303999\pi$
$242$	−10.9991	−0.707052
$243$	−11.4300	−0.733237
$244$	−4.45587	−0.285258
$245$	−4.22490	−0.269919
$246$	2.81076	0.179207
$247$	−49.6436	−3.15875
$248$	2.80401	0.178055
$249$	−3.28378	−0.208101
$250$	−8.09527	−0.511990
$251$	13.6955	0.864455	0.432227	−	0.901765i	$-0.357728\pi$
$251$	13.6955	0.864455	0.432227	+	0.901765i	$0.357728\pi$
$252$	1.74324	0.109814
$253$	−0.128258	−0.00806353
$254$	12.9094	0.810007
$255$	7.95220	0.497986
$256$	1.00000	0.0625000
$257$	5.40478	0.337141	0.168570	−	0.985690i	$-0.446085\pi$
$257$	5.40478	0.337141	0.168570	+	0.985690i	$0.446085\pi$
$258$	−0.975159	−0.0607107
$259$	−7.03507	−0.437138
$260$	5.42464	0.336422
$261$	−3.53708	−0.218940
$262$	−3.55452	−0.219599
$263$	−20.8073	−1.28303	−0.641517	−	0.767109i	$-0.721694\pi$
$263$	−20.8073	−1.28303	−0.641517	+	0.767109i	$0.721694\pi$
$264$	0.0395229	0.00243247
$265$	10.2814	0.631579
$266$	11.8563	0.726955
$267$	−0.788890	−0.0482793
$268$	12.9003	0.788010
$269$	7.08465	0.431959	0.215979	−	0.976398i	$-0.430706\pi$
$269$	7.08465	0.431959	0.215979	+	0.976398i	$0.430706\pi$
$270$	4.94537	0.300965
$271$	28.9776	1.76027	0.880133	−	0.474727i	$-0.157453\pi$
$271$	28.9776	1.76027	0.880133	+	0.474727i	$0.157453\pi$
$272$	−6.72151	−0.407552
$273$	12.3278	0.746113
$274$	12.4968	0.754959
$275$	0.123951	0.00747452
$276$	−5.90614	−0.355508
$277$	−14.5476	−0.874080	−0.437040	−	0.899442i	$-0.643973\pi$
$277$	−14.5476	−0.874080	−0.437040	+	0.899442i	$0.643973\pi$
$278$	−4.50141	−0.269976
$279$	−3.30878	−0.198092
$280$	−1.29555	−0.0774241
$281$	10.3190	0.615581	0.307790	−	0.951454i	$-0.400410\pi$
$281$	10.3190	0.615581	0.307790	+	0.951454i	$0.400410\pi$
$282$	−11.1567	−0.664374
$283$	21.3666	1.27011	0.635055	−	0.772467i	$-0.280977\pi$
$283$	21.3666	1.27011	0.635055	+	0.772467i	$0.280977\pi$
$284$	−8.68095	−0.515119
$285$	9.49511	0.562442
$286$	−0.181217	−0.0107156
$287$	3.07792	0.181684
$288$	−1.18002	−0.0695332
$289$	28.1787	1.65757
$290$	2.62871	0.154363
$291$	−1.97900	−0.116011
$292$	6.27841	0.367416
$293$	3.68617	0.215349	0.107674	−	0.994186i	$-0.465660\pi$
$293$	3.68617	0.215349	0.107674	+	0.994186i	$0.465660\pi$
$294$	6.49925	0.379044
$295$	0.148918	0.00867033
$296$	4.76212	0.276792
$297$	−0.165207	−0.00958625
$298$	16.4971	0.955649
$299$	27.0803	1.56610
$300$	5.70779	0.329539
$301$	−1.06785	−0.0615498
$302$	−1.80492	−0.103862
$303$	18.3660	1.05510
$304$	−8.02564	−0.460302
$305$	−3.90768	−0.223753
$306$	7.93150	0.453414
$307$	−7.88521	−0.450033	−0.225016	−	0.974355i	$-0.572244\pi$
$307$	−7.88521	−0.450033	−0.225016	+	0.974355i	$0.572244\pi$
$308$	0.0432797	0.00246609
$309$	−12.6180	−0.717814
$310$	2.45905	0.139664
$311$	−7.31365	−0.414719	−0.207359	−	0.978265i	$-0.566487\pi$
$311$	−7.31365	−0.414719	−0.207359	+	0.978265i	$0.566487\pi$
$312$	−8.34483	−0.472433
$313$	−21.8428	−1.23463	−0.617314	−	0.786717i	$-0.711779\pi$
$313$	−21.8428	−1.23463	−0.617314	+	0.786717i	$0.711779\pi$
$314$	−6.84260	−0.386150
$315$	1.52877	0.0861367
$316$	16.4909	0.927687
$317$	32.4405	1.82204	0.911021	−	0.412360i	$-0.135295\pi$
$317$	32.4405	1.82204	0.911021	+	0.412360i	$0.135295\pi$
$318$	−15.8160	−0.886919
$319$	−0.0878157	−0.00491674
$320$	0.876974	0.0490243
$321$	−2.89121	−0.161372
$322$	−6.46753	−0.360421
$323$	53.9444	3.00155
$324$	−4.06751	−0.225973
$325$	−26.1709	−1.45170
$326$	15.6027	0.864152
$327$	11.8676	0.656278
$328$	−2.08348	−0.115041
$329$	−12.2172	−0.673556
$330$	0.0346606	0.00190800
$331$	−6.82134	−0.374935	−0.187467	−	0.982271i	$-0.560028\pi$
$331$	−6.82134	−0.374935	−0.187467	+	0.982271i	$0.560028\pi$
$332$	2.43411	0.133589
$333$	−5.61938	−0.307940
$334$	25.2705	1.38274
$335$	11.3132	0.618107
$336$	1.99298	0.108726
$337$	−6.51775	−0.355045	−0.177522	−	0.984117i	$-0.556808\pi$
$337$	−6.51775	−0.355045	−0.177522	+	0.984117i	$0.556808\pi$
$338$	25.2620	1.37407
$339$	3.33222	0.180981
$340$	−5.89459	−0.319679
$341$	−0.0821477	−0.00444855
$342$	9.47040	0.512100
$343$	17.4581	0.942649
$344$	0.722839	0.0389729
$345$	−5.17953	−0.278857
$346$	−4.94218	−0.265693
$347$	−19.9693	−1.07201	−0.536005	−	0.844215i	$-0.680067\pi$
$347$	−19.9693	−1.07201	−0.536005	+	0.844215i	$0.680067\pi$
$348$	−4.04381	−0.216771
$349$	−20.6786	−1.10690	−0.553449	−	0.832883i	$-0.686689\pi$
$349$	−20.6786	−1.10690	−0.553449	+	0.832883i	$0.686689\pi$
$350$	6.25033	0.334094
$351$	34.8815	1.86184
$352$	−0.0292965	−0.00156151
$353$	8.88999	0.473166	0.236583	−	0.971611i	$-0.423972\pi$
$353$	8.88999	0.473166	0.236583	+	0.971611i	$0.423972\pi$
$354$	−0.229083	−0.0121756
$355$	−7.61296	−0.404054
$356$	0.584767	0.0309926
$357$	−13.3958	−0.708981
$358$	−2.63896	−0.139473
$359$	25.0732	1.32331	0.661657	−	0.749807i	$-0.269854\pi$
$359$	25.0732	1.32331	0.661657	+	0.749807i	$0.269854\pi$
$360$	−1.03484	−0.0545411
$361$	45.4109	2.39005
$362$	10.0058	0.525893
$363$	14.8386	0.778824
$364$	−9.13802	−0.478962
$365$	5.50600	0.288197
$366$	6.01126	0.314214
$367$	−25.4414	−1.32803	−0.664015	−	0.747719i	$-0.731149\pi$
$367$	−25.4414	−1.32803	−0.664015	+	0.747719i	$0.731149\pi$
$368$	4.37794	0.228216
$369$	2.45854	0.127987
$370$	4.17625	0.217113
$371$	−17.3194	−0.899178
$372$	−3.78280	−0.196129
$373$	18.3520	0.950229	0.475114	−	0.879924i	$-0.342407\pi$
$373$	18.3520	0.950229	0.475114	+	0.879924i	$0.342407\pi$
$374$	0.196917	0.0101823
$375$	10.9211	0.563962
$376$	8.26996	0.426490
$377$	18.5413	0.954926
$378$	−8.33067	−0.428484
$379$	31.7836	1.63261	0.816306	−	0.577619i	$-0.196018\pi$
$379$	31.7836	1.63261	0.816306	+	0.577619i	$0.196018\pi$
$380$	−7.03828	−0.361056
$381$	−17.4156	−0.892230
$382$	0.461403	0.0236075
$383$	10.2717	0.524857	0.262428	−	0.964951i	$-0.415477\pi$
$383$	10.2717	0.524857	0.262428	+	0.964951i	$0.415477\pi$
$384$	−1.34907	−0.0688443
$385$	0.0379551	0.00193437
$386$	−0.516765	−0.0263027
$387$	−0.852963	−0.0433585
$388$	1.46694	0.0744727
$389$	−0.511903	−0.0259545	−0.0129773	−	0.999916i	$-0.504131\pi$
$389$	−0.511903	−0.0259545	−0.0129773	+	0.999916i	$0.504131\pi$
$390$	−7.31820	−0.370571
$391$	−29.4264	−1.48816
$392$	−4.81759	−0.243325
$393$	4.79528	0.241890
$394$	−6.80637	−0.342900
$395$	14.4621	0.727668
$396$	0.0345704	0.00173723
$397$	17.8554	0.896135	0.448067	−	0.894000i	$-0.352112\pi$
$397$	17.8554	0.896135	0.448067	+	0.894000i	$0.352112\pi$
$398$	−3.66385	−0.183652
$399$	−15.9949	−0.800747
$400$	−4.23092	−0.211546
$401$	−10.3647	−0.517589	−0.258794	−	0.965932i	$-0.583325\pi$
$401$	−10.3647	−0.517589	−0.258794	+	0.965932i	$0.583325\pi$
$402$	−17.4034	−0.868000
$403$	17.3446	0.863995
$404$	−13.6139	−0.677316
$405$	−3.56710	−0.177251
$406$	−4.42818	−0.219767
$407$	−0.139513	−0.00691541
$408$	9.06777	0.448922
$409$	−7.47695	−0.369711	−0.184856	−	0.982766i	$-0.559182\pi$
$409$	−7.47695	−0.369711	−0.184856	+	0.982766i	$0.559182\pi$
$410$	−1.82716	−0.0902369
$411$	−16.8590	−0.831594
$412$	9.35314	0.460796
$413$	−0.250858	−0.0123439
$414$	−5.16605	−0.253897
$415$	2.13465	0.104786
$416$	6.18563	0.303275
$417$	6.07270	0.297381
$418$	0.235123	0.0115002
$419$	−12.8829	−0.629373	−0.314687	−	0.949196i	$-0.601899\pi$
$419$	−12.8829	−0.629373	−0.314687	+	0.949196i	$0.601899\pi$
$420$	1.74779	0.0852833
$421$	−35.5689	−1.73352	−0.866760	−	0.498726i	$-0.833802\pi$
$421$	−35.5689	−1.73352	−0.866760	+	0.498726i	$0.833802\pi$
$422$	−9.53355	−0.464086
$423$	−9.75869	−0.474484
$424$	11.7237	0.569352
$425$	28.4382	1.37945
$426$	11.7112	0.567409
$427$	6.58265	0.318557
$428$	2.14312	0.103591
$429$	0.244474	0.0118033
$430$	0.633911	0.0305699
$431$	−13.6744	−0.658672	−0.329336	−	0.944213i	$-0.606825\pi$
$431$	−13.6744	−0.658672	−0.329336	+	0.944213i	$0.606825\pi$
$432$	5.63912	0.271313
$433$	−34.9999	−1.68199	−0.840993	−	0.541045i	$-0.818029\pi$
$433$	−34.9999	−1.68199	−0.840993	+	0.541045i	$0.818029\pi$
$434$	−4.14237	−0.198840
$435$	−3.54631	−0.170033
$436$	−8.79687	−0.421294
$437$	−35.1358	−1.68077
$438$	−8.47000	−0.404712
$439$	−24.6220	−1.17514	−0.587571	−	0.809173i	$-0.699916\pi$
$439$	−24.6220	−1.17514	−0.587571	+	0.809173i	$0.699916\pi$
$440$	−0.0256923	−0.00122483
$441$	5.68484	0.270707
$442$	−41.5768	−1.97761
$443$	0.154174	0.00732505	0.00366252	−	0.999993i	$-0.498834\pi$
$443$	0.154174	0.00732505	0.00366252	+	0.999993i	$0.498834\pi$
$444$	−6.42442	−0.304889
$445$	0.512825	0.0243103
$446$	18.7881	0.889642
$447$	−22.2557	−1.05266
$448$	−1.47730	−0.0697958
$449$	10.4439	0.492878	0.246439	−	0.969158i	$-0.420740\pi$
$449$	10.4439	0.492878	0.246439	+	0.969158i	$0.420740\pi$
$450$	4.99255	0.235351
$451$	0.0610387	0.00287420
$452$	−2.47002	−0.116180
$453$	2.43497	0.114405
$454$	2.43014	0.114052
$455$	−8.01381	−0.375693
$456$	10.8271	0.507027
$457$	−31.4648	−1.47186	−0.735931	−	0.677057i	$-0.763255\pi$
$457$	−31.4648	−1.47186	−0.735931	+	0.677057i	$0.763255\pi$
$458$	−5.70478	−0.266567
$459$	−37.9034	−1.76918
$460$	3.83934	0.179010
$461$	−25.7406	−1.19886	−0.599429	−	0.800428i	$-0.704605\pi$
$461$	−25.7406	−1.19886	−0.599429	+	0.800428i	$0.704605\pi$
$462$	−0.0583872	−0.00271642
$463$	29.4959	1.37079	0.685396	−	0.728171i	$-0.259629\pi$
$463$	29.4959	1.37079	0.685396	+	0.728171i	$0.259629\pi$
$464$	2.99748	0.139155
$465$	−3.31742	−0.153842
$466$	19.3581	0.896748
$467$	−10.2838	−0.475878	−0.237939	−	0.971280i	$-0.576472\pi$
$467$	−10.2838	−0.475878	−0.237939	+	0.971280i	$0.576472\pi$
$468$	−7.29915	−0.337403
$469$	−19.0576	−0.879997
$470$	7.25254	0.334535
$471$	9.23112	0.425348
$472$	0.169809	0.00781608
$473$	−0.0211766	−0.000973703 0
$474$	−22.2474	−1.02186
$475$	33.9558	1.55800
$476$	9.92968	0.455126
$477$	−13.8341	−0.633422
$478$	14.6166	0.668547
$479$	23.9369	1.09371	0.546853	−	0.837228i	$-0.315825\pi$
$479$	23.9369	1.09371	0.546853	+	0.837228i	$0.315825\pi$
$480$	−1.18310	−0.0540008
$481$	29.4567	1.34311
$482$	17.9327	0.816814
$483$	8.72513	0.397007
$484$	−10.9991	−0.499961
$485$	1.28647	0.0584156
$486$	−11.4300	−0.518477
$487$	−26.1293	−1.18403	−0.592016	−	0.805926i	$-0.701668\pi$
$487$	−26.1293	−1.18403	−0.592016	+	0.805926i	$0.701668\pi$
$488$	−4.45587	−0.201708
$489$	−21.0490	−0.951871
$490$	−4.22490	−0.190862
$491$	9.69302	0.437440	0.218720	−	0.975788i	$-0.429812\pi$
$491$	9.69302	0.437440	0.218720	+	0.975788i	$0.429812\pi$
$492$	2.81076	0.126719
$493$	−20.1476	−0.907403
$494$	−49.6436	−2.23357
$495$	0.0303173	0.00136266
$496$	2.80401	0.125904
$497$	12.8244	0.575251
$498$	−3.28378	−0.147150
$499$	22.6696	1.01483	0.507416	−	0.861701i	$-0.330601\pi$
$499$	22.6696	1.01483	0.507416	+	0.861701i	$0.330601\pi$
$500$	−8.09527	−0.362032
$501$	−34.0917	−1.52310
$502$	13.6955	0.611262
$503$	−4.57746	−0.204099	−0.102049	−	0.994779i	$-0.532540\pi$
$503$	−4.57746	−0.204099	−0.102049	+	0.994779i	$0.532540\pi$
$504$	1.74324	0.0776500
$505$	−11.9390	−0.531279
$506$	−0.128258	−0.00570178
$507$	−34.0801	−1.51355
$508$	12.9094	0.572762
$509$	7.46166	0.330732	0.165366	−	0.986232i	$-0.447119\pi$
$509$	7.46166	0.330732	0.165366	+	0.986232i	$0.447119\pi$
$510$	7.95220	0.352129
$511$	−9.27509	−0.410306
$512$	1.00000	0.0441942
$513$	−45.2576	−1.99817
$514$	5.40478	0.238394
$515$	8.20246	0.361444
$516$	−0.975159	−0.0429290
$517$	−0.242281	−0.0106555
$518$	−7.03507	−0.309103
$519$	6.66733	0.292663
$520$	5.42464	0.237886
$521$	4.52244	0.198132	0.0990659	−	0.995081i	$-0.468415\pi$
$521$	4.52244	0.198132	0.0990659	+	0.995081i	$0.468415\pi$
$522$	−3.53708	−0.154814
$523$	7.40598	0.323841	0.161921	−	0.986804i	$-0.448231\pi$
$523$	7.40598	0.323841	0.161921	+	0.986804i	$0.448231\pi$
$524$	−3.55452	−0.155280
$525$	−8.43211	−0.368008
$526$	−20.8073	−0.907241
$527$	−18.8472	−0.820997
$528$	0.0395229	0.00172001
$529$	−3.83362	−0.166679
$530$	10.2814	0.446594
$531$	−0.200377	−0.00869563
$532$	11.8563	0.514035
$533$	−12.8876	−0.558226
$534$	−0.788890	−0.0341386
$535$	1.87946	0.0812561
$536$	12.9003	0.557207
$537$	3.56013	0.153631
$538$	7.08465	0.305441
$539$	0.141138	0.00607926
$540$	4.94537	0.212815
$541$	26.3134	1.13130	0.565651	−	0.824645i	$-0.308625\pi$
$541$	26.3134	1.13130	0.565651	+	0.824645i	$0.308625\pi$
$542$	28.9776	1.24470
$543$	−13.4985	−0.579276
$544$	−6.72151	−0.288182
$545$	−7.71463	−0.330458
$546$	12.3278	0.527581
$547$	−19.2367	−0.822502	−0.411251	−	0.911522i	$-0.634908\pi$
$547$	−19.2367	−0.822502	−0.411251	+	0.911522i	$0.634908\pi$
$548$	12.4968	0.533836
$549$	5.25800	0.224406
$550$	0.123951	0.00528529
$551$	−24.0567	−1.02485
$552$	−5.90614	−0.251382
$553$	−24.3620	−1.03598
$554$	−14.5476	−0.618068
$555$	−5.63405	−0.239152
$556$	−4.50141	−0.190902
$557$	−12.8755	−0.545554	−0.272777	−	0.962077i	$-0.587942\pi$
$557$	−12.8755	−0.545554	−0.272777	+	0.962077i	$0.587942\pi$
$558$	−3.30878	−0.140072
$559$	4.47121	0.189112
$560$	−1.29555	−0.0547471
$561$	−0.265654	−0.0112159
$562$	10.3190	0.435281
$563$	17.3571	0.731516	0.365758	−	0.930710i	$-0.380810\pi$
$563$	17.3571	0.731516	0.365758	+	0.930710i	$0.380810\pi$
$564$	−11.1567	−0.469783
$565$	−2.16614	−0.0911303
$566$	21.3666	0.898103
$567$	6.00892	0.252351
$568$	−8.68095	−0.364244
$569$	−7.34078	−0.307741	−0.153871	−	0.988091i	$-0.549174\pi$
$569$	−7.34078	−0.307741	−0.153871	+	0.988091i	$0.549174\pi$
$570$	9.49511	0.397707
$571$	1.47807	0.0618554	0.0309277	−	0.999522i	$-0.490154\pi$
$571$	1.47807	0.0618554	0.0309277	+	0.999522i	$0.490154\pi$
$572$	−0.181217	−0.00757707
$573$	−0.622464	−0.0260038
$574$	3.07792	0.128470
$575$	−18.5227	−0.772450
$576$	−1.18002	−0.0491674
$577$	32.6965	1.36118	0.680588	−	0.732667i	$-0.261725\pi$
$577$	32.6965	1.36118	0.680588	+	0.732667i	$0.261725\pi$
$578$	28.1787	1.17208
$579$	0.697151	0.0289726
$580$	2.62871	0.109151
$581$	−3.59591	−0.149183
$582$	−1.97900	−0.0820323
$583$	−0.343463	−0.0142248
$584$	6.27841	0.259802
$585$	−6.40116	−0.264656
$586$	3.68617	0.152274
$587$	5.40621	0.223138	0.111569	−	0.993757i	$-0.464412\pi$
$587$	5.40621	0.223138	0.111569	+	0.993757i	$0.464412\pi$
$588$	6.49925	0.268025
$589$	−22.5040	−0.927261
$590$	0.148918	0.00613085
$591$	9.18225	0.377707
$592$	4.76212	0.195722
$593$	−9.23946	−0.379419	−0.189710	−	0.981840i	$-0.560755\pi$
$593$	−9.23946	−0.379419	−0.189710	+	0.981840i	$0.560755\pi$
$594$	−0.165207	−0.00677851
$595$	8.70807	0.356996
$596$	16.4971	0.675746
$597$	4.94278	0.202294
$598$	27.0803	1.10740
$599$	36.2607	1.48157	0.740785	−	0.671742i	$-0.234454\pi$
$599$	36.2607	1.48157	0.740785	+	0.671742i	$0.234454\pi$
$600$	5.70779	0.233020
$601$	26.3061	1.07305	0.536524	−	0.843885i	$-0.319737\pi$
$601$	26.3061	1.07305	0.536524	+	0.843885i	$0.319737\pi$
$602$	−1.06785	−0.0435223
$603$	−15.2226	−0.619911
$604$	−1.80492	−0.0734414
$605$	−9.64596	−0.392164
$606$	18.3660	0.746069
$607$	−0.150776	−0.00611983	−0.00305991	−	0.999995i	$-0.500974\pi$
$607$	−0.150776	−0.00611983	−0.00305991	+	0.999995i	$0.500974\pi$
$608$	−8.02564	−0.325483
$609$	5.97391	0.242075
$610$	−3.90768	−0.158217
$611$	51.1549	2.06950
$612$	7.93150	0.320612
$613$	−22.4485	−0.906687	−0.453343	−	0.891336i	$-0.649769\pi$
$613$	−22.4485	−0.906687	−0.453343	+	0.891336i	$0.649769\pi$
$614$	−7.88521	−0.318221
$615$	2.46496	0.0993968
$616$	0.0432797	0.00174379
$617$	40.4120	1.62693	0.813464	−	0.581616i	$-0.197579\pi$
$617$	40.4120	1.62693	0.813464	+	0.581616i	$0.197579\pi$
$618$	−12.6180	−0.507571
$619$	−11.0970	−0.446027	−0.223014	−	0.974815i	$-0.571589\pi$
$619$	−11.0970	−0.446027	−0.223014	+	0.974815i	$0.571589\pi$
$620$	2.45905	0.0987577
$621$	24.6878	0.990686
$622$	−7.31365	−0.293250
$623$	−0.863875	−0.0346104
$624$	−8.34483	−0.334060
$625$	14.0552	0.562209
$626$	−21.8428	−0.873014
$627$	−0.317197	−0.0126676
$628$	−6.84260	−0.273049
$629$	−32.0086	−1.27627
$630$	1.52877	0.0609078
$631$	−7.02296	−0.279580	−0.139790	−	0.990181i	$-0.544643\pi$
$631$	−7.02296	−0.279580	−0.139790	+	0.990181i	$0.544643\pi$
$632$	16.4909	0.655974
$633$	12.8614	0.511195
$634$	32.4405	1.28838
$635$	11.3212	0.449268
$636$	−15.8160	−0.627147
$637$	−29.7998	−1.18071
$638$	−0.0878157	−0.00347666
$639$	10.2437	0.405233
$640$	0.876974	0.0346654
$641$	−25.7844	−1.01842	−0.509210	−	0.860642i	$-0.670063\pi$
$641$	−25.7844	−1.01842	−0.509210	+	0.860642i	$0.670063\pi$
$642$	−2.89121	−0.114107
$643$	26.1900	1.03283	0.516417	−	0.856337i	$-0.327265\pi$
$643$	26.1900	1.03283	0.516417	+	0.856337i	$0.327265\pi$
$644$	−6.46753	−0.254856
$645$	−0.855189	−0.0336730
$646$	53.9444	2.12242
$647$	−43.1250	−1.69542	−0.847709	−	0.530461i	$-0.822019\pi$
$647$	−43.1250	−1.69542	−0.847709	+	0.530461i	$0.822019\pi$
$648$	−4.06751	−0.159787
$649$	−0.00497480	−0.000195278 0
$650$	−26.1709	−1.02651
$651$	5.58833	0.219024
$652$	15.6027	0.611048
$653$	−10.9988	−0.430418	−0.215209	−	0.976568i	$-0.569043\pi$
$653$	−10.9988	−0.430418	−0.215209	+	0.976568i	$0.569043\pi$
$654$	11.8676	0.464059
$655$	−3.11722	−0.121800
$656$	−2.08348	−0.0813463
$657$	−7.40863	−0.289038
$658$	−12.2172	−0.476276
$659$	−13.8603	−0.539921	−0.269961	−	0.962871i	$-0.587011\pi$
$659$	−13.8603	−0.539921	−0.269961	+	0.962871i	$0.587011\pi$
$660$	0.0346606	0.00134916
$661$	1.90516	0.0741023	0.0370512	−	0.999313i	$-0.488204\pi$
$661$	1.90516	0.0741023	0.0370512	+	0.999313i	$0.488204\pi$
$662$	−6.82134	−0.265119
$663$	56.0899	2.17835
$664$	2.43411	0.0944617
$665$	10.3976	0.403203
$666$	−5.61938	−0.217747
$667$	13.1228	0.508117
$668$	25.2705	0.977747
$669$	−25.3464	−0.979948
$670$	11.3132	0.437068
$671$	0.130541	0.00503949
$672$	1.99298	0.0768807
$673$	−40.2334	−1.55088	−0.775442	−	0.631419i	$-0.782473\pi$
$673$	−40.2334	−1.55088	−0.775442	+	0.631419i	$0.782473\pi$
$674$	−6.51775	−0.251055
$675$	−23.8587	−0.918320
$676$	25.2620	0.971615
$677$	41.2207	1.58424	0.792120	−	0.610366i	$-0.208977\pi$
$677$	41.2207	1.58424	0.792120	+	0.610366i	$0.208977\pi$
$678$	3.33222	0.127973
$679$	−2.16711	−0.0831661
$680$	−5.89459	−0.226047
$681$	−3.27843	−0.125630
$682$	−0.0821477	−0.00314560
$683$	24.6111	0.941719	0.470859	−	0.882208i	$-0.343944\pi$
$683$	24.6111	0.941719	0.470859	+	0.882208i	$0.343944\pi$
$684$	9.47040	0.362110
$685$	10.9594	0.418736
$686$	17.4581	0.666554
$687$	7.69614	0.293626
$688$	0.722839	0.0275580
$689$	72.5183	2.76273
$690$	−5.17953	−0.197181
$691$	−45.1140	−1.71622	−0.858108	−	0.513469i	$-0.828360\pi$
$691$	−45.1140	−1.71622	−0.858108	+	0.513469i	$0.828360\pi$
$692$	−4.94218	−0.187873
$693$	−0.0510707	−0.00194002
$694$	−19.9693	−0.758025
$695$	−3.94762	−0.149742
$696$	−4.04381	−0.153280
$697$	14.0041	0.530445
$698$	−20.6786	−0.782695
$699$	−26.1154	−0.987776
$700$	6.25033	0.236240
$701$	−37.7640	−1.42633	−0.713164	−	0.700997i	$-0.752739\pi$
$701$	−37.7640	−1.42633	−0.713164	+	0.700997i	$0.752739\pi$
$702$	34.8815	1.31652
$703$	−38.2190	−1.44146
$704$	−0.0292965	−0.00110415
$705$	−9.78416	−0.368493
$706$	8.88999	0.334579
$707$	20.1118	0.756381
$708$	−0.229083	−0.00860948
$709$	−37.8345	−1.42090	−0.710452	−	0.703745i	$-0.751510\pi$
$709$	−37.8345	−1.42090	−0.710452	+	0.703745i	$0.751510\pi$
$710$	−7.61296	−0.285709
$711$	−19.4596	−0.729791
$712$	0.584767	0.0219151
$713$	12.2758	0.459733
$714$	−13.3958	−0.501326
$715$	−0.158923	−0.00594337
$716$	−2.63896	−0.0986225
$717$	−19.7187	−0.736410
$718$	25.0732	0.935724
$719$	−48.8316	−1.82111	−0.910556	−	0.413387i	$-0.864346\pi$
$719$	−48.8316	−1.82111	−0.910556	+	0.413387i	$0.864346\pi$
$720$	−1.03484	−0.0385664
$721$	−13.8174	−0.514586
$722$	45.4109	1.69002
$723$	−24.1925	−0.899728
$724$	10.0058	0.371863
$725$	−12.6821	−0.471001
$726$	14.8386	0.550711
$727$	−31.6433	−1.17359	−0.586793	−	0.809737i	$-0.699610\pi$
$727$	−31.6433	−1.17359	−0.586793	+	0.809737i	$0.699610\pi$
$728$	−9.13802	−0.338678
$729$	27.6224	1.02305
$730$	5.50600	0.203786
$731$	−4.85857	−0.179701
$732$	6.01126	0.222183
$733$	11.0761	0.409105	0.204553	−	0.978856i	$-0.434426\pi$
$733$	11.0761	0.409105	0.204553	+	0.978856i	$0.434426\pi$
$734$	−25.4414	−0.939059
$735$	5.69967	0.210236
$736$	4.37794	0.161373
$737$	−0.377933	−0.0139213
$738$	2.45854	0.0905002
$739$	36.5482	1.34445	0.672224	−	0.740347i	$-0.265339\pi$
$739$	36.5482	1.34445	0.672224	+	0.740347i	$0.265339\pi$
$740$	4.17625	0.153522
$741$	66.9726	2.46030
$742$	−17.3194	−0.635815
$743$	−40.7396	−1.49459	−0.747295	−	0.664493i	$-0.768648\pi$
$743$	−40.7396	−1.49459	−0.747295	+	0.664493i	$0.768648\pi$
$744$	−3.78280	−0.138684
$745$	14.4675	0.530048
$746$	18.3520	0.671913
$747$	−2.87229	−0.105092
$748$	0.196917	0.00719999
$749$	−3.16602	−0.115684
$750$	10.9211	0.398781
$751$	42.6557	1.55653	0.778265	−	0.627936i	$-0.216100\pi$
$751$	42.6557	1.55653	0.778265	+	0.627936i	$0.216100\pi$
$752$	8.26996	0.301574
$753$	−18.4762	−0.673310
$754$	18.5413	0.675235
$755$	−1.58287	−0.0576066
$756$	−8.33067	−0.302984
$757$	1.49764	0.0544327	0.0272164	−	0.999630i	$-0.491336\pi$
$757$	1.49764	0.0544327	0.0272164	+	0.999630i	$0.491336\pi$
$758$	31.7836	1.15443
$759$	0.173029	0.00628056
$760$	−7.03828	−0.255305
$761$	−16.9914	−0.615939	−0.307969	−	0.951396i	$-0.599649\pi$
$761$	−16.9914	−0.615939	−0.307969	+	0.951396i	$0.599649\pi$
$762$	−17.4156	−0.630902
$763$	12.9956	0.470473
$764$	0.461403	0.0166930
$765$	6.95572	0.251485
$766$	10.2717	0.371130
$767$	1.05037	0.0379268
$768$	−1.34907	−0.0486803
$769$	51.1306	1.84381	0.921907	−	0.387410i	$-0.126630\pi$
$769$	51.1306	1.84381	0.921907	+	0.387410i	$0.126630\pi$
$770$	0.0379551	0.00136781
$771$	−7.29141	−0.262594
$772$	−0.516765	−0.0185988
$773$	−1.24338	−0.0447213	−0.0223607	−	0.999750i	$-0.507118\pi$
$773$	−1.24338	−0.0447213	−0.0223607	+	0.999750i	$0.507118\pi$
$774$	−0.852963	−0.0306591
$775$	−11.8635	−0.426151
$776$	1.46694	0.0526601
$777$	9.49078	0.340480
$778$	−0.511903	−0.0183526
$779$	16.7213	0.599102
$780$	−7.31820	−0.262034
$781$	0.254321	0.00910033
$782$	−29.4264	−1.05229
$783$	16.9032	0.604070
$784$	−4.81759	−0.172057
$785$	−6.00078	−0.214177
$786$	4.79528	0.171042
$787$	33.2329	1.18462	0.592312	−	0.805709i	$-0.298215\pi$
$787$	33.2329	1.18462	0.592312	+	0.805709i	$0.298215\pi$
$788$	−6.80637	−0.242467
$789$	28.0704	0.999334
$790$	14.4621	0.514539
$791$	3.64896	0.129742
$792$	0.0345704	0.00122840
$793$	−27.5623	−0.978767
$794$	17.8554	0.633663
$795$	−13.8703	−0.491927
$796$	−3.66385	−0.129862
$797$	3.68711	0.130604	0.0653021	−	0.997866i	$-0.479199\pi$
$797$	3.68711	0.130604	0.0653021	+	0.997866i	$0.479199\pi$
$798$	−15.9949	−0.566214
$799$	−55.5866	−1.96651
$800$	−4.23092	−0.149585
$801$	−0.690035	−0.0243812
$802$	−10.3647	−0.365991
$803$	−0.183935	−0.00649093
$804$	−17.4034	−0.613769
$805$	−5.67186	−0.199907
$806$	17.3446	0.610937
$807$	−9.55767	−0.336446
$808$	−13.6139	−0.478934
$809$	−10.2942	−0.361924	−0.180962	−	0.983490i	$-0.557921\pi$
$809$	−10.2942	−0.361924	−0.180962	+	0.983490i	$0.557921\pi$
$810$	−3.56710	−0.125335
$811$	13.5913	0.477255	0.238627	−	0.971111i	$-0.423303\pi$
$811$	13.5913	0.477255	0.238627	+	0.971111i	$0.423303\pi$
$812$	−4.42818	−0.155399
$813$	−39.0928	−1.37104
$814$	−0.139513	−0.00488994
$815$	13.6831	0.479299
$816$	9.06777	0.317436
$817$	−5.80125	−0.202960
$818$	−7.47695	−0.261425
$819$	10.7830	0.376789
$820$	−1.82716	−0.0638072
$821$	28.6009	0.998180	0.499090	−	0.866550i	$-0.333668\pi$
$821$	28.6009	0.998180	0.499090	+	0.866550i	$0.333668\pi$
$822$	−16.8590	−0.588026
$823$	0.855000	0.0298034	0.0149017	−	0.999889i	$-0.495256\pi$
$823$	0.855000	0.0298034	0.0149017	+	0.999889i	$0.495256\pi$
$824$	9.35314	0.325832
$825$	−0.167218	−0.00582179
$826$	−0.250858	−0.00872848
$827$	−49.8267	−1.73264	−0.866321	−	0.499487i	$-0.833522\pi$
$827$	−49.8267	−1.73264	−0.866321	+	0.499487i	$0.833522\pi$
$828$	−5.16605	−0.179533
$829$	31.8024	1.10454	0.552272	−	0.833664i	$-0.313761\pi$
$829$	31.8024	1.10454	0.552272	+	0.833664i	$0.313761\pi$
$830$	2.13465	0.0740948
$831$	19.6257	0.680808
$832$	6.18563	0.214448
$833$	32.3815	1.12195
$834$	6.07270	0.210280
$835$	22.1616	0.766934
$836$	0.235123	0.00813190
$837$	15.8122	0.546549
$838$	−12.8829	−0.445034
$839$	9.78312	0.337751	0.168875	−	0.985637i	$-0.445986\pi$
$839$	9.78312	0.337751	0.168875	+	0.985637i	$0.445986\pi$
$840$	1.74779	0.0603044
$841$	−20.0151	−0.690176
$842$	−35.5689	−1.22578
$843$	−13.9210	−0.479466
$844$	−9.53355	−0.328158
$845$	22.1541	0.762125
$846$	−9.75869	−0.335511
$847$	16.2490	0.558323
$848$	11.7237	0.402593
$849$	−28.8249	−0.989269
$850$	28.4382	0.975421
$851$	20.8483	0.714670
$852$	11.7112	0.401218
$853$	−39.3722	−1.34808	−0.674039	−	0.738696i	$-0.735442\pi$
$853$	−39.3722	−1.34808	−0.674039	+	0.738696i	$0.735442\pi$
$854$	6.58265	0.225254
$855$	8.30529	0.284035
$856$	2.14312	0.0732502
$857$	−14.5699	−0.497700	−0.248850	−	0.968542i	$-0.580053\pi$
$857$	−14.5699	−0.497700	−0.248850	+	0.968542i	$0.580053\pi$
$858$	0.244474	0.00834621
$859$	36.3073	1.23879	0.619395	−	0.785080i	$-0.287378\pi$
$859$	36.3073	1.23879	0.619395	+	0.785080i	$0.287378\pi$
$860$	0.633911	0.0216162
$861$	−4.15233	−0.141511
$862$	−13.6744	−0.465751
$863$	−25.4998	−0.868024	−0.434012	−	0.900907i	$-0.642902\pi$
$863$	−25.4998	−0.868024	−0.434012	+	0.900907i	$0.642902\pi$
$864$	5.63912	0.191847
$865$	−4.33416	−0.147366
$866$	−34.9999	−1.18934
$867$	−38.0150	−1.29106
$868$	−4.14237	−0.140601
$869$	−0.483126	−0.0163889
$870$	−3.54631	−0.120231
$871$	79.7964	2.70380
$872$	−8.79687	−0.297900
$873$	−1.73102	−0.0585860
$874$	−35.1358	−1.18849
$875$	11.9591	0.404293
$876$	−8.47000	−0.286175
$877$	−13.1130	−0.442796	−0.221398	−	0.975184i	$-0.571062\pi$
$877$	−13.1130	−0.442796	−0.221398	+	0.975184i	$0.571062\pi$
$878$	−24.6220	−0.830951
$879$	−4.97290	−0.167732
$880$	−0.0256923	−0.000866086 0
$881$	34.8059	1.17264	0.586320	−	0.810080i	$-0.300576\pi$
$881$	34.8059	1.17264	0.586320	+	0.810080i	$0.300576\pi$
$882$	5.68484	0.191418
$883$	21.8777	0.736242	0.368121	−	0.929778i	$-0.380001\pi$
$883$	21.8777	0.736242	0.368121	+	0.929778i	$0.380001\pi$
$884$	−41.5768	−1.39838
$885$	−0.200900	−0.00675319
$886$	0.154174	0.00517959
$887$	44.6339	1.49866	0.749329	−	0.662197i	$-0.230376\pi$
$887$	44.6339	1.49866	0.749329	+	0.662197i	$0.230376\pi$
$888$	−6.42442	−0.215589
$889$	−19.0710	−0.639622
$890$	0.512825	0.0171899
$891$	0.119164	0.00399213
$892$	18.7881	0.629072
$893$	−66.3717	−2.22104
$894$	−22.2557	−0.744340
$895$	−2.31430	−0.0773584
$896$	−1.47730	−0.0493531
$897$	−36.5332	−1.21981
$898$	10.4439	0.348517
$899$	8.40498	0.280322
$900$	4.99255	0.166418
$901$	−78.8009	−2.62524
$902$	0.0610387	0.00203237
$903$	1.44060	0.0479402
$904$	−2.47002	−0.0821516
$905$	8.77483	0.291685
$906$	2.43497	0.0808963
$907$	−28.3024	−0.939767	−0.469884	−	0.882728i	$-0.655704\pi$
$907$	−28.3024	−0.939767	−0.469884	+	0.882728i	$0.655704\pi$
$908$	2.43014	0.0806471
$909$	16.0646	0.532829
$910$	−8.01381	−0.265655
$911$	−13.0713	−0.433071	−0.216536	−	0.976275i	$-0.569476\pi$
$911$	−13.0713	−0.433071	−0.216536	+	0.976275i	$0.569476\pi$
$912$	10.8271	0.358522
$913$	−0.0713108	−0.00236004
$914$	−31.4648	−1.04076
$915$	5.27172	0.174278
$916$	−5.70478	−0.188491
$917$	5.25109	0.173406
$918$	−37.9034	−1.25100
$919$	16.2939	0.537485	0.268742	−	0.963212i	$-0.413392\pi$
$919$	16.2939	0.537485	0.268742	+	0.963212i	$0.413392\pi$
$920$	3.83934	0.126579
$921$	10.6377	0.350524
$922$	−25.7406	−0.847720
$923$	−53.6971	−1.76746
$924$	−0.0583872	−0.00192080
$925$	−20.1481	−0.662466
$926$	29.4959	0.969296
$927$	−11.0369	−0.362498
$928$	2.99748	0.0983972
$929$	−44.7810	−1.46922	−0.734608	−	0.678492i	$-0.762634\pi$
$929$	−44.7810	−1.46922	−0.734608	+	0.678492i	$0.762634\pi$
$930$	−3.31742	−0.108782
$931$	38.6642	1.26717
$932$	19.3581	0.634097
$933$	9.86660	0.323018
$934$	−10.2838	−0.336496
$935$	0.172691	0.00564759
$936$	−7.29915	−0.238580
$937$	−41.4372	−1.35370	−0.676848	−	0.736123i	$-0.736654\pi$
$937$	−41.4372	−1.35370	−0.676848	+	0.736123i	$0.736654\pi$
$938$	−19.0576	−0.622252
$939$	29.4674	0.961633
$940$	7.25254	0.236552
$941$	4.68463	0.152715	0.0763573	−	0.997081i	$-0.475671\pi$
$941$	4.68463	0.152715	0.0763573	+	0.997081i	$0.475671\pi$
$942$	9.23112	0.300766
$943$	−9.12136	−0.297032
$944$	0.169809	0.00552680
$945$	−7.30578	−0.237657
$946$	−0.0211766	−0.000688512 0
$947$	14.8931	0.483962	0.241981	−	0.970281i	$-0.422203\pi$
$947$	14.8931	0.483962	0.241981	+	0.970281i	$0.422203\pi$
$948$	−22.2474	−0.722561
$949$	38.8359	1.26067
$950$	33.9558	1.10167
$951$	−43.7645	−1.41916
$952$	9.92968	0.321823
$953$	47.4089	1.53572	0.767862	−	0.640615i	$-0.221321\pi$
$953$	47.4089	1.53572	0.767862	+	0.640615i	$0.221321\pi$
$954$	−13.8341	−0.447897
$955$	0.404639	0.0130938
$956$	14.6166	0.472734
$957$	0.118469	0.00382957
$958$	23.9369	0.773368
$959$	−18.4615	−0.596153
$960$	−1.18310	−0.0381843
$961$	−23.1375	−0.746371
$962$	29.4567	0.949721
$963$	−2.52892	−0.0814932
$964$	17.9327	0.577575
$965$	−0.453190	−0.0145887
$966$	8.72513	0.280727
$967$	−18.9999	−0.610995	−0.305497	−	0.952193i	$-0.598823\pi$
$967$	−18.9999	−0.610995	−0.305497	+	0.952193i	$0.598823\pi$
$968$	−10.9991	−0.353526
$969$	−72.7747	−2.33786
$970$	1.28647	0.0413061
$971$	−13.2615	−0.425582	−0.212791	−	0.977098i	$-0.568255\pi$
$971$	−13.2615	−0.425582	−0.212791	+	0.977098i	$0.568255\pi$
$972$	−11.4300	−0.366618
$973$	6.64992	0.213187
$974$	−26.1293	−0.837237
$975$	35.3063	1.13071
$976$	−4.45587	−0.142629
$977$	13.6565	0.436910	0.218455	−	0.975847i	$-0.429898\pi$
$977$	13.6565	0.436910	0.218455	+	0.975847i	$0.429898\pi$
$978$	−21.0490	−0.673074
$979$	−0.0171316	−0.000547529 0
$980$	−4.22490	−0.134959
$981$	10.3805	0.331423
$982$	9.69302	0.309317
$983$	−16.7096	−0.532953	−0.266477	−	0.963841i	$-0.585859\pi$
$983$	−16.7096	−0.532953	−0.266477	+	0.963841i	$0.585859\pi$
$984$	2.81076	0.0896036
$985$	−5.96901	−0.190189
$986$	−20.1476	−0.641631
$987$	16.4818	0.524622
$988$	−49.6436	−1.57937
$989$	3.16455	0.100627
$990$	0.0303173	0.000963547 0
$991$	18.4190	0.585099	0.292549	−	0.956250i	$-0.405496\pi$
$991$	18.4190	0.585099	0.292549	+	0.956250i	$0.405496\pi$
$992$	2.80401	0.0890275
$993$	9.20245	0.292031
$994$	12.8244	0.406764
$995$	−3.21310	−0.101862
$996$	−3.28378	−0.104050
$997$	32.5258	1.03010	0.515052	−	0.857159i	$-0.327773\pi$
$997$	32.5258	1.03010	0.515052	+	0.857159i	$0.327773\pi$
$998$	22.6696	0.717595
$999$	26.8542	0.849628

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6046.2.a.f.1.18	✓	67

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6046.2.a.f.1.18	✓	67	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:	\( N \)	\(=\)	\( 6046 = 2 \cdot 3023 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6046.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.34907 q^{3} +1.00000 q^{4} +0.876974 q^{5} -1.34907 q^{6} -1.47730 q^{7} +1.00000 q^{8} -1.18002 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.34907 q^{3} +1.00000 q^{4} +0.876974 q^{5} -1.34907 q^{6} -1.47730 q^{7} +1.00000 q^{8} -1.18002 q^{9} +0.876974 q^{10} -0.0292965 q^{11} -1.34907 q^{12} +6.18563 q^{13} -1.47730 q^{14} -1.18310 q^{15} +1.00000 q^{16} -6.72151 q^{17} -1.18002 q^{18} -8.02564 q^{19} +0.876974 q^{20} +1.99298 q^{21} -0.0292965 q^{22} +4.37794 q^{23} -1.34907 q^{24} -4.23092 q^{25} +6.18563 q^{26} +5.63912 q^{27} -1.47730 q^{28} +2.99748 q^{29} -1.18310 q^{30} +2.80401 q^{31} +1.00000 q^{32} +0.0395229 q^{33} -6.72151 q^{34} -1.29555 q^{35} -1.18002 q^{36} +4.76212 q^{37} -8.02564 q^{38} -8.34483 q^{39} +0.876974 q^{40} -2.08348 q^{41} +1.99298 q^{42} +0.722839 q^{43} -0.0292965 q^{44} -1.03484 q^{45} +4.37794 q^{46} +8.26996 q^{47} -1.34907 q^{48} -4.81759 q^{49} -4.23092 q^{50} +9.06777 q^{51} +6.18563 q^{52} +11.7237 q^{53} +5.63912 q^{54} -0.0256923 q^{55} -1.47730 q^{56} +10.8271 q^{57} +2.99748 q^{58} +0.169809 q^{59} -1.18310 q^{60} -4.45587 q^{61} +2.80401 q^{62} +1.74324 q^{63} +1.00000 q^{64} +5.42464 q^{65} +0.0395229 q^{66} +12.9003 q^{67} -6.72151 q^{68} -5.90614 q^{69} -1.29555 q^{70} -8.68095 q^{71} -1.18002 q^{72} +6.27841 q^{73} +4.76212 q^{74} +5.70779 q^{75} -8.02564 q^{76} +0.0432797 q^{77} -8.34483 q^{78} +16.4909 q^{79} +0.876974 q^{80} -4.06751 q^{81} -2.08348 q^{82} +2.43411 q^{83} +1.99298 q^{84} -5.89459 q^{85} +0.722839 q^{86} -4.04381 q^{87} -0.0292965 q^{88} +0.584767 q^{89} -1.03484 q^{90} -9.13802 q^{91} +4.37794 q^{92} -3.78280 q^{93} +8.26996 q^{94} -7.03828 q^{95} -1.34907 q^{96} +1.46694 q^{97} -4.81759 q^{98} +0.0345704 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 67 q + 67 q^{2} + 21 q^{3} + 67 q^{4} + 21 q^{5} + 21 q^{6} + 38 q^{7} + 67 q^{8} + 90 q^{9}+O(q^{10}) \) 67 * q + 67 * q^2 + 21 * q^3 + 67 * q^4 + 21 * q^5 + 21 * q^6 + 38 * q^7 + 67 * q^8 + 90 * q^9	\( 67 q + 67 q^{2} + 21 q^{3} + 67 q^{4} + 21 q^{5} + 21 q^{6} + 38 q^{7} + 67 q^{8} + 90 q^{9} + 21 q^{10} + 56 q^{11} + 21 q^{12} + 33 q^{13} + 38 q^{14} + 25 q^{15} + 67 q^{16} + 30 q^{17} + 90 q^{18} + 36 q^{19} + 21 q^{20} + 20 q^{21} + 56 q^{22} + 65 q^{23} + 21 q^{24} + 72 q^{25} + 33 q^{26} + 57 q^{27} + 38 q^{28} + 84 q^{29} + 25 q^{30} + 52 q^{31} + 67 q^{32} - 9 q^{33} + 30 q^{34} + 30 q^{35} + 90 q^{36} + 52 q^{37} + 36 q^{38} + 41 q^{39} + 21 q^{40} + 46 q^{41} + 20 q^{42} + 61 q^{43} + 56 q^{44} + 23 q^{45} + 65 q^{46} + 51 q^{47} + 21 q^{48} + 81 q^{49} + 72 q^{50} + 33 q^{51} + 33 q^{52} + 72 q^{53} + 57 q^{54} + 14 q^{55} + 38 q^{56} - 26 q^{57} + 84 q^{58} + 71 q^{59} + 25 q^{60} + 42 q^{61} + 52 q^{62} + 63 q^{63} + 67 q^{64} - 2 q^{65} - 9 q^{66} + 70 q^{67} + 30 q^{68} + 21 q^{69} + 30 q^{70} + 104 q^{71} + 90 q^{72} - 31 q^{73} + 52 q^{74} + 69 q^{75} + 36 q^{76} + 48 q^{77} + 41 q^{78} + 79 q^{79} + 21 q^{80} + 123 q^{81} + 46 q^{82} + 41 q^{83} + 20 q^{84} + 6 q^{85} + 61 q^{86} + 19 q^{87} + 56 q^{88} + 58 q^{89} + 23 q^{90} + 31 q^{91} + 65 q^{92} + 13 q^{93} + 51 q^{94} + 77 q^{95} + 21 q^{96} - 8 q^{97} + 81 q^{98} + 129 q^{99}+O(q^{100}) \) 67 * q + 67 * q^2 + 21 * q^3 + 67 * q^4 + 21 * q^5 + 21 * q^6 + 38 * q^7 + 67 * q^8 + 90 * q^9 + 21 * q^10 + 56 * q^11 + 21 * q^12 + 33 * q^13 + 38 * q^14 + 25 * q^15 + 67 * q^16 + 30 * q^17 + 90 * q^18 + 36 * q^19 + 21 * q^20 + 20 * q^21 + 56 * q^22 + 65 * q^23 + 21 * q^24 + 72 * q^25 + 33 * q^26 + 57 * q^27 + 38 * q^28 + 84 * q^29 + 25 * q^30 + 52 * q^31 + 67 * q^32 - 9 * q^33 + 30 * q^34 + 30 * q^35 + 90 * q^36 + 52 * q^37 + 36 * q^38 + 41 * q^39 + 21 * q^40 + 46 * q^41 + 20 * q^42 + 61 * q^43 + 56 * q^44 + 23 * q^45 + 65 * q^46 + 51 * q^47 + 21 * q^48 + 81 * q^49 + 72 * q^50 + 33 * q^51 + 33 * q^52 + 72 * q^53 + 57 * q^54 + 14 * q^55 + 38 * q^56 - 26 * q^57 + 84 * q^58 + 71 * q^59 + 25 * q^60 + 42 * q^61 + 52 * q^62 + 63 * q^63 + 67 * q^64 - 2 * q^65 - 9 * q^66 + 70 * q^67 + 30 * q^68 + 21 * q^69 + 30 * q^70 + 104 * q^71 + 90 * q^72 - 31 * q^73 + 52 * q^74 + 69 * q^75 + 36 * q^76 + 48 * q^77 + 41 * q^78 + 79 * q^79 + 21 * q^80 + 123 * q^81 + 46 * q^82 + 41 * q^83 + 20 * q^84 + 6 * q^85 + 61 * q^86 + 19 * q^87 + 56 * q^88 + 58 * q^89 + 23 * q^90 + 31 * q^91 + 65 * q^92 + 13 * q^93 + 51 * q^94 + 77 * q^95 + 21 * q^96 - 8 * q^97 + 81 * q^98 + 129 * q^99

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