LMFDB - Embedded newform 4009.2.a.c.1.53

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4009,2,Mod(1,4009)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4009, 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("4009.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4009 = 19 \cdot 211 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4009.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:	$32.0120261703$
Analytic rank:	$1$
Dimension:	$71$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.53
Character	$\chi$	$=$	4009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.33140 q^{2} -1.88132 q^{3} -0.227382 q^{4} -0.789831 q^{5} -2.50478 q^{6} -3.27277 q^{7} -2.96553 q^{8} +0.539350 q^{9} +O(q^{10})$	\(q+1.33140 q^{2} -1.88132 q^{3} -0.227382 q^{4} -0.789831 q^{5} -2.50478 q^{6} -3.27277 q^{7} -2.96553 q^{8} +0.539350 q^{9} -1.05158 q^{10} +2.55484 q^{11} +0.427778 q^{12} +5.09980 q^{13} -4.35735 q^{14} +1.48592 q^{15} -3.49353 q^{16} +0.495976 q^{17} +0.718089 q^{18} +1.00000 q^{19} +0.179593 q^{20} +6.15711 q^{21} +3.40150 q^{22} +4.92095 q^{23} +5.57910 q^{24} -4.37617 q^{25} +6.78985 q^{26} +4.62926 q^{27} +0.744169 q^{28} +2.77227 q^{29} +1.97835 q^{30} +9.26640 q^{31} +1.27978 q^{32} -4.80646 q^{33} +0.660341 q^{34} +2.58493 q^{35} -0.122639 q^{36} +8.44067 q^{37} +1.33140 q^{38} -9.59433 q^{39} +2.34227 q^{40} -9.44782 q^{41} +8.19756 q^{42} -11.6513 q^{43} -0.580924 q^{44} -0.425995 q^{45} +6.55174 q^{46} -5.71903 q^{47} +6.57244 q^{48} +3.71100 q^{49} -5.82642 q^{50} -0.933088 q^{51} -1.15960 q^{52} -10.7169 q^{53} +6.16338 q^{54} -2.01789 q^{55} +9.70549 q^{56} -1.88132 q^{57} +3.69099 q^{58} -5.10132 q^{59} -0.337872 q^{60} -0.448807 q^{61} +12.3373 q^{62} -1.76517 q^{63} +8.69096 q^{64} -4.02797 q^{65} -6.39930 q^{66} +0.632961 q^{67} -0.112776 q^{68} -9.25786 q^{69} +3.44157 q^{70} -7.00931 q^{71} -1.59946 q^{72} -5.81923 q^{73} +11.2379 q^{74} +8.23295 q^{75} -0.227382 q^{76} -8.36139 q^{77} -12.7739 q^{78} +10.3589 q^{79} +2.75930 q^{80} -10.3272 q^{81} -12.5788 q^{82} -6.80596 q^{83} -1.40002 q^{84} -0.391737 q^{85} -15.5125 q^{86} -5.21552 q^{87} -7.57645 q^{88} -1.25184 q^{89} -0.567168 q^{90} -16.6904 q^{91} -1.11894 q^{92} -17.4330 q^{93} -7.61430 q^{94} -0.789831 q^{95} -2.40767 q^{96} -2.38402 q^{97} +4.94082 q^{98} +1.37795 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9}+O(q^{10}) $ 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9	\( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9} - 10 q^{10} - 52 q^{11} - 9 q^{12} - 15 q^{13} - 53 q^{14} - 33 q^{15} + 53 q^{16} - 10 q^{17} - 35 q^{18} + 71 q^{19} - 33 q^{20} - 38 q^{21} - 6 q^{22} - 65 q^{23} - 30 q^{24} + 51 q^{25} - 4 q^{26} - 23 q^{27} - 29 q^{28} - 97 q^{29} - 27 q^{30} - 53 q^{31} - 78 q^{32} - 17 q^{33} - 24 q^{34} - 38 q^{35} + 24 q^{36} - 33 q^{37} - 15 q^{38} - 86 q^{39} + 25 q^{40} - 69 q^{41} + 64 q^{42} - 10 q^{43} - 94 q^{44} - 34 q^{45} - 6 q^{46} - 37 q^{47} - q^{48} + 74 q^{49} - 41 q^{50} - 46 q^{51} - 30 q^{52} - 50 q^{53} - 17 q^{54} - 30 q^{55} - 116 q^{56} - 8 q^{57} + 11 q^{58} - 93 q^{59} - 56 q^{60} - 18 q^{61} - q^{62} - 84 q^{63} + 93 q^{64} - 78 q^{65} - 53 q^{66} - 5 q^{67} - 9 q^{68} - 69 q^{69} - 10 q^{70} - 221 q^{71} - 73 q^{72} - 34 q^{73} - 58 q^{74} - 70 q^{75} + 69 q^{76} - 2 q^{77} + 7 q^{78} - 68 q^{79} - 71 q^{80} + 39 q^{81} + 26 q^{82} - 45 q^{83} - 10 q^{84} - 44 q^{85} - 80 q^{86} - 7 q^{87} - 46 q^{88} - 143 q^{89} + 41 q^{90} - 30 q^{91} - 46 q^{92} + 32 q^{93} + 41 q^{94} - 18 q^{95} - 140 q^{96} - 18 q^{97} - 97 q^{98} - 142 q^{99}+O(q^{100}) \) 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9 - 10 * q^10 - 52 * q^11 - 9 * q^12 - 15 * q^13 - 53 * q^14 - 33 * q^15 + 53 * q^16 - 10 * q^17 - 35 * q^18 + 71 * q^19 - 33 * q^20 - 38 * q^21 - 6 * q^22 - 65 * q^23 - 30 * q^24 + 51 * q^25 - 4 * q^26 - 23 * q^27 - 29 * q^28 - 97 * q^29 - 27 * q^30 - 53 * q^31 - 78 * q^32 - 17 * q^33 - 24 * q^34 - 38 * q^35 + 24 * q^36 - 33 * q^37 - 15 * q^38 - 86 * q^39 + 25 * q^40 - 69 * q^41 + 64 * q^42 - 10 * q^43 - 94 * q^44 - 34 * q^45 - 6 * q^46 - 37 * q^47 - q^48 + 74 * q^49 - 41 * q^50 - 46 * q^51 - 30 * q^52 - 50 * q^53 - 17 * q^54 - 30 * q^55 - 116 * q^56 - 8 * q^57 + 11 * q^58 - 93 * q^59 - 56 * q^60 - 18 * q^61 - q^62 - 84 * q^63 + 93 * q^64 - 78 * q^65 - 53 * q^66 - 5 * q^67 - 9 * q^68 - 69 * q^69 - 10 * q^70 - 221 * q^71 - 73 * q^72 - 34 * q^73 - 58 * q^74 - 70 * q^75 + 69 * q^76 - 2 * q^77 + 7 * q^78 - 68 * q^79 - 71 * q^80 + 39 * q^81 + 26 * q^82 - 45 * q^83 - 10 * q^84 - 44 * q^85 - 80 * q^86 - 7 * q^87 - 46 * q^88 - 143 * q^89 + 41 * q^90 - 30 * q^91 - 46 * q^92 + 32 * q^93 + 41 * q^94 - 18 * q^95 - 140 * q^96 - 18 * q^97 - 97 * q^98 - 142 * 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 $

$2$	1.33140	0.941440	0.470720	−	0.882283i	$-0.343994\pi$
$2$	1.33140	0.941440	0.470720	+	0.882283i	$0.343994\pi$
$3$	−1.88132	−1.08618	−0.543089	−	0.839675i	$-0.682745\pi$
$3$	−1.88132	−1.08618	−0.543089	+	0.839675i	$0.682745\pi$
$4$	−0.227382	−0.113691
$5$	−0.789831	−0.353223	−0.176611	−	0.984281i	$-0.556514\pi$
$5$	−0.789831	−0.353223	−0.176611	+	0.984281i	$0.556514\pi$
$6$	−2.50478	−1.02257
$7$	−3.27277	−1.23699	−0.618495	−	0.785789i	$-0.712257\pi$
$7$	−3.27277	−1.23699	−0.618495	+	0.785789i	$0.712257\pi$
$8$	−2.96553	−1.04847
$9$	0.539350	0.179783
$10$	−1.05158	−0.332538
$11$	2.55484	0.770312	0.385156	−	0.922851i	$-0.374148\pi$
$11$	2.55484	0.770312	0.385156	+	0.922851i	$0.374148\pi$
$12$	0.427778	0.123489
$13$	5.09980	1.41443	0.707214	−	0.706999i	$-0.249952\pi$
$13$	5.09980	1.41443	0.707214	+	0.706999i	$0.249952\pi$
$14$	−4.35735	−1.16455
$15$	1.48592	0.383663
$16$	−3.49353	−0.873383
$17$	0.495976	0.120292	0.0601460	−	0.998190i	$-0.480843\pi$
$17$	0.495976	0.120292	0.0601460	+	0.998190i	$0.480843\pi$
$18$	0.718089	0.169255
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0.179593	0.0401583
$21$	6.15711	1.34359
$22$	3.40150	0.725203
$23$	4.92095	1.02609	0.513045	−	0.858362i	$-0.328518\pi$
$23$	4.92095	1.02609	0.513045	+	0.858362i	$0.328518\pi$
$24$	5.57910	1.13883
$25$	−4.37617	−0.875234
$26$	6.78985	1.33160
$27$	4.62926	0.890902
$28$	0.744169	0.140635
$29$	2.77227	0.514798	0.257399	−	0.966305i	$-0.417135\pi$
$29$	2.77227	0.514798	0.257399	+	0.966305i	$0.417135\pi$
$30$	1.97835	0.361196
$31$	9.26640	1.66430	0.832148	−	0.554554i	$-0.187111\pi$
$31$	9.26640	1.66430	0.832148	+	0.554554i	$0.187111\pi$
$32$	1.27978	0.226235
$33$	−4.80646	−0.836697
$34$	0.660341	0.113248
$35$	2.58493	0.436933
$36$	−0.122639	−0.0204398
$37$	8.44067	1.38764	0.693819	−	0.720150i	$-0.255927\pi$
$37$	8.44067	1.38764	0.693819	+	0.720150i	$0.255927\pi$
$38$	1.33140	0.215981
$39$	−9.59433	−1.53632
$40$	2.34227	0.370345
$41$	−9.44782	−1.47550	−0.737751	−	0.675073i	$-0.764112\pi$
$41$	−9.44782	−1.47550	−0.737751	+	0.675073i	$0.764112\pi$
$42$	8.19756	1.26491
$43$	−11.6513	−1.77680	−0.888402	−	0.459067i	$-0.848184\pi$
$43$	−11.6513	−1.77680	−0.888402	+	0.459067i	$0.848184\pi$
$44$	−0.580924	−0.0875777
$45$	−0.425995	−0.0635036
$46$	6.55174	0.966001
$47$	−5.71903	−0.834207	−0.417103	−	0.908859i	$-0.636955\pi$
$47$	−5.71903	−0.834207	−0.417103	+	0.908859i	$0.636955\pi$
$48$	6.57244	0.948650
$49$	3.71100	0.530143
$50$	−5.82642	−0.823980
$51$	−0.933088	−0.130659
$52$	−1.15960	−0.160808
$53$	−10.7169	−1.47207	−0.736037	−	0.676941i	$-0.763305\pi$
$53$	−10.7169	−1.47207	−0.736037	+	0.676941i	$0.763305\pi$
$54$	6.16338	0.838730
$55$	−2.01789	−0.272092
$56$	9.70549	1.29695
$57$	−1.88132	−0.249186
$58$	3.69099	0.484651
$59$	−5.10132	−0.664135	−0.332068	−	0.943256i	$-0.607746\pi$
$59$	−5.10132	−0.664135	−0.332068	+	0.943256i	$0.607746\pi$
$60$	−0.337872	−0.0436191
$61$	−0.448807	−0.0574638	−0.0287319	−	0.999587i	$-0.509147\pi$
$61$	−0.448807	−0.0574638	−0.0287319	+	0.999587i	$0.509147\pi$
$62$	12.3373	1.56683
$63$	−1.76517	−0.222390
$64$	8.69096	1.08637
$65$	−4.02797	−0.499609
$66$	−6.39930	−0.787700
$67$	0.632961	0.0773284	0.0386642	−	0.999252i	$-0.487690\pi$
$67$	0.632961	0.0773284	0.0386642	+	0.999252i	$0.487690\pi$
$68$	−0.112776	−0.0136761
$69$	−9.25786	−1.11452
$70$	3.44157	0.411346
$71$	−7.00931	−0.831852	−0.415926	−	0.909399i	$-0.636542\pi$
$71$	−7.00931	−0.831852	−0.415926	+	0.909399i	$0.636542\pi$
$72$	−1.59946	−0.188498
$73$	−5.81923	−0.681090	−0.340545	−	0.940228i	$-0.610611\pi$
$73$	−5.81923	−0.681090	−0.340545	+	0.940228i	$0.610611\pi$
$74$	11.2379	1.30638
$75$	8.23295	0.950660
$76$	−0.227382	−0.0260825
$77$	−8.36139	−0.952868
$78$	−12.7739	−1.44635
$79$	10.3589	1.16547	0.582734	−	0.812663i	$-0.301983\pi$
$79$	10.3589	1.16547	0.582734	+	0.812663i	$0.301983\pi$
$80$	2.75930	0.308499
$81$	−10.3272	−1.14746
$82$	−12.5788	−1.38910
$83$	−6.80596	−0.747051	−0.373526	−	0.927620i	$-0.621851\pi$
$83$	−6.80596	−0.747051	−0.373526	+	0.927620i	$0.621851\pi$
$84$	−1.40002	−0.152754
$85$	−0.391737	−0.0424899
$86$	−15.5125	−1.67275
$87$	−5.21552	−0.559162
$88$	−7.57645	−0.807652
$89$	−1.25184	−0.132695	−0.0663475	−	0.997797i	$-0.521135\pi$
$89$	−1.25184	−0.132695	−0.0663475	+	0.997797i	$0.521135\pi$
$90$	−0.567168	−0.0597848
$91$	−16.6904	−1.74963
$92$	−1.11894	−0.116657
$93$	−17.4330	−1.80772
$94$	−7.61430	−0.785355
$95$	−0.789831	−0.0810349
$96$	−2.40767	−0.245732
$97$	−2.38402	−0.242061	−0.121030	−	0.992649i	$-0.538620\pi$
$97$	−2.38402	−0.242061	−0.121030	+	0.992649i	$0.538620\pi$
$98$	4.94082	0.499098
$99$	1.37795	0.138489
$100$	0.995063	0.0995063
$101$	−1.40759	−0.140061	−0.0700304	−	0.997545i	$-0.522310\pi$
$101$	−1.40759	−0.140061	−0.0700304	+	0.997545i	$0.522310\pi$
$102$	−1.24231	−0.123007
$103$	6.00995	0.592178	0.296089	−	0.955160i	$-0.404317\pi$
$103$	6.00995	0.592178	0.296089	+	0.955160i	$0.404317\pi$
$104$	−15.1236	−1.48299
$105$	−4.86307	−0.474587
$106$	−14.2684	−1.38587
$107$	1.50503	0.145497	0.0727485	−	0.997350i	$-0.476823\pi$
$107$	1.50503	0.145497	0.0727485	+	0.997350i	$0.476823\pi$
$108$	−1.05261	−0.101288
$109$	6.49047	0.621675	0.310837	−	0.950463i	$-0.399391\pi$
$109$	6.49047	0.621675	0.310837	+	0.950463i	$0.399391\pi$
$110$	−2.68661	−0.256158
$111$	−15.8796	−1.50722
$112$	11.4335	1.08037
$113$	17.2564	1.62335	0.811674	−	0.584110i	$-0.198556\pi$
$113$	17.2564	1.62335	0.811674	+	0.584110i	$0.198556\pi$
$114$	−2.50478	−0.234594
$115$	−3.88672	−0.362438
$116$	−0.630365	−0.0585279
$117$	2.75057	0.254291
$118$	−6.79188	−0.625243
$119$	−1.62322	−0.148800
$120$	−4.40654	−0.402260
$121$	−4.47281	−0.406619
$122$	−0.597540	−0.0540987
$123$	17.7743	1.60266
$124$	−2.10701	−0.189216
$125$	7.40558	0.662376
$126$	−2.35014	−0.209367
$127$	−7.42958	−0.659269	−0.329634	−	0.944109i	$-0.606925\pi$
$127$	−7.42958	−0.659269	−0.329634	+	0.944109i	$0.606925\pi$
$128$	9.01156	0.796517
$129$	21.9197	1.92992
$130$	−5.36283	−0.470352
$131$	−9.07736	−0.793092	−0.396546	−	0.918015i	$-0.629791\pi$
$131$	−9.07736	−0.793092	−0.396546	+	0.918015i	$0.629791\pi$
$132$	1.09290	0.0951250
$133$	−3.27277	−0.283785
$134$	0.842722	0.0728001
$135$	−3.65633	−0.314687
$136$	−1.47083	−0.126123
$137$	11.6661	0.996703	0.498351	−	0.866975i	$-0.333939\pi$
$137$	11.6661	0.996703	0.498351	+	0.866975i	$0.333939\pi$
$138$	−12.3259	−1.04925
$139$	−11.7855	−0.999632	−0.499816	−	0.866132i	$-0.666599\pi$
$139$	−11.7855	−0.999632	−0.499816	+	0.866132i	$0.666599\pi$
$140$	−0.587767	−0.0496754
$141$	10.7593	0.906097
$142$	−9.33217	−0.783138
$143$	13.0291	1.08955
$144$	−1.88424	−0.157020
$145$	−2.18962	−0.181838
$146$	−7.74771	−0.641205
$147$	−6.98157	−0.575830
$148$	−1.91926	−0.157762
$149$	8.20457	0.672144	0.336072	−	0.941836i	$-0.390901\pi$
$149$	8.20457	0.672144	0.336072	+	0.941836i	$0.390901\pi$
$150$	10.9613	0.894989
$151$	9.18459	0.747431	0.373716	−	0.927543i	$-0.378084\pi$
$151$	9.18459	0.747431	0.373716	+	0.927543i	$0.378084\pi$
$152$	−2.96553	−0.240536
$153$	0.267505	0.0216265
$154$	−11.1323	−0.897068
$155$	−7.31889	−0.587867
$156$	2.18158	0.174666
$157$	−17.4986	−1.39654	−0.698269	−	0.715835i	$-0.746046\pi$
$157$	−17.4986	−1.39654	−0.698269	+	0.715835i	$0.746046\pi$
$158$	13.7918	1.09722
$159$	20.1618	1.59894
$160$	−1.01081	−0.0799115
$161$	−16.1051	−1.26926
$162$	−13.7495	−1.08027
$163$	−6.16954	−0.483236	−0.241618	−	0.970371i	$-0.577678\pi$
$163$	−6.16954	−0.483236	−0.241618	+	0.970371i	$0.577678\pi$
$164$	2.14827	0.167751
$165$	3.79629	0.295540
$166$	−9.06144	−0.703304
$167$	5.14008	0.397752	0.198876	−	0.980025i	$-0.436271\pi$
$167$	5.14008	0.397752	0.198876	+	0.980025i	$0.436271\pi$
$168$	−18.2591	−1.40872
$169$	13.0079	1.00061
$170$	−0.521558	−0.0400017
$171$	0.539350	0.0412451
$172$	2.64929	0.202007
$173$	15.4568	1.17516	0.587579	−	0.809167i	$-0.300081\pi$
$173$	15.4568	1.17516	0.587579	+	0.809167i	$0.300081\pi$
$174$	−6.94392	−0.526417
$175$	14.3222	1.08265
$176$	−8.92541	−0.672778
$177$	9.59719	0.721369
$178$	−1.66670	−0.124924
$179$	−20.4592	−1.52919	−0.764597	−	0.644509i	$-0.777062\pi$
$179$	−20.4592	−1.52919	−0.764597	+	0.644509i	$0.777062\pi$
$180$	0.0968636	0.00721979
$181$	−20.3232	−1.51061	−0.755305	−	0.655373i	$-0.772511\pi$
$181$	−20.3232	−1.51061	−0.755305	+	0.655373i	$0.772511\pi$
$182$	−22.2216	−1.64717
$183$	0.844347	0.0624159
$184$	−14.5932	−1.07583
$185$	−6.66670	−0.490146
$186$	−23.2103	−1.70186
$187$	1.26714	0.0926624
$188$	1.30041	0.0948418
$189$	−15.1505	−1.10204
$190$	−1.05158	−0.0762895
$191$	−22.1691	−1.60410	−0.802051	−	0.597256i	$-0.796258\pi$
$191$	−22.1691	−1.60410	−0.802051	+	0.597256i	$0.796258\pi$
$192$	−16.3504	−1.17999
$193$	24.3614	1.75357	0.876787	−	0.480880i	$-0.159683\pi$
$193$	24.3614	1.75357	0.876787	+	0.480880i	$0.159683\pi$
$194$	−3.17408	−0.227885
$195$	7.57789	0.542664
$196$	−0.843816	−0.0602726
$197$	8.67239	0.617882	0.308941	−	0.951081i	$-0.400025\pi$
$197$	8.67239	0.617882	0.308941	+	0.951081i	$0.400025\pi$
$198$	1.83460	0.130379
$199$	−13.7331	−0.973516	−0.486758	−	0.873537i	$-0.661821\pi$
$199$	−13.7331	−0.973516	−0.486758	+	0.873537i	$0.661821\pi$
$200$	12.9777	0.917659
$201$	−1.19080	−0.0839925
$202$	−1.87407	−0.131859
$203$	−9.07299	−0.636799
$204$	0.212168	0.0148547
$205$	7.46218	0.521181
$206$	8.00163	0.557500
$207$	2.65411	0.184474
$208$	−17.8163	−1.23534
$209$	2.55484	0.176722
$210$	−6.47468	−0.446795
$211$	1.00000	0.0688428
$211$	1.00000	0.0688428
$212$	2.43682	0.167362
$213$	13.1867	0.903539
$214$	2.00379	0.136977
$215$	9.20253	0.627608
$216$	−13.7282	−0.934086
$217$	−30.3268	−2.05872
$218$	8.64140	0.585269
$219$	10.9478	0.739785
$220$	0.458832	0.0309344
$221$	2.52938	0.170144
$222$	−21.1420	−1.41896
$223$	−13.6294	−0.912693	−0.456346	−	0.889802i	$-0.650842\pi$
$223$	−13.6294	−0.912693	−0.456346	+	0.889802i	$0.650842\pi$
$224$	−4.18842	−0.279851
$225$	−2.36028	−0.157352
$226$	22.9752	1.52828
$227$	12.8664	0.853974	0.426987	−	0.904258i	$-0.359575\pi$
$227$	12.8664	0.853974	0.426987	+	0.904258i	$0.359575\pi$
$228$	0.427778	0.0283303
$229$	27.9205	1.84504	0.922519	−	0.385952i	$-0.126127\pi$
$229$	27.9205	1.84504	0.922519	+	0.385952i	$0.126127\pi$
$230$	−5.17476	−0.341214
$231$	15.7304	1.03498
$232$	−8.22125	−0.539752
$233$	−7.38498	−0.483806	−0.241903	−	0.970300i	$-0.577772\pi$
$233$	−7.38498	−0.483806	−0.241903	+	0.970300i	$0.577772\pi$
$234$	3.66210	0.239399
$235$	4.51707	0.294661
$236$	1.15995	0.0755062
$237$	−19.4884	−1.26591
$238$	−2.16114	−0.140086
$239$	−14.9539	−0.967285	−0.483643	−	0.875266i	$-0.660687\pi$
$239$	−14.9539	−0.967285	−0.483643	+	0.875266i	$0.660687\pi$
$240$	−5.19111	−0.335085
$241$	3.51689	0.226543	0.113271	−	0.993564i	$-0.463867\pi$
$241$	3.51689	0.226543	0.113271	+	0.993564i	$0.463867\pi$
$242$	−5.95508	−0.382807
$243$	5.54085	0.355446
$244$	0.102051	0.00653312
$245$	−2.93106	−0.187259
$246$	23.6647	1.50881
$247$	5.09980	0.324492
$248$	−27.4798	−1.74497
$249$	12.8042	0.811431
$250$	9.85977	0.623587
$251$	−23.2382	−1.46678	−0.733391	−	0.679807i	$-0.762064\pi$
$251$	−23.2382	−1.46678	−0.733391	+	0.679807i	$0.762064\pi$
$252$	0.401367	0.0252838
$253$	12.5722	0.790409
$254$	−9.89172	−0.620662
$255$	0.736982	0.0461516
$256$	−5.38396	−0.336498
$257$	−24.4822	−1.52716	−0.763578	−	0.645715i	$-0.776559\pi$
$257$	−24.4822	−1.52716	−0.763578	+	0.645715i	$0.776559\pi$
$258$	29.1839	1.81691
$259$	−27.6243	−1.71649
$260$	0.915890	0.0568011
$261$	1.49522	0.0925520
$262$	−12.0856	−0.746649
$263$	−9.58091	−0.590784	−0.295392	−	0.955376i	$-0.595450\pi$
$263$	−9.58091	−0.590784	−0.295392	+	0.955376i	$0.595450\pi$
$264$	14.2537	0.877254
$265$	8.46451	0.519970
$266$	−4.35735	−0.267166
$267$	2.35511	0.144130
$268$	−0.143924	−0.00879155
$269$	0.0381960	0.00232885	0.00116443	−	0.999999i	$-0.499629\pi$
$269$	0.0381960	0.00232885	0.00116443	+	0.999999i	$0.499629\pi$
$270$	−4.86803	−0.296259
$271$	−18.2614	−1.10930	−0.554652	−	0.832083i	$-0.687148\pi$
$271$	−18.2614	−1.10930	−0.554652	+	0.832083i	$0.687148\pi$
$272$	−1.73271	−0.105061
$273$	31.4000	1.90041
$274$	15.5322	0.938336
$275$	−11.1804	−0.674203
$276$	2.10507	0.126711
$277$	−0.111960	−0.00672700	−0.00336350	−	0.999994i	$-0.501071\pi$
$277$	−0.111960	−0.00672700	−0.00336350	+	0.999994i	$0.501071\pi$
$278$	−15.6912	−0.941093
$279$	4.99783	0.299212
$280$	−7.66569	−0.458113
$281$	26.4086	1.57540	0.787702	−	0.616056i	$-0.211271\pi$
$281$	26.4086	1.57540	0.787702	+	0.616056i	$0.211271\pi$
$282$	14.3249	0.853036
$283$	−21.7392	−1.29226	−0.646130	−	0.763228i	$-0.723614\pi$
$283$	−21.7392	−1.29226	−0.646130	+	0.763228i	$0.723614\pi$
$284$	1.59379	0.0945741
$285$	1.48592	0.0880184
$286$	17.3470	1.02575
$287$	30.9205	1.82518
$288$	0.690249	0.0406733
$289$	−16.7540	−0.985530
$290$	−2.91526	−0.171190
$291$	4.48509	0.262921
$292$	1.32319	0.0774338
$293$	18.6945	1.09214	0.546072	−	0.837738i	$-0.316123\pi$
$293$	18.6945	1.09214	0.546072	+	0.837738i	$0.316123\pi$
$294$	−9.29524	−0.542109
$295$	4.02918	0.234588
$296$	−25.0311	−1.45490
$297$	11.8270	0.686273
$298$	10.9235	0.632783
$299$	25.0958	1.45133
$300$	−1.87203	−0.108082
$301$	38.1319	2.19789
$302$	12.2283	0.703661
$303$	2.64813	0.152131
$304$	−3.49353	−0.200368
$305$	0.354481	0.0202975
$306$	0.356155	0.0203600
$307$	25.1713	1.43660	0.718302	−	0.695731i	$-0.244920\pi$
$307$	25.1713	1.43660	0.718302	+	0.695731i	$0.244920\pi$
$308$	1.90123	0.108333
$309$	−11.3066	−0.643211
$310$	−9.74435	−0.553442
$311$	−13.7880	−0.781846	−0.390923	−	0.920423i	$-0.627844\pi$
$311$	−13.7880	−0.781846	−0.390923	+	0.920423i	$0.627844\pi$
$312$	28.4523	1.61079
$313$	−26.5730	−1.50200	−0.750999	−	0.660304i	$-0.770428\pi$
$313$	−26.5730	−1.50200	−0.750999	+	0.660304i	$0.770428\pi$
$314$	−23.2976	−1.31476
$315$	1.39418	0.0785532
$316$	−2.35543	−0.132503
$317$	−16.7547	−0.941039	−0.470520	−	0.882390i	$-0.655933\pi$
$317$	−16.7547	−0.941039	−0.470520	+	0.882390i	$0.655933\pi$
$318$	26.8434	1.50530
$319$	7.08270	0.396555
$320$	−6.86439	−0.383731
$321$	−2.83144	−0.158036
$322$	−21.4423	−1.19493
$323$	0.495976	0.0275969
$324$	2.34821	0.130456
$325$	−22.3176	−1.23796
$326$	−8.21411	−0.454937
$327$	−12.2106	−0.675249
$328$	28.0178	1.54702
$329$	18.7171	1.03190
$330$	5.05436	0.278234
$331$	4.59038	0.252310	0.126155	−	0.992011i	$-0.459736\pi$
$331$	4.59038	0.252310	0.126155	+	0.992011i	$0.459736\pi$
$332$	1.54755	0.0849331
$333$	4.55247	0.249474
$334$	6.84349	0.374459
$335$	−0.499932	−0.0273142
$336$	−21.5101	−1.17347
$337$	11.9971	0.653522	0.326761	−	0.945107i	$-0.394043\pi$
$337$	11.9971	0.653522	0.326761	+	0.945107i	$0.394043\pi$
$338$	17.3187	0.942013
$339$	−32.4648	−1.76325
$340$	0.0890741	0.00483072
$341$	23.6742	1.28203
$342$	0.718089	0.0388298
$343$	10.7641	0.581208
$344$	34.5522	1.86293
$345$	7.31214	0.393673
$346$	20.5791	1.10634
$347$	−32.9501	−1.76885	−0.884426	−	0.466680i	$-0.845450\pi$
$347$	−32.9501	−1.76885	−0.884426	+	0.466680i	$0.845450\pi$
$348$	1.18592	0.0635717
$349$	35.1895	1.88365	0.941826	−	0.336101i	$-0.109108\pi$
$349$	35.1895	1.88365	0.941826	+	0.336101i	$0.109108\pi$
$350$	19.0685	1.01925
$351$	23.6083	1.26012
$352$	3.26963	0.174272
$353$	2.93350	0.156134	0.0780671	−	0.996948i	$-0.475125\pi$
$353$	2.93350	0.156134	0.0780671	+	0.996948i	$0.475125\pi$
$354$	12.7777	0.679126
$355$	5.53616	0.293829
$356$	0.284647	0.0150862
$357$	3.05378	0.161623
$358$	−27.2393	−1.43964
$359$	−14.4810	−0.764280	−0.382140	−	0.924105i	$-0.624813\pi$
$359$	−14.4810	−0.764280	−0.382140	+	0.924105i	$0.624813\pi$
$360$	1.26330	0.0665818
$361$	1.00000	0.0526316
$362$	−27.0582	−1.42215
$363$	8.41476	0.441660
$364$	3.79511	0.198918
$365$	4.59621	0.240576
$366$	1.12416	0.0587608
$367$	17.8206	0.930227	0.465113	−	0.885251i	$-0.346014\pi$
$367$	17.8206	0.930227	0.465113	+	0.885251i	$0.346014\pi$
$368$	−17.1915	−0.896169
$369$	−5.09568	−0.265270
$370$	−8.87602	−0.461443
$371$	35.0738	1.82094
$372$	3.96396	0.205522
$373$	−33.7727	−1.74869	−0.874343	−	0.485308i	$-0.838708\pi$
$373$	−33.7727	−1.74869	−0.874343	+	0.485308i	$0.838708\pi$
$374$	1.68707	0.0872361
$375$	−13.9322	−0.719458
$376$	16.9600	0.874643
$377$	14.1380	0.728145
$378$	−20.1713	−1.03750
$379$	−30.9866	−1.59167	−0.795836	−	0.605512i	$-0.792969\pi$
$379$	−30.9866	−1.59167	−0.795836	+	0.605512i	$0.792969\pi$
$380$	0.179593	0.00921295
$381$	13.9774	0.716083
$382$	−29.5159	−1.51017
$383$	9.51060	0.485969	0.242985	−	0.970030i	$-0.421874\pi$
$383$	9.51060	0.485969	0.242985	+	0.970030i	$0.421874\pi$
$384$	−16.9536	−0.865159
$385$	6.60408	0.336575
$386$	32.4347	1.65088
$387$	−6.28411	−0.319439
$388$	0.542084	0.0275201
$389$	−7.63693	−0.387208	−0.193604	−	0.981080i	$-0.562018\pi$
$389$	−7.63693	−0.387208	−0.193604	+	0.981080i	$0.562018\pi$
$390$	10.0892	0.510886
$391$	2.44068	0.123430
$392$	−11.0051	−0.555841
$393$	17.0774	0.861440
$394$	11.5464	0.581699
$395$	−8.18178	−0.411670
$396$	−0.313321	−0.0157450
$397$	−11.1467	−0.559435	−0.279718	−	0.960082i	$-0.590241\pi$
$397$	−11.1467	−0.559435	−0.279718	+	0.960082i	$0.590241\pi$
$398$	−18.2842	−0.916506
$399$	6.15711	0.308241
$400$	15.2883	0.764414
$401$	−4.68311	−0.233863	−0.116932	−	0.993140i	$-0.537306\pi$
$401$	−4.68311	−0.233863	−0.116932	+	0.993140i	$0.537306\pi$
$402$	−1.58543	−0.0790739
$403$	47.2568	2.35403
$404$	0.320062	0.0159237
$405$	8.15670	0.405310
$406$	−12.0798	−0.599508
$407$	21.5645	1.06891
$408$	2.76710	0.136992
$409$	24.4257	1.20777	0.603887	−	0.797070i	$-0.293618\pi$
$409$	24.4257	1.20777	0.603887	+	0.797070i	$0.293618\pi$
$410$	9.93512	0.490661
$411$	−21.9476	−1.08260
$412$	−1.36656	−0.0673254
$413$	16.6954	0.821528
$414$	3.53368	0.173671
$415$	5.37556	0.263876
$416$	6.52662	0.319994
$417$	22.1722	1.08578
$418$	3.40150	0.166373
$419$	−34.6848	−1.69446	−0.847231	−	0.531224i	$-0.821732\pi$
$419$	−34.6848	−1.69446	−0.847231	+	0.531224i	$0.821732\pi$
$420$	1.10578	0.0539563
$421$	−35.8162	−1.74557	−0.872786	−	0.488102i	$-0.837689\pi$
$421$	−35.8162	−1.74557	−0.872786	+	0.488102i	$0.837689\pi$
$422$	1.33140	0.0648114
$423$	−3.08456	−0.149976
$424$	31.7812	1.54343
$425$	−2.17048	−0.105284
$426$	17.5568	0.850628
$427$	1.46884	0.0710821
$428$	−0.342217	−0.0165417
$429$	−24.5119	−1.18345
$430$	12.2522	0.590855
$431$	−6.95175	−0.334854	−0.167427	−	0.985884i	$-0.553546\pi$
$431$	−6.95175	−0.334854	−0.167427	+	0.985884i	$0.553546\pi$
$432$	−16.1725	−0.778099
$433$	−25.9894	−1.24897	−0.624486	−	0.781036i	$-0.714692\pi$
$433$	−25.9894	−1.24897	−0.624486	+	0.781036i	$0.714692\pi$
$434$	−40.3770	−1.93816
$435$	4.11937	0.197509
$436$	−1.47582	−0.0706789
$437$	4.92095	0.235401
$438$	14.5759	0.696463
$439$	33.8757	1.61680	0.808400	−	0.588634i	$-0.200334\pi$
$439$	33.8757	1.61680	0.808400	+	0.588634i	$0.200334\pi$
$440$	5.98411	0.285281
$441$	2.00153	0.0953108
$442$	3.36761	0.160181
$443$	−27.8844	−1.32483	−0.662414	−	0.749138i	$-0.730468\pi$
$443$	−27.8844	−1.32483	−0.662414	+	0.749138i	$0.730468\pi$
$444$	3.61073	0.171358
$445$	0.988743	0.0468709
$446$	−18.1462	−0.859245
$447$	−15.4354	−0.730068
$448$	−28.4435	−1.34383
$449$	−31.2673	−1.47560	−0.737798	−	0.675022i	$-0.764134\pi$
$449$	−31.2673	−1.47560	−0.737798	+	0.675022i	$0.764134\pi$
$450$	−3.14248	−0.148138
$451$	−24.1376	−1.13660
$452$	−3.92381	−0.184560
$453$	−17.2791	−0.811843
$454$	17.1303	0.803965
$455$	13.1826	0.618011
$456$	5.57910	0.261265
$457$	−20.5142	−0.959614	−0.479807	−	0.877374i	$-0.659293\pi$
$457$	−20.5142	−0.959614	−0.479807	+	0.877374i	$0.659293\pi$
$458$	37.1733	1.73699
$459$	2.29600	0.107168
$460$	0.883770	0.0412060
$461$	32.5257	1.51487	0.757436	−	0.652910i	$-0.226452\pi$
$461$	32.5257	1.51487	0.757436	+	0.652910i	$0.226452\pi$
$462$	20.9434	0.974376
$463$	−4.55318	−0.211604	−0.105802	−	0.994387i	$-0.533741\pi$
$463$	−4.55318	−0.211604	−0.105802	+	0.994387i	$0.533741\pi$
$464$	−9.68502	−0.449616
$465$	13.7691	0.638529
$466$	−9.83234	−0.455474
$467$	10.9384	0.506170	0.253085	−	0.967444i	$-0.418555\pi$
$467$	10.9384	0.506170	0.253085	+	0.967444i	$0.418555\pi$
$468$	−0.625431	−0.0289106
$469$	−2.07153	−0.0956545
$470$	6.01401	0.277406
$471$	32.9204	1.51689
$472$	15.1281	0.696328
$473$	−29.7671	−1.36869
$474$	−25.9468	−1.19177
$475$	−4.37617	−0.200792
$476$	0.369090	0.0169172
$477$	−5.78014	−0.264654
$478$	−19.9095	−0.910641
$479$	17.7247	0.809861	0.404930	−	0.914347i	$-0.367296\pi$
$479$	17.7247	0.809861	0.404930	+	0.914347i	$0.367296\pi$
$480$	1.90165	0.0867982
$481$	43.0457	1.96271
$482$	4.68237	0.213276
$483$	30.2988	1.37864
$484$	1.01704	0.0462289
$485$	1.88297	0.0855013
$486$	7.37707	0.334631
$487$	6.65691	0.301653	0.150827	−	0.988560i	$-0.451806\pi$
$487$	6.65691	0.301653	0.150827	+	0.988560i	$0.451806\pi$
$488$	1.33095	0.0602492
$489$	11.6069	0.524880
$490$	−3.90241	−0.176293
$491$	−29.9419	−1.35126	−0.675630	−	0.737241i	$-0.736128\pi$
$491$	−29.9419	−1.35126	−0.675630	+	0.737241i	$0.736128\pi$
$492$	−4.04157	−0.182208
$493$	1.37498	0.0619260
$494$	6.78985	0.305490
$495$	−1.08835	−0.0489176
$496$	−32.3725	−1.45357
$497$	22.9398	1.02899
$498$	17.0474	0.763913
$499$	25.2598	1.13078	0.565392	−	0.824823i	$-0.308725\pi$
$499$	25.2598	1.13078	0.565392	+	0.824823i	$0.308725\pi$
$500$	−1.68390	−0.0753062
$501$	−9.67012	−0.432029
$502$	−30.9393	−1.38089
$503$	−5.05785	−0.225518	−0.112759	−	0.993622i	$-0.535969\pi$
$503$	−5.05785	−0.225518	−0.112759	+	0.993622i	$0.535969\pi$
$504$	5.23465	0.233170
$505$	1.11176	0.0494727
$506$	16.7386	0.744123
$507$	−24.4720	−1.08684
$508$	1.68935	0.0749530
$509$	29.9204	1.32620	0.663099	−	0.748532i	$-0.269241\pi$
$509$	29.9204	1.32620	0.663099	+	0.748532i	$0.269241\pi$
$510$	0.981215	0.0434489
$511$	19.0450	0.842501
$512$	−25.1913	−1.11331
$513$	4.62926	0.204387
$514$	−32.5955	−1.43773
$515$	−4.74684	−0.209171
$516$	−4.98416	−0.219415
$517$	−14.6112	−0.642600
$518$	−36.7790	−1.61598
$519$	−29.0791	−1.27643
$520$	11.9451	0.523826
$521$	−5.34501	−0.234169	−0.117085	−	0.993122i	$-0.537355\pi$
$521$	−5.34501	−0.234169	−0.117085	+	0.993122i	$0.537355\pi$
$522$	1.99074	0.0871321
$523$	−28.2294	−1.23439	−0.617193	−	0.786812i	$-0.711730\pi$
$523$	−28.2294	−1.23439	−0.617193	+	0.786812i	$0.711730\pi$
$524$	2.06403	0.0901675
$525$	−26.9445	−1.17596
$526$	−12.7560	−0.556188
$527$	4.59592	0.200201
$528$	16.7915	0.730757
$529$	1.21575	0.0528589
$530$	11.2696	0.489521
$531$	−2.75139	−0.119400
$532$	0.744169	0.0322638
$533$	−48.1819	−2.08699
$534$	3.13559	0.135690
$535$	−1.18872	−0.0513929
$536$	−1.87706	−0.0810768
$537$	38.4902	1.66098
$538$	0.0508541	0.00219248
$539$	9.48101	0.408376
$540$	0.831385	0.0357771
$541$	−23.7694	−1.02193	−0.510963	−	0.859603i	$-0.670711\pi$
$541$	−23.7694	−1.02193	−0.510963	+	0.859603i	$0.670711\pi$
$542$	−24.3132	−1.04434
$543$	38.2343	1.64079
$544$	0.634741	0.0272143
$545$	−5.12637	−0.219590
$546$	41.8059	1.78913
$547$	−26.2422	−1.12203	−0.561017	−	0.827804i	$-0.689590\pi$
$547$	−26.2422	−1.12203	−0.561017	+	0.827804i	$0.689590\pi$
$548$	−2.65266	−0.113316
$549$	−0.242064	−0.0103310
$550$	−14.8855	−0.634722
$551$	2.77227	0.118103
$552$	27.4545	1.16854
$553$	−33.9023	−1.44167
$554$	−0.149063	−0.00633306
$555$	12.5422	0.532385
$556$	2.67981	0.113649
$557$	11.1455	0.472250	0.236125	−	0.971723i	$-0.424122\pi$
$557$	11.1455	0.472250	0.236125	+	0.971723i	$0.424122\pi$
$558$	6.65410	0.281690
$559$	−59.4191	−2.51316
$560$	−9.03054	−0.381610
$561$	−2.38389	−0.100648
$562$	35.1603	1.48315
$563$	−16.5253	−0.696460	−0.348230	−	0.937409i	$-0.613217\pi$
$563$	−16.5253	−0.696460	−0.348230	+	0.937409i	$0.613217\pi$
$564$	−2.44647	−0.103015
$565$	−13.6297	−0.573404
$566$	−28.9435	−1.21658
$567$	33.7984	1.41940
$568$	20.7863	0.872174
$569$	−1.94855	−0.0816876	−0.0408438	−	0.999166i	$-0.513005\pi$
$569$	−1.94855	−0.0816876	−0.0408438	+	0.999166i	$0.513005\pi$
$570$	1.97835	0.0828640
$571$	−43.1304	−1.80495	−0.902476	−	0.430739i	$-0.858253\pi$
$571$	−43.1304	−1.80495	−0.902476	+	0.430739i	$0.858253\pi$
$572$	−2.96260	−0.123872
$573$	41.7071	1.74234
$574$	41.1675	1.71830
$575$	−21.5349	−0.898068
$576$	4.68747	0.195311
$577$	18.4933	0.769886	0.384943	−	0.922940i	$-0.374221\pi$
$577$	18.4933	0.769886	0.384943	+	0.922940i	$0.374221\pi$
$578$	−22.3062	−0.927817
$579$	−45.8315	−1.90469
$580$	0.497881	0.0206734
$581$	22.2743	0.924095
$582$	5.97144	0.247524
$583$	−27.3798	−1.13396
$584$	17.2571	0.714104
$585$	−2.17249	−0.0898213
$586$	24.8898	1.02819
$587$	−2.19604	−0.0906401	−0.0453201	−	0.998973i	$-0.514431\pi$
$587$	−2.19604	−0.0906401	−0.0453201	+	0.998973i	$0.514431\pi$
$588$	1.58748	0.0654667
$589$	9.26640	0.381815
$590$	5.36443	0.220850
$591$	−16.3155	−0.671130
$592$	−29.4878	−1.21194
$593$	−20.0546	−0.823546	−0.411773	−	0.911287i	$-0.635090\pi$
$593$	−20.0546	−0.823546	−0.411773	+	0.911287i	$0.635090\pi$
$594$	15.7464	0.646084
$595$	1.28206	0.0525595
$596$	−1.86557	−0.0764168
$597$	25.8364	1.05741
$598$	33.4125	1.36634
$599$	−19.7870	−0.808475	−0.404237	−	0.914654i	$-0.632463\pi$
$599$	−19.7870	−0.808475	−0.404237	+	0.914654i	$0.632463\pi$
$600$	−24.4151	−0.996741
$601$	−15.8497	−0.646521	−0.323260	−	0.946310i	$-0.604779\pi$
$601$	−15.8497	−0.646521	−0.323260	+	0.946310i	$0.604779\pi$
$602$	50.7687	2.06918
$603$	0.341387	0.0139024
$604$	−2.08841	−0.0849763
$605$	3.53276	0.143627
$606$	3.52571	0.143222
$607$	27.7093	1.12469	0.562343	−	0.826904i	$-0.309900\pi$
$607$	27.7093	1.12469	0.562343	+	0.826904i	$0.309900\pi$
$608$	1.27978	0.0519020
$609$	17.0692	0.691678
$610$	0.471955	0.0191089
$611$	−29.1659	−1.17993
$612$	−0.0608258	−0.00245874
$613$	−19.3109	−0.779959	−0.389980	−	0.920824i	$-0.627518\pi$
$613$	−19.3109	−0.779959	−0.389980	+	0.920824i	$0.627518\pi$
$614$	33.5130	1.35248
$615$	−14.0387	−0.566096
$616$	24.7959	0.999057
$617$	24.7579	0.996714	0.498357	−	0.866972i	$-0.333937\pi$
$617$	24.7579	0.996714	0.498357	+	0.866972i	$0.333937\pi$
$618$	−15.0536	−0.605545
$619$	−20.5209	−0.824804	−0.412402	−	0.911002i	$-0.635310\pi$
$619$	−20.5209	−0.824804	−0.412402	+	0.911002i	$0.635310\pi$
$620$	1.66418	0.0668353
$621$	22.7804	0.914144
$622$	−18.3573	−0.736061
$623$	4.09699	0.164142
$624$	33.5181	1.34180
$625$	16.0317	0.641267
$626$	−35.3793	−1.41404
$627$	−4.80646	−0.191951
$628$	3.97886	0.158774
$629$	4.18637	0.166922
$630$	1.85621	0.0739532
$631$	23.4644	0.934102	0.467051	−	0.884230i	$-0.345316\pi$
$631$	23.4644	0.934102	0.467051	+	0.884230i	$0.345316\pi$
$632$	−30.7196	−1.22196
$633$	−1.88132	−0.0747756
$634$	−22.3072	−0.885932
$635$	5.86811	0.232869
$636$	−4.58444	−0.181785
$637$	18.9253	0.749850
$638$	9.42989	0.373333
$639$	−3.78047	−0.149553
$640$	−7.11760	−0.281348
$641$	9.09726	0.359320	0.179660	−	0.983729i	$-0.442500\pi$
$641$	9.09726	0.359320	0.179660	+	0.983729i	$0.442500\pi$
$642$	−3.76977	−0.148781
$643$	−7.90449	−0.311723	−0.155861	−	0.987779i	$-0.549815\pi$
$643$	−7.90449	−0.311723	−0.155861	+	0.987779i	$0.549815\pi$
$644$	3.66202	0.144304
$645$	−17.3129	−0.681694
$646$	0.660341	0.0259808
$647$	17.6017	0.691993	0.345996	−	0.938236i	$-0.387541\pi$
$647$	17.6017	0.691993	0.345996	+	0.938236i	$0.387541\pi$
$648$	30.6255	1.20308
$649$	−13.0330	−0.511592
$650$	−29.7135	−1.16546
$651$	57.0542	2.23613
$652$	1.40284	0.0549396
$653$	12.5054	0.489373	0.244686	−	0.969602i	$-0.421315\pi$
$653$	12.5054	0.489373	0.244686	+	0.969602i	$0.421315\pi$
$654$	−16.2572	−0.635707
$655$	7.16957	0.280138
$656$	33.0063	1.28868
$657$	−3.13860	−0.122448
$658$	24.9198	0.971476
$659$	−23.2067	−0.904005	−0.452002	−	0.892017i	$-0.649290\pi$
$659$	−23.2067	−0.904005	−0.452002	+	0.892017i	$0.649290\pi$
$660$	−0.863208	−0.0336003
$661$	−6.66340	−0.259176	−0.129588	−	0.991568i	$-0.541365\pi$
$661$	−6.66340	−0.259176	−0.129588	+	0.991568i	$0.541365\pi$
$662$	6.11161	0.237535
$663$	−4.75856	−0.184807
$664$	20.1833	0.783263
$665$	2.58493	0.100239
$666$	6.06115	0.234865
$667$	13.6422	0.528228
$668$	−1.16876	−0.0452208
$669$	25.6412	0.991347
$670$	−0.665607	−0.0257147
$671$	−1.14663	−0.0442651
$672$	7.87975	0.303968
$673$	33.8408	1.30447	0.652234	−	0.758017i	$-0.273832\pi$
$673$	33.8408	1.30447	0.652234	+	0.758017i	$0.273832\pi$
$674$	15.9729	0.615252
$675$	−20.2584	−0.779747
$676$	−2.95777	−0.113760
$677$	19.0978	0.733990	0.366995	−	0.930223i	$-0.380387\pi$
$677$	19.0978	0.733990	0.366995	+	0.930223i	$0.380387\pi$
$678$	−43.2235	−1.65999
$679$	7.80234	0.299426
$680$	1.16171	0.0445495
$681$	−24.2058	−0.927568
$682$	31.5197	1.20695
$683$	4.69442	0.179627	0.0898135	−	0.995959i	$-0.471373\pi$
$683$	4.69442	0.179627	0.0898135	+	0.995959i	$0.471373\pi$
$684$	−0.122639	−0.00468920
$685$	−9.21425	−0.352058
$686$	14.3313	0.547172
$687$	−52.5273	−2.00404
$688$	40.7041	1.55183
$689$	−54.6538	−2.08214
$690$	9.73537	0.370619
$691$	−24.8278	−0.944494	−0.472247	−	0.881466i	$-0.656557\pi$
$691$	−24.8278	−0.944494	−0.472247	+	0.881466i	$0.656557\pi$
$692$	−3.51460	−0.133605
$693$	−4.50971	−0.171310
$694$	−43.8696	−1.66527
$695$	9.30854	0.353093
$696$	15.4668	0.586266
$697$	−4.68590	−0.177491
$698$	46.8512	1.77335
$699$	13.8935	0.525500
$700$	−3.25661	−0.123088
$701$	1.43200	0.0540859	0.0270429	−	0.999634i	$-0.491391\pi$
$701$	1.43200	0.0540859	0.0270429	+	0.999634i	$0.491391\pi$
$702$	31.4320	1.18632
$703$	8.44067	0.318346
$704$	22.2040	0.836844
$705$	−8.49803	−0.320054
$706$	3.90565	0.146991
$707$	4.60673	0.173254
$708$	−2.18223	−0.0820132
$709$	−33.5266	−1.25912	−0.629558	−	0.776953i	$-0.716764\pi$
$709$	−33.5266	−1.25912	−0.629558	+	0.776953i	$0.716764\pi$
$710$	7.37083	0.276622
$711$	5.58707	0.209532
$712$	3.71238	0.139127
$713$	45.5995	1.70772
$714$	4.06579	0.152159
$715$	−10.2908	−0.384855
$716$	4.65206	0.173856
$717$	28.1330	1.05064
$718$	−19.2800	−0.719523
$719$	51.4198	1.91764	0.958818	−	0.284022i	$-0.0916690\pi$
$719$	51.4198	1.91764	0.958818	+	0.284022i	$0.0916690\pi$
$720$	1.48823	0.0554630
$721$	−19.6692	−0.732518
$722$	1.33140	0.0495495
$723$	−6.61638	−0.246066
$724$	4.62113	0.171743
$725$	−12.1319	−0.450568
$726$	11.2034	0.415797
$727$	−10.1773	−0.377456	−0.188728	−	0.982029i	$-0.560436\pi$
$727$	−10.1773	−0.377456	−0.188728	+	0.982029i	$0.560436\pi$
$728$	49.4960	1.83444
$729$	20.5574	0.761384
$730$	6.11938	0.226488
$731$	−5.77876	−0.213735
$732$	−0.191989	−0.00709613
$733$	−2.37892	−0.0878675	−0.0439338	−	0.999034i	$-0.513989\pi$
$733$	−2.37892	−0.0878675	−0.0439338	+	0.999034i	$0.513989\pi$
$734$	23.7263	0.875752
$735$	5.51426	0.203396
$736$	6.29774	0.232138
$737$	1.61711	0.0595671
$738$	−6.78437	−0.249736
$739$	−16.4782	−0.606161	−0.303081	−	0.952965i	$-0.598015\pi$
$739$	−16.4782	−0.606161	−0.303081	+	0.952965i	$0.598015\pi$
$740$	1.51589	0.0557252
$741$	−9.59433	−0.352456
$742$	46.6971	1.71431
$743$	−24.4466	−0.896858	−0.448429	−	0.893818i	$-0.648016\pi$
$743$	−24.4466	−0.896858	−0.448429	+	0.893818i	$0.648016\pi$
$744$	51.6982	1.89535
$745$	−6.48022	−0.237417
$746$	−44.9649	−1.64628
$747$	−3.67079	−0.134307
$748$	−0.288125	−0.0105349
$749$	−4.92562	−0.179978
$750$	−18.5493	−0.677326
$751$	−13.8855	−0.506690	−0.253345	−	0.967376i	$-0.581531\pi$
$751$	−13.8855	−0.506690	−0.253345	+	0.967376i	$0.581531\pi$
$752$	19.9796	0.728582
$753$	43.7184	1.59319
$754$	18.8233	0.685504
$755$	−7.25427	−0.264010
$756$	3.44495	0.125292
$757$	19.8394	0.721075	0.360538	−	0.932745i	$-0.382593\pi$
$757$	19.8394	0.721075	0.360538	+	0.932745i	$0.382593\pi$
$758$	−41.2554	−1.49846
$759$	−23.6523	−0.858525
$760$	2.34227	0.0849629
$761$	20.9965	0.761121	0.380560	−	0.924756i	$-0.375731\pi$
$761$	20.9965	0.761121	0.380560	+	0.924756i	$0.375731\pi$
$762$	18.6095	0.674149
$763$	−21.2418	−0.769005
$764$	5.04086	0.182372
$765$	−0.211283	−0.00763897
$766$	12.6624	0.457511
$767$	−26.0157	−0.939372
$768$	10.1289	0.365496
$769$	−27.7816	−1.00183	−0.500915	−	0.865497i	$-0.667003\pi$
$769$	−27.7816	−1.00183	−0.500915	+	0.865497i	$0.667003\pi$
$770$	8.79265	0.316865
$771$	46.0587	1.65876
$772$	−5.53935	−0.199366
$773$	25.8797	0.930829	0.465414	−	0.885093i	$-0.345905\pi$
$773$	25.8797	0.930829	0.465414	+	0.885093i	$0.345905\pi$
$774$	−8.36665	−0.300733
$775$	−40.5513	−1.45665
$776$	7.06988	0.253794
$777$	51.9701	1.86442
$778$	−10.1678	−0.364533
$779$	−9.44782	−0.338503
$780$	−1.72308	−0.0616961
$781$	−17.9076	−0.640786
$782$	3.24951	0.116202
$783$	12.8336	0.458634
$784$	−12.9645	−0.463018
$785$	13.8209	0.493290
$786$	22.7368	0.810994
$787$	45.9254	1.63706	0.818532	−	0.574461i	$-0.194788\pi$
$787$	45.9254	1.63706	0.818532	+	0.574461i	$0.194788\pi$
$788$	−1.97195	−0.0702477
$789$	18.0247	0.641697
$790$	−10.8932	−0.387562
$791$	−56.4763	−2.00807
$792$	−4.08635	−0.145202
$793$	−2.28882	−0.0812784
$794$	−14.8406	−0.526675
$795$	−15.9244	−0.564781
$796$	3.12267	0.110680
$797$	39.5854	1.40219	0.701093	−	0.713069i	$-0.252696\pi$
$797$	39.5854	1.40219	0.701093	+	0.713069i	$0.252696\pi$
$798$	8.19756	0.290190
$799$	−2.83650	−0.100348
$800$	−5.60053	−0.198009
$801$	−0.675181	−0.0238563
$802$	−6.23508	−0.220168
$803$	−14.8672	−0.524652
$804$	0.270766	0.00954919
$805$	12.7203	0.448332
$806$	62.9175	2.21617
$807$	−0.0718588	−0.00252955
$808$	4.17426	0.146850
$809$	9.40004	0.330488	0.165244	−	0.986253i	$-0.447159\pi$
$809$	9.40004	0.330488	0.165244	+	0.986253i	$0.447159\pi$
$810$	10.8598	0.381575
$811$	17.0652	0.599241	0.299621	−	0.954058i	$-0.403140\pi$
$811$	17.0652	0.599241	0.299621	+	0.954058i	$0.403140\pi$
$812$	2.06304	0.0723984
$813$	34.3555	1.20490
$814$	28.7110	1.00632
$815$	4.87289	0.170690
$816$	3.25977	0.114115
$817$	−11.6513	−0.407627
$818$	32.5203	1.13705
$819$	−9.00198	−0.314555
$820$	−1.69677	−0.0592536
$821$	54.4005	1.89859	0.949296	−	0.314383i	$-0.101798\pi$
$821$	54.4005	1.89859	0.949296	+	0.314383i	$0.101798\pi$
$822$	−29.2210	−1.01920
$823$	−5.20566	−0.181458	−0.0907290	−	0.995876i	$-0.528920\pi$
$823$	−5.20566	−0.181458	−0.0907290	+	0.995876i	$0.528920\pi$
$824$	−17.8227	−0.620883
$825$	21.0339	0.732305
$826$	22.2282	0.773419
$827$	−53.2921	−1.85315	−0.926575	−	0.376111i	$-0.877261\pi$
$827$	−53.2921	−1.85315	−0.926575	+	0.376111i	$0.877261\pi$
$828$	−0.603498	−0.0209730
$829$	10.1184	0.351427	0.175713	−	0.984441i	$-0.443777\pi$
$829$	10.1184	0.351427	0.175713	+	0.984441i	$0.443777\pi$
$830$	7.15700	0.248423
$831$	0.210631	0.00730672
$832$	44.3221	1.53659
$833$	1.84057	0.0637720
$834$	29.5200	1.02220
$835$	−4.05980	−0.140495
$836$	−0.580924	−0.0200917
$837$	42.8966	1.48272
$838$	−46.1792	−1.59523
$839$	27.1518	0.937385	0.468693	−	0.883361i	$-0.344725\pi$
$839$	27.1518	0.937385	0.468693	+	0.883361i	$0.344725\pi$
$840$	14.4216	0.497592
$841$	−21.3145	−0.734983
$842$	−47.6855	−1.64335
$843$	−49.6829	−1.71117
$844$	−0.227382	−0.00782682
$845$	−10.2740	−0.353438
$846$	−4.10677	−0.141194
$847$	14.6385	0.502983
$848$	37.4397	1.28569
$849$	40.8982	1.40362
$850$	−2.88977	−0.0991181
$851$	41.5361	1.42384
$852$	−2.99843	−0.102724
$853$	41.4043	1.41766	0.708828	−	0.705381i	$-0.249224\pi$
$853$	41.4043	1.41766	0.708828	+	0.705381i	$0.249224\pi$
$854$	1.95561	0.0669195
$855$	−0.425995	−0.0145687
$856$	−4.46322	−0.152550
$857$	−32.3304	−1.10439	−0.552193	−	0.833716i	$-0.686209\pi$
$857$	−32.3304	−1.10439	−0.552193	+	0.833716i	$0.686209\pi$
$858$	−32.6351	−1.11414
$859$	−6.48888	−0.221398	−0.110699	−	0.993854i	$-0.535309\pi$
$859$	−6.48888	−0.221398	−0.110699	+	0.993854i	$0.535309\pi$
$860$	−2.09249	−0.0713534
$861$	−58.1712	−1.98247
$862$	−9.25553	−0.315245
$863$	35.7680	1.21756	0.608779	−	0.793340i	$-0.291660\pi$
$863$	35.7680	1.21756	0.608779	+	0.793340i	$0.291660\pi$
$864$	5.92444	0.201553
$865$	−12.2082	−0.415093
$866$	−34.6023	−1.17583
$867$	31.5196	1.07046
$868$	6.89577	0.234058
$869$	26.4653	0.897774
$870$	5.48452	0.185943
$871$	3.22797	0.109376
$872$	−19.2477	−0.651809
$873$	−1.28582	−0.0435184
$874$	6.55174	0.221616
$875$	−24.2367	−0.819352
$876$	−2.48934	−0.0841069
$877$	−7.41247	−0.250301	−0.125151	−	0.992138i	$-0.539941\pi$
$877$	−7.41247	−0.250301	−0.125151	+	0.992138i	$0.539941\pi$
$878$	45.1020	1.52212
$879$	−35.1702	−1.18626
$880$	7.04956	0.237641
$881$	5.95406	0.200597	0.100299	−	0.994957i	$-0.468020\pi$
$881$	5.95406	0.200597	0.100299	+	0.994957i	$0.468020\pi$
$882$	2.66483	0.0897294
$883$	−41.5047	−1.39674	−0.698372	−	0.715735i	$-0.746092\pi$
$883$	−41.5047	−1.39674	−0.698372	+	0.715735i	$0.746092\pi$
$884$	−0.575136	−0.0193439
$885$	−7.58016	−0.254804
$886$	−37.1252	−1.24725
$887$	14.6393	0.491538	0.245769	−	0.969328i	$-0.420960\pi$
$887$	14.6393	0.491538	0.245769	+	0.969328i	$0.420960\pi$
$888$	47.0913	1.58028
$889$	24.3153	0.815508
$890$	1.31641	0.0441262
$891$	−26.3842	−0.883904
$892$	3.09908	0.103765
$893$	−5.71903	−0.191380
$894$	−20.5506	−0.687316
$895$	16.1593	0.540146
$896$	−29.4927	−0.985283
$897$	−47.2132	−1.57640
$898$	−41.6292	−1.38918
$899$	25.6890	0.856775
$900$	0.536687	0.0178896
$901$	−5.31531	−0.177079
$902$	−32.1368	−1.07004
$903$	−71.7382	−2.38730
$904$	−51.1745	−1.70204
$905$	16.0519	0.533582
$906$	−23.0054	−0.764302
$907$	34.1691	1.13456	0.567282	−	0.823523i	$-0.307995\pi$
$907$	34.1691	1.13456	0.567282	+	0.823523i	$0.307995\pi$
$908$	−2.92559	−0.0970893
$909$	−0.759185	−0.0251806
$910$	17.5513	0.581820
$911$	−33.2677	−1.10221	−0.551105	−	0.834436i	$-0.685794\pi$
$911$	−33.2677	−1.10221	−0.551105	+	0.834436i	$0.685794\pi$
$912$	6.57244	0.217635
$913$	−17.3881	−0.575463
$914$	−27.3125	−0.903418
$915$	−0.666891	−0.0220467
$916$	−6.34862	−0.209764
$917$	29.7081	0.981047
$918$	3.05689	0.100892
$919$	−9.37241	−0.309167	−0.154584	−	0.987980i	$-0.549404\pi$
$919$	−9.37241	−0.309167	−0.154584	+	0.987980i	$0.549404\pi$
$920$	11.5262	0.380007
$921$	−47.3552	−1.56041
$922$	43.3046	1.42616
$923$	−35.7460	−1.17659
$924$	−3.57681	−0.117669
$925$	−36.9378	−1.21451
$926$	−6.06210	−0.199213
$927$	3.24147	0.106464
$928$	3.54790	0.116465
$929$	1.94881	0.0639383	0.0319692	−	0.999489i	$-0.489822\pi$
$929$	1.94881	0.0639383	0.0319692	+	0.999489i	$0.489822\pi$
$930$	18.3322	0.601136
$931$	3.71100	0.121623
$932$	1.67921	0.0550044
$933$	25.9396	0.849224
$934$	14.5634	0.476529
$935$	−1.00083	−0.0327305
$936$	−8.15691	−0.266617
$937$	−37.4287	−1.22274	−0.611371	−	0.791344i	$-0.709381\pi$
$937$	−37.4287	−1.22274	−0.611371	+	0.791344i	$0.709381\pi$
$938$	−2.75803	−0.0900529
$939$	49.9923	1.63144
$940$	−1.02710	−0.0335003
$941$	−42.2276	−1.37658	−0.688291	−	0.725435i	$-0.741639\pi$
$941$	−42.2276	−1.37658	−0.688291	+	0.725435i	$0.741639\pi$
$942$	43.8301	1.42806
$943$	−46.4923	−1.51400
$944$	17.8216	0.580045
$945$	11.9663	0.389264
$946$	−39.6318	−1.28854
$947$	50.9655	1.65616	0.828078	−	0.560614i	$-0.189435\pi$
$947$	50.9655	1.65616	0.828078	+	0.560614i	$0.189435\pi$
$948$	4.43131	0.143922
$949$	−29.6769	−0.963353
$950$	−5.82642	−0.189034
$951$	31.5209	1.02214
$952$	4.81369	0.156013
$953$	−21.8862	−0.708964	−0.354482	−	0.935063i	$-0.615343\pi$
$953$	−21.8862	−0.708964	−0.354482	+	0.935063i	$0.615343\pi$
$954$	−7.69566	−0.249156
$955$	17.5099	0.566606
$956$	3.40024	0.109972
$957$	−13.3248	−0.430729
$958$	23.5986	0.762435
$959$	−38.1804	−1.23291
$960$	12.9141	0.416800
$961$	54.8662	1.76988
$962$	57.3109	1.84778
$963$	0.811738	0.0261579
$964$	−0.799678	−0.0257559
$965$	−19.2414	−0.619402
$966$	40.3398	1.29791
$967$	12.5330	0.403033	0.201516	−	0.979485i	$-0.435413\pi$
$967$	12.5330	0.403033	0.201516	+	0.979485i	$0.435413\pi$
$968$	13.2642	0.426329
$969$	−0.933088	−0.0299751
$970$	2.50698	0.0804944
$971$	−39.4499	−1.26601	−0.633003	−	0.774149i	$-0.718178\pi$
$971$	−39.4499	−1.26601	−0.633003	+	0.774149i	$0.718178\pi$
$972$	−1.25989	−0.0404110
$973$	38.5712	1.23653
$974$	8.86298	0.283988
$975$	41.9864	1.34464
$976$	1.56792	0.0501879
$977$	−37.4606	−1.19847	−0.599236	−	0.800573i	$-0.704529\pi$
$977$	−37.4606	−1.19847	−0.599236	+	0.800573i	$0.704529\pi$
$978$	15.4533	0.494143
$979$	−3.19825	−0.102217
$980$	0.666471	0.0212896
$981$	3.50063	0.111767
$982$	−39.8646	−1.27213
$983$	−2.86125	−0.0912597	−0.0456298	−	0.998958i	$-0.514529\pi$
$983$	−2.86125	−0.0912597	−0.0456298	+	0.998958i	$0.514529\pi$
$984$	−52.7103	−1.68034
$985$	−6.84972	−0.218250
$986$	1.83065	0.0582996
$987$	−35.2127	−1.12083
$988$	−1.15960	−0.0368919
$989$	−57.3354	−1.82316
$990$	−1.44902	−0.0460530
$991$	51.0449	1.62150	0.810748	−	0.585395i	$-0.199061\pi$
$991$	51.0449	1.62150	0.810748	+	0.585395i	$0.199061\pi$
$992$	11.8590	0.376522
$993$	−8.63595	−0.274054
$994$	30.5420	0.968734
$995$	10.8468	0.343868
$996$	−2.91144	−0.0922525
$997$	14.3161	0.453394	0.226697	−	0.973965i	$-0.427207\pi$
$997$	14.3161	0.453394	0.226697	+	0.973965i	$0.427207\pi$
$998$	33.6308	1.06456
$999$	39.0741	1.23625

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4009.2.a.c.1.53	✓	71

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4009.2.a.c.1.53	✓	71	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:	\( N \)	\(=\)	\( 4009 = 19 \cdot 211 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4009.a (trivial)

\(f(q)\)	\(=\)	\(q+1.33140 q^{2} -1.88132 q^{3} -0.227382 q^{4} -0.789831 q^{5} -2.50478 q^{6} -3.27277 q^{7} -2.96553 q^{8} +0.539350 q^{9} +O(q^{10})\)	\(q+1.33140 q^{2} -1.88132 q^{3} -0.227382 q^{4} -0.789831 q^{5} -2.50478 q^{6} -3.27277 q^{7} -2.96553 q^{8} +0.539350 q^{9} -1.05158 q^{10} +2.55484 q^{11} +0.427778 q^{12} +5.09980 q^{13} -4.35735 q^{14} +1.48592 q^{15} -3.49353 q^{16} +0.495976 q^{17} +0.718089 q^{18} +1.00000 q^{19} +0.179593 q^{20} +6.15711 q^{21} +3.40150 q^{22} +4.92095 q^{23} +5.57910 q^{24} -4.37617 q^{25} +6.78985 q^{26} +4.62926 q^{27} +0.744169 q^{28} +2.77227 q^{29} +1.97835 q^{30} +9.26640 q^{31} +1.27978 q^{32} -4.80646 q^{33} +0.660341 q^{34} +2.58493 q^{35} -0.122639 q^{36} +8.44067 q^{37} +1.33140 q^{38} -9.59433 q^{39} +2.34227 q^{40} -9.44782 q^{41} +8.19756 q^{42} -11.6513 q^{43} -0.580924 q^{44} -0.425995 q^{45} +6.55174 q^{46} -5.71903 q^{47} +6.57244 q^{48} +3.71100 q^{49} -5.82642 q^{50} -0.933088 q^{51} -1.15960 q^{52} -10.7169 q^{53} +6.16338 q^{54} -2.01789 q^{55} +9.70549 q^{56} -1.88132 q^{57} +3.69099 q^{58} -5.10132 q^{59} -0.337872 q^{60} -0.448807 q^{61} +12.3373 q^{62} -1.76517 q^{63} +8.69096 q^{64} -4.02797 q^{65} -6.39930 q^{66} +0.632961 q^{67} -0.112776 q^{68} -9.25786 q^{69} +3.44157 q^{70} -7.00931 q^{71} -1.59946 q^{72} -5.81923 q^{73} +11.2379 q^{74} +8.23295 q^{75} -0.227382 q^{76} -8.36139 q^{77} -12.7739 q^{78} +10.3589 q^{79} +2.75930 q^{80} -10.3272 q^{81} -12.5788 q^{82} -6.80596 q^{83} -1.40002 q^{84} -0.391737 q^{85} -15.5125 q^{86} -5.21552 q^{87} -7.57645 q^{88} -1.25184 q^{89} -0.567168 q^{90} -16.6904 q^{91} -1.11894 q^{92} -17.4330 q^{93} -7.61430 q^{94} -0.789831 q^{95} -2.40767 q^{96} -2.38402 q^{97} +4.94082 q^{98} +1.37795 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9}+O(q^{10}) \) 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9	\( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9} - 10 q^{10} - 52 q^{11} - 9 q^{12} - 15 q^{13} - 53 q^{14} - 33 q^{15} + 53 q^{16} - 10 q^{17} - 35 q^{18} + 71 q^{19} - 33 q^{20} - 38 q^{21} - 6 q^{22} - 65 q^{23} - 30 q^{24} + 51 q^{25} - 4 q^{26} - 23 q^{27} - 29 q^{28} - 97 q^{29} - 27 q^{30} - 53 q^{31} - 78 q^{32} - 17 q^{33} - 24 q^{34} - 38 q^{35} + 24 q^{36} - 33 q^{37} - 15 q^{38} - 86 q^{39} + 25 q^{40} - 69 q^{41} + 64 q^{42} - 10 q^{43} - 94 q^{44} - 34 q^{45} - 6 q^{46} - 37 q^{47} - q^{48} + 74 q^{49} - 41 q^{50} - 46 q^{51} - 30 q^{52} - 50 q^{53} - 17 q^{54} - 30 q^{55} - 116 q^{56} - 8 q^{57} + 11 q^{58} - 93 q^{59} - 56 q^{60} - 18 q^{61} - q^{62} - 84 q^{63} + 93 q^{64} - 78 q^{65} - 53 q^{66} - 5 q^{67} - 9 q^{68} - 69 q^{69} - 10 q^{70} - 221 q^{71} - 73 q^{72} - 34 q^{73} - 58 q^{74} - 70 q^{75} + 69 q^{76} - 2 q^{77} + 7 q^{78} - 68 q^{79} - 71 q^{80} + 39 q^{81} + 26 q^{82} - 45 q^{83} - 10 q^{84} - 44 q^{85} - 80 q^{86} - 7 q^{87} - 46 q^{88} - 143 q^{89} + 41 q^{90} - 30 q^{91} - 46 q^{92} + 32 q^{93} + 41 q^{94} - 18 q^{95} - 140 q^{96} - 18 q^{97} - 97 q^{98} - 142 q^{99}+O(q^{100}) \) 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9 - 10 * q^10 - 52 * q^11 - 9 * q^12 - 15 * q^13 - 53 * q^14 - 33 * q^15 + 53 * q^16 - 10 * q^17 - 35 * q^18 + 71 * q^19 - 33 * q^20 - 38 * q^21 - 6 * q^22 - 65 * q^23 - 30 * q^24 + 51 * q^25 - 4 * q^26 - 23 * q^27 - 29 * q^28 - 97 * q^29 - 27 * q^30 - 53 * q^31 - 78 * q^32 - 17 * q^33 - 24 * q^34 - 38 * q^35 + 24 * q^36 - 33 * q^37 - 15 * q^38 - 86 * q^39 + 25 * q^40 - 69 * q^41 + 64 * q^42 - 10 * q^43 - 94 * q^44 - 34 * q^45 - 6 * q^46 - 37 * q^47 - q^48 + 74 * q^49 - 41 * q^50 - 46 * q^51 - 30 * q^52 - 50 * q^53 - 17 * q^54 - 30 * q^55 - 116 * q^56 - 8 * q^57 + 11 * q^58 - 93 * q^59 - 56 * q^60 - 18 * q^61 - q^62 - 84 * q^63 + 93 * q^64 - 78 * q^65 - 53 * q^66 - 5 * q^67 - 9 * q^68 - 69 * q^69 - 10 * q^70 - 221 * q^71 - 73 * q^72 - 34 * q^73 - 58 * q^74 - 70 * q^75 + 69 * q^76 - 2 * q^77 + 7 * q^78 - 68 * q^79 - 71 * q^80 + 39 * q^81 + 26 * q^82 - 45 * q^83 - 10 * q^84 - 44 * q^85 - 80 * q^86 - 7 * q^87 - 46 * q^88 - 143 * q^89 + 41 * q^90 - 30 * q^91 - 46 * q^92 + 32 * q^93 + 41 * q^94 - 18 * q^95 - 140 * q^96 - 18 * q^97 - 97 * q^98 - 142 * q^99

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