LMFDB - Embedded newform 4009.2.a.c.1.34

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			1.34
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)

$f(q)$	$=$	$q-0.440540 q^{2} +3.33608 q^{3} -1.80592 q^{4} -2.91639 q^{5} -1.46968 q^{6} -4.19739 q^{7} +1.67666 q^{8} +8.12943 q^{9} +O(q^{10})$	\(q-0.440540 q^{2} +3.33608 q^{3} -1.80592 q^{4} -2.91639 q^{5} -1.46968 q^{6} -4.19739 q^{7} +1.67666 q^{8} +8.12943 q^{9} +1.28479 q^{10} +3.74359 q^{11} -6.02471 q^{12} -1.49535 q^{13} +1.84912 q^{14} -9.72930 q^{15} +2.87321 q^{16} +2.00713 q^{17} -3.58134 q^{18} +1.00000 q^{19} +5.26677 q^{20} -14.0028 q^{21} -1.64920 q^{22} -7.25048 q^{23} +5.59348 q^{24} +3.50531 q^{25} +0.658762 q^{26} +17.1122 q^{27} +7.58017 q^{28} +0.228630 q^{29} +4.28615 q^{30} -2.96672 q^{31} -4.61909 q^{32} +12.4889 q^{33} -0.884224 q^{34} +12.2412 q^{35} -14.6811 q^{36} +7.91649 q^{37} -0.440540 q^{38} -4.98861 q^{39} -4.88980 q^{40} -7.44308 q^{41} +6.16882 q^{42} -5.14436 q^{43} -6.76064 q^{44} -23.7085 q^{45} +3.19413 q^{46} -0.713111 q^{47} +9.58526 q^{48} +10.6181 q^{49} -1.54423 q^{50} +6.69596 q^{51} +2.70049 q^{52} +11.7808 q^{53} -7.53860 q^{54} -10.9178 q^{55} -7.03762 q^{56} +3.33608 q^{57} -0.100721 q^{58} +2.37068 q^{59} +17.5704 q^{60} -10.3868 q^{61} +1.30696 q^{62} -34.1224 q^{63} -3.71152 q^{64} +4.36102 q^{65} -5.50187 q^{66} -3.24158 q^{67} -3.62473 q^{68} -24.1882 q^{69} -5.39275 q^{70} -7.64803 q^{71} +13.6303 q^{72} -15.6512 q^{73} -3.48753 q^{74} +11.6940 q^{75} -1.80592 q^{76} -15.7133 q^{77} +2.19768 q^{78} -14.8461 q^{79} -8.37939 q^{80} +32.6993 q^{81} +3.27898 q^{82} -6.05255 q^{83} +25.2881 q^{84} -5.85358 q^{85} +2.26630 q^{86} +0.762726 q^{87} +6.27674 q^{88} -2.45406 q^{89} +10.4446 q^{90} +6.27657 q^{91} +13.0938 q^{92} -9.89720 q^{93} +0.314154 q^{94} -2.91639 q^{95} -15.4097 q^{96} +5.54721 q^{97} -4.67771 q^{98} +30.4332 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.440540	−0.311509	−0.155755	−	0.987796i	$-0.549781\pi$
$2$	−0.440540	−0.311509	−0.155755	+	0.987796i	$0.549781\pi$
$3$	3.33608	1.92609	0.963043	−	0.269347i	$-0.0868079\pi$
$3$	3.33608	1.92609	0.963043	+	0.269347i	$0.0868079\pi$
$4$	−1.80592	−0.902962
$5$	−2.91639	−1.30425	−0.652124	−	0.758112i	$-0.726122\pi$
$5$	−2.91639	−1.30425	−0.652124	+	0.758112i	$0.726122\pi$
$6$	−1.46968	−0.599993
$7$	−4.19739	−1.58647	−0.793233	−	0.608919i	$-0.791604\pi$
$7$	−4.19739	−1.58647	−0.793233	+	0.608919i	$0.791604\pi$
$8$	1.67666	0.592790
$9$	8.12943	2.70981
$10$	1.28479	0.406285
$11$	3.74359	1.12874	0.564368	−	0.825524i	$-0.309120\pi$
$11$	3.74359	1.12874	0.564368	+	0.825524i	$0.309120\pi$
$12$	−6.02471	−1.73918
$13$	−1.49535	−0.414735	−0.207368	−	0.978263i	$-0.566490\pi$
$13$	−1.49535	−0.414735	−0.207368	+	0.978263i	$0.566490\pi$
$14$	1.84912	0.494198
$15$	−9.72930	−2.51209
$16$	2.87321	0.718303
$17$	2.00713	0.486802	0.243401	−	0.969926i	$-0.421737\pi$
$17$	2.00713	0.486802	0.243401	+	0.969926i	$0.421737\pi$
$18$	−3.58134	−0.844130
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	5.26677	1.17769
$21$	−14.0028	−3.05567
$22$	−1.64920	−0.351611
$23$	−7.25048	−1.51183	−0.755915	−	0.654670i	$-0.772808\pi$
$23$	−7.25048	−1.51183	−0.755915	+	0.654670i	$0.772808\pi$
$24$	5.59348	1.14176
$25$	3.50531	0.701062
$26$	0.658762	0.129194
$27$	17.1122	3.29324
$28$	7.58017	1.43252
$29$	0.228630	0.0424554	0.0212277	−	0.999775i	$-0.493242\pi$
$29$	0.228630	0.0424554	0.0212277	+	0.999775i	$0.493242\pi$
$30$	4.28615	0.782540
$31$	−2.96672	−0.532838	−0.266419	−	0.963857i	$-0.585840\pi$
$31$	−2.96672	−0.532838	−0.266419	+	0.963857i	$0.585840\pi$
$32$	−4.61909	−0.816548
$33$	12.4889	2.17404
$34$	−0.884224	−0.151643
$35$	12.2412	2.06914
$36$	−14.6811	−2.44685
$37$	7.91649	1.30146	0.650731	−	0.759308i	$-0.274462\pi$
$37$	7.91649	1.30146	0.650731	+	0.759308i	$0.274462\pi$
$38$	−0.440540	−0.0714651
$39$	−4.98861	−0.798816
$40$	−4.88980	−0.773145
$41$	−7.44308	−1.16241	−0.581207	−	0.813756i	$-0.697419\pi$
$41$	−7.44308	−1.16241	−0.581207	+	0.813756i	$0.697419\pi$
$42$	6.16882	0.951869
$43$	−5.14436	−0.784507	−0.392253	−	0.919857i	$-0.628304\pi$
$43$	−5.14436	−0.784507	−0.392253	+	0.919857i	$0.628304\pi$
$44$	−6.76064	−1.01921
$45$	−23.7085	−3.53426
$46$	3.19413	0.470949
$47$	−0.713111	−0.104018	−0.0520090	−	0.998647i	$-0.516562\pi$
$47$	−0.713111	−0.104018	−0.0520090	+	0.998647i	$0.516562\pi$
$48$	9.58526	1.38351
$49$	10.6181	1.51687
$50$	−1.54423	−0.218387
$51$	6.69596	0.937622
$52$	2.70049	0.374490
$53$	11.7808	1.61822	0.809108	−	0.587659i	$-0.199951\pi$
$53$	11.7808	1.61822	0.809108	+	0.587659i	$0.199951\pi$
$54$	−7.53860	−1.02587
$55$	−10.9178	−1.47215
$56$	−7.03762	−0.940441
$57$	3.33608	0.441875
$58$	−0.100721	−0.0132253
$59$	2.37068	0.308636	0.154318	−	0.988021i	$-0.450682\pi$
$59$	2.37068	0.308636	0.154318	+	0.988021i	$0.450682\pi$
$60$	17.5704	2.26833
$61$	−10.3868	−1.32989	−0.664945	−	0.746892i	$-0.731545\pi$
$61$	−10.3868	−1.32989	−0.664945	+	0.746892i	$0.731545\pi$
$62$	1.30696	0.165984
$63$	−34.1224	−4.29902
$64$	−3.71152	−0.463941
$65$	4.36102	0.540918
$66$	−5.50187	−0.677234
$67$	−3.24158	−0.396022	−0.198011	−	0.980200i	$-0.563448\pi$
$67$	−3.24158	−0.396022	−0.198011	+	0.980200i	$0.563448\pi$
$68$	−3.62473	−0.439563
$69$	−24.1882	−2.91192
$70$	−5.39275	−0.644557
$71$	−7.64803	−0.907654	−0.453827	−	0.891090i	$-0.649942\pi$
$71$	−7.64803	−0.907654	−0.453827	+	0.891090i	$0.649942\pi$
$72$	13.6303	1.60635
$73$	−15.6512	−1.83183	−0.915917	−	0.401368i	$-0.868535\pi$
$73$	−15.6512	−1.83183	−0.915917	+	0.401368i	$0.868535\pi$
$74$	−3.48753	−0.405417
$75$	11.6940	1.35031
$76$	−1.80592	−0.207154
$77$	−15.7133	−1.79070
$78$	2.19768	0.248839
$79$	−14.8461	−1.67031	−0.835157	−	0.550011i	$-0.814623\pi$
$79$	−14.8461	−1.67031	−0.835157	+	0.550011i	$0.814623\pi$
$80$	−8.37939	−0.936844
$81$	32.6993	3.63325
$82$	3.27898	0.362103
$83$	−6.05255	−0.664354	−0.332177	−	0.943217i	$-0.607783\pi$
$83$	−6.05255	−0.664354	−0.332177	+	0.943217i	$0.607783\pi$
$84$	25.2881	2.75915
$85$	−5.85358	−0.634910
$86$	2.26630	0.244381
$87$	0.762726	0.0817728
$88$	6.27674	0.669103
$89$	−2.45406	−0.260130	−0.130065	−	0.991505i	$-0.541519\pi$
$89$	−2.45406	−0.260130	−0.130065	+	0.991505i	$0.541519\pi$
$90$	10.4446	1.10095
$91$	6.27657	0.657964
$92$	13.0938	1.36513
$93$	−9.89720	−1.02629
$94$	0.314154	0.0324025
$95$	−2.91639	−0.299215
$96$	−15.4097	−1.57274
$97$	5.54721	0.563234	0.281617	−	0.959527i	$-0.409129\pi$
$97$	5.54721	0.563234	0.281617	+	0.959527i	$0.409129\pi$
$98$	−4.67771	−0.472520
$99$	30.4332	3.05866
$100$	−6.33032	−0.633032
$101$	13.2280	1.31623	0.658116	−	0.752917i	$-0.271354\pi$
$101$	13.2280	1.31623	0.658116	+	0.752917i	$0.271354\pi$
$102$	−2.94984	−0.292078
$103$	−3.65598	−0.360234	−0.180117	−	0.983645i	$-0.557648\pi$
$103$	−3.65598	−0.360234	−0.180117	+	0.983645i	$0.557648\pi$
$104$	−2.50720	−0.245851
$105$	40.8377	3.98535
$106$	−5.18992	−0.504089
$107$	−15.8082	−1.52823	−0.764116	−	0.645079i	$-0.776824\pi$
$107$	−15.8082	−1.52823	−0.764116	+	0.645079i	$0.776824\pi$
$108$	−30.9033	−2.97367
$109$	−9.81921	−0.940510	−0.470255	−	0.882531i	$-0.655838\pi$
$109$	−9.81921	−0.940510	−0.470255	+	0.882531i	$0.655838\pi$
$110$	4.80971	0.458588
$111$	26.4100	2.50673
$112$	−12.0600	−1.13956
$113$	2.06663	0.194412	0.0972062	−	0.995264i	$-0.469009\pi$
$113$	2.06663	0.194412	0.0972062	+	0.995264i	$0.469009\pi$
$114$	−1.46968	−0.137648
$115$	21.1452	1.97180
$116$	−0.412888	−0.0383357
$117$	−12.1563	−1.12385
$118$	−1.04438	−0.0961430
$119$	−8.42473	−0.772294
$120$	−16.3128	−1.48914
$121$	3.01448	0.274043
$122$	4.57579	0.414273
$123$	−24.8307	−2.23891
$124$	5.35766	0.481132
$125$	4.35909	0.389889
$126$	15.0323	1.33918
$127$	2.11926	0.188054	0.0940269	−	0.995570i	$-0.470026\pi$
$127$	2.11926	0.188054	0.0940269	+	0.995570i	$0.470026\pi$
$128$	10.8733	0.961070
$129$	−17.1620	−1.51103
$130$	−1.92120	−0.168501
$131$	−6.39731	−0.558936	−0.279468	−	0.960155i	$-0.590158\pi$
$131$	−6.39731	−0.558936	−0.279468	+	0.960155i	$0.590158\pi$
$132$	−22.5540	−1.96308
$133$	−4.19739	−0.363960
$134$	1.42805	0.123364
$135$	−49.9057	−4.29520
$136$	3.36529	0.288571
$137$	6.97373	0.595806	0.297903	−	0.954596i	$-0.403713\pi$
$137$	6.97373	0.595806	0.297903	+	0.954596i	$0.403713\pi$
$138$	10.6559	0.907088
$139$	11.2699	0.955901	0.477950	−	0.878387i	$-0.341380\pi$
$139$	11.2699	0.955901	0.477950	+	0.878387i	$0.341380\pi$
$140$	−22.1067	−1.86836
$141$	−2.37899	−0.200347
$142$	3.36926	0.282742
$143$	−5.59798	−0.468127
$144$	23.3575	1.94646
$145$	−0.666772	−0.0553724
$146$	6.89498	0.570633
$147$	35.4229	2.92163
$148$	−14.2966	−1.17517
$149$	−11.2726	−0.923484	−0.461742	−	0.887014i	$-0.652775\pi$
$149$	−11.2726	−0.923484	−0.461742	+	0.887014i	$0.652775\pi$
$150$	−5.15168	−0.420633
$151$	2.64453	0.215209	0.107604	−	0.994194i	$-0.465682\pi$
$151$	2.64453	0.215209	0.107604	+	0.994194i	$0.465682\pi$
$152$	1.67666	0.135995
$153$	16.3168	1.31914
$154$	6.92235	0.557819
$155$	8.65209	0.694953
$156$	9.00904	0.721301
$157$	−2.28467	−0.182336	−0.0911681	−	0.995836i	$-0.529060\pi$
$157$	−2.28467	−0.182336	−0.0911681	+	0.995836i	$0.529060\pi$
$158$	6.54030	0.520318
$159$	39.3017	3.11683
$160$	13.4711	1.06498
$161$	30.4331	2.39847
$162$	−14.4054	−1.13179
$163$	−19.1227	−1.49780	−0.748902	−	0.662681i	$-0.769419\pi$
$163$	−19.1227	−1.49780	−0.748902	+	0.662681i	$0.769419\pi$
$164$	13.4416	1.04962
$165$	−36.4225	−2.83549
$166$	2.66639	0.206952
$167$	−20.6284	−1.59628	−0.798138	−	0.602475i	$-0.794181\pi$
$167$	−20.6284	−1.59628	−0.798138	+	0.602475i	$0.794181\pi$
$168$	−23.4780	−1.81137
$169$	−10.7639	−0.827994
$170$	2.57874	0.197780
$171$	8.12943	0.621673
$172$	9.29032	0.708380
$173$	0.0945027	0.00718491	0.00359245	−	0.999994i	$-0.498856\pi$
$173$	0.0945027	0.00718491	0.00359245	+	0.999994i	$0.498856\pi$
$174$	−0.336012	−0.0254730
$175$	−14.7132	−1.11221
$176$	10.7561	0.810774
$177$	7.90878	0.594460
$178$	1.08111	0.0810329
$179$	−3.37935	−0.252584	−0.126292	−	0.991993i	$-0.540308\pi$
$179$	−3.37935	−0.252584	−0.126292	+	0.991993i	$0.540308\pi$
$180$	42.8158	3.19130
$181$	−13.4940	−1.00300	−0.501500	−	0.865157i	$-0.667218\pi$
$181$	−13.4940	−1.00300	−0.501500	+	0.865157i	$0.667218\pi$
$182$	−2.76508	−0.204962
$183$	−34.6511	−2.56148
$184$	−12.1566	−0.896198
$185$	−23.0875	−1.69743
$186$	4.36012	0.319699
$187$	7.51389	0.549470
$188$	1.28782	0.0939242
$189$	−71.8265	−5.22461
$190$	1.28479	0.0932082
$191$	13.3782	0.968011	0.484006	−	0.875065i	$-0.339181\pi$
$191$	13.3782	0.968011	0.484006	+	0.875065i	$0.339181\pi$
$192$	−12.3819	−0.893589
$193$	−21.8535	−1.57305	−0.786524	−	0.617560i	$-0.788121\pi$
$193$	−21.8535	−1.57305	−0.786524	+	0.617560i	$0.788121\pi$
$194$	−2.44377	−0.175452
$195$	14.5487	1.04185
$196$	−19.1755	−1.36968
$197$	2.70138	0.192465	0.0962326	−	0.995359i	$-0.469321\pi$
$197$	2.70138	0.192465	0.0962326	+	0.995359i	$0.469321\pi$
$198$	−13.4071	−0.952799
$199$	−26.4667	−1.87617	−0.938086	−	0.346402i	$-0.887403\pi$
$199$	−26.4667	−1.87617	−0.938086	+	0.346402i	$0.887403\pi$
$200$	5.87722	0.415583
$201$	−10.8142	−0.762772
$202$	−5.82745	−0.410018
$203$	−0.959648	−0.0673541
$204$	−12.0924	−0.846637
$205$	21.7069	1.51608
$206$	1.61061	0.112216
$207$	−58.9423	−4.09677
$208$	−4.29646	−0.297906
$209$	3.74359	0.258950
$210$	−17.9906	−1.24147
$211$	1.00000	0.0688428
$211$	1.00000	0.0688428
$212$	−21.2752	−1.46119
$213$	−25.5144	−1.74822
$214$	6.96413	0.476058
$215$	15.0029	1.02319
$216$	28.6913	1.95220
$217$	12.4525	0.845329
$218$	4.32576	0.292977
$219$	−52.2136	−3.52827
$220$	19.7166	1.32930
$221$	−3.00137	−0.201894
$222$	−11.6347	−0.780869
$223$	6.93515	0.464412	0.232206	−	0.972667i	$-0.425406\pi$
$223$	6.93515	0.464412	0.232206	+	0.972667i	$0.425406\pi$
$224$	19.3881	1.29542
$225$	28.4962	1.89974
$226$	−0.910435	−0.0605613
$227$	−18.1468	−1.20445	−0.602223	−	0.798328i	$-0.705718\pi$
$227$	−18.1468	−1.20445	−0.602223	+	0.798328i	$0.705718\pi$
$228$	−6.02471	−0.398996
$229$	19.4659	1.28634	0.643171	−	0.765722i	$-0.277618\pi$
$229$	19.4659	1.28634	0.643171	+	0.765722i	$0.277618\pi$
$230$	−9.31532	−0.614234
$231$	−52.4209	−3.44904
$232$	0.383335	0.0251672
$233$	−0.000729422 0	−4.77860e−5 0	−2.38930e−5	−	1.00000i	$-0.500008\pi$
$233$	−0.000729422 0	−4.77860e−5 0	−2.38930e−5		1.00000i	$0.500008\pi$
$234$	5.35536	0.350091
$235$	2.07971	0.135665
$236$	−4.28127	−0.278687
$237$	−49.5277	−3.21717
$238$	3.71144	0.240577
$239$	−19.5772	−1.26635	−0.633173	−	0.774010i	$-0.718248\pi$
$239$	−19.5772	−1.26635	−0.633173	+	0.774010i	$0.718248\pi$
$240$	−27.9543	−1.80444
$241$	−7.85032	−0.505684	−0.252842	−	0.967508i	$-0.581365\pi$
$241$	−7.85032	−0.505684	−0.252842	+	0.967508i	$0.581365\pi$
$242$	−1.32800	−0.0853670
$243$	57.7509	3.70472
$244$	18.7577	1.20084
$245$	−30.9665	−1.97838
$246$	10.9389	0.697441
$247$	−1.49535	−0.0951468
$248$	−4.97418	−0.315861
$249$	−20.1918	−1.27960
$250$	−1.92036	−0.121454
$251$	−5.93972	−0.374912	−0.187456	−	0.982273i	$-0.560024\pi$
$251$	−5.93972	−0.374912	−0.187456	+	0.982273i	$0.560024\pi$
$252$	61.6225	3.88185
$253$	−27.1428	−1.70646
$254$	−0.933620	−0.0585805
$255$	−19.5280	−1.22289
$256$	2.63294	0.164559
$257$	27.0600	1.68795	0.843977	−	0.536380i	$-0.180208\pi$
$257$	27.0600	1.68795	0.843977	+	0.536380i	$0.180208\pi$
$258$	7.56054	0.470699
$259$	−33.2286	−2.06473
$260$	−7.87567	−0.488428
$261$	1.85863	0.115046
$262$	2.81827	0.174113
$263$	22.7088	1.40028	0.700141	−	0.714004i	$-0.253120\pi$
$263$	22.7088	1.40028	0.700141	+	0.714004i	$0.253120\pi$
$264$	20.9397	1.28875
$265$	−34.3574	−2.11056
$266$	1.84912	0.113377
$267$	−8.18695	−0.501033
$268$	5.85404	0.357593
$269$	1.03447	0.0630728	0.0315364	−	0.999503i	$-0.489960\pi$
$269$	1.03447	0.0630728	0.0315364	+	0.999503i	$0.489960\pi$
$270$	21.9855	1.33799
$271$	2.23523	0.135781	0.0678903	−	0.997693i	$-0.478373\pi$
$271$	2.23523	0.135781	0.0678903	+	0.997693i	$0.478373\pi$
$272$	5.76692	0.349671
$273$	20.9391	1.26729
$274$	−3.07221	−0.185599
$275$	13.1224	0.791313
$276$	43.6820	2.62935
$277$	23.3038	1.40019	0.700094	−	0.714051i	$-0.253141\pi$
$277$	23.3038	1.40019	0.700094	+	0.714051i	$0.253141\pi$
$278$	−4.96485	−0.297772
$279$	−24.1177	−1.44389
$280$	20.5244	1.22657
$281$	20.9591	1.25031	0.625157	−	0.780499i	$-0.285035\pi$
$281$	20.9591	1.25031	0.625157	+	0.780499i	$0.285035\pi$
$282$	1.04804	0.0624101
$283$	4.07948	0.242500	0.121250	−	0.992622i	$-0.461310\pi$
$283$	4.07948	0.242500	0.121250	+	0.992622i	$0.461310\pi$
$284$	13.8118	0.819577
$285$	−9.72930	−0.576314
$286$	2.46614	0.145826
$287$	31.2415	1.84413
$288$	−37.5506	−2.21269
$289$	−12.9714	−0.763024
$290$	0.293740	0.0172490
$291$	18.5059	1.08484
$292$	28.2649	1.65408
$293$	−17.9210	−1.04696	−0.523478	−	0.852039i	$-0.675366\pi$
$293$	−17.9210	−1.04696	−0.523478	+	0.852039i	$0.675366\pi$
$294$	−15.6052	−0.910114
$295$	−6.91382	−0.402538
$296$	13.2733	0.771494
$297$	64.0610	3.71719
$298$	4.96602	0.287674
$299$	10.8420	0.627010
$300$	−21.1185	−1.21928
$301$	21.5929	1.24459
$302$	−1.16502	−0.0670394
$303$	44.1295	2.53518
$304$	2.87321	0.164790
$305$	30.2918	1.73451
$306$	−7.18823	−0.410924
$307$	−11.6468	−0.664721	−0.332360	−	0.943152i	$-0.607845\pi$
$307$	−11.6468	−0.664721	−0.332360	+	0.943152i	$0.607845\pi$
$308$	28.3771	1.61693
$309$	−12.1966	−0.693842
$310$	−3.81160	−0.216484
$311$	28.2861	1.60396	0.801980	−	0.597351i	$-0.203780\pi$
$311$	28.2861	1.60396	0.801980	+	0.597351i	$0.203780\pi$
$312$	−8.36421	−0.473530
$313$	23.6543	1.33702	0.668510	−	0.743704i	$-0.266933\pi$
$313$	23.6543	1.33702	0.668510	+	0.743704i	$0.266933\pi$
$314$	1.00649	0.0567994
$315$	99.5141	5.60698
$316$	26.8109	1.50823
$317$	14.0950	0.791655	0.395827	−	0.918325i	$-0.370458\pi$
$317$	14.0950	0.791655	0.395827	+	0.918325i	$0.370458\pi$
$318$	−17.3140	−0.970919
$319$	0.855896	0.0479210
$320$	10.8242	0.605093
$321$	−52.7373	−2.94351
$322$	−13.4070	−0.747144
$323$	2.00713	0.111680
$324$	−59.0524	−3.28069
$325$	−5.24167	−0.290755
$326$	8.42431	0.466579
$327$	−32.7577	−1.81150
$328$	−12.4795	−0.689067
$329$	2.99321	0.165021
$330$	16.0456	0.883281
$331$	−25.0299	−1.37577	−0.687885	−	0.725820i	$-0.741461\pi$
$331$	−25.0299	−1.37577	−0.687885	+	0.725820i	$0.741461\pi$
$332$	10.9304	0.599886
$333$	64.3565	3.52671
$334$	9.08765	0.497254
$335$	9.45370	0.516511
$336$	−40.2331	−2.19490
$337$	10.4091	0.567022	0.283511	−	0.958969i	$-0.408501\pi$
$337$	10.4091	0.567022	0.283511	+	0.958969i	$0.408501\pi$
$338$	4.74195	0.257928
$339$	6.89445	0.374455
$340$	10.5711	0.573300
$341$	−11.1062	−0.601433
$342$	−3.58134	−0.193657
$343$	−15.1866	−0.820001
$344$	−8.62535	−0.465048
$345$	70.5421	3.79786
$346$	−0.0416323	−0.00223816
$347$	−22.6512	−1.21598	−0.607989	−	0.793945i	$-0.708024\pi$
$347$	−22.6512	−1.21598	−0.607989	+	0.793945i	$0.708024\pi$
$348$	−1.37743	−0.0738378
$349$	17.4916	0.936305	0.468152	−	0.883648i	$-0.344920\pi$
$349$	17.4916	0.936305	0.468152	+	0.883648i	$0.344920\pi$
$350$	6.48174	0.346464
$351$	−25.5887	−1.36582
$352$	−17.2920	−0.921666
$353$	−19.8403	−1.05599	−0.527997	−	0.849246i	$-0.677057\pi$
$353$	−19.8403	−1.05599	−0.527997	+	0.849246i	$0.677057\pi$
$354$	−3.48414	−0.185180
$355$	22.3046	1.18381
$356$	4.43185	0.234888
$357$	−28.1056	−1.48750
$358$	1.48874	0.0786823
$359$	−25.6712	−1.35487	−0.677436	−	0.735582i	$-0.736909\pi$
$359$	−25.6712	−1.35487	−0.677436	+	0.735582i	$0.736909\pi$
$360$	−39.7512	−2.09507
$361$	1.00000	0.0526316
$362$	5.94465	0.312444
$363$	10.0565	0.527831
$364$	−11.3350	−0.594116
$365$	45.6449	2.38916
$366$	15.2652	0.797925
$367$	12.0590	0.629476	0.314738	−	0.949179i	$-0.398083\pi$
$367$	12.0590	0.629476	0.314738	+	0.949179i	$0.398083\pi$
$368$	−20.8322	−1.08595
$369$	−60.5080	−3.14992
$370$	10.1710	0.528765
$371$	−49.4486	−2.56725
$372$	17.8736	0.926703
$373$	9.98729	0.517122	0.258561	−	0.965995i	$-0.416752\pi$
$373$	9.98729	0.517122	0.258561	+	0.965995i	$0.416752\pi$
$374$	−3.31017	−0.171165
$375$	14.5423	0.750960
$376$	−1.19565	−0.0616608
$377$	−0.341881	−0.0176078
$378$	31.6425	1.62751
$379$	−19.3438	−0.993626	−0.496813	−	0.867858i	$-0.665496\pi$
$379$	−19.3438	−0.993626	−0.496813	+	0.867858i	$0.665496\pi$
$380$	5.26677	0.270180
$381$	7.07002	0.362208
$382$	−5.89363	−0.301544
$383$	19.1095	0.976449	0.488224	−	0.872718i	$-0.337645\pi$
$383$	19.1095	0.976449	0.488224	+	0.872718i	$0.337645\pi$
$384$	36.2741	1.85110
$385$	45.8261	2.33552
$386$	9.62734	0.490019
$387$	−41.8207	−2.12586
$388$	−10.0178	−0.508579
$389$	−11.1814	−0.566922	−0.283461	−	0.958984i	$-0.591483\pi$
$389$	−11.1814	−0.566922	−0.283461	+	0.958984i	$0.591483\pi$
$390$	−6.40929	−0.324547
$391$	−14.5527	−0.735961
$392$	17.8030	0.899187
$393$	−21.3419	−1.07656
$394$	−1.19007	−0.0599547
$395$	43.2969	2.17850
$396$	−54.9601	−2.76185
$397$	26.6199	1.33602	0.668008	−	0.744154i	$-0.267147\pi$
$397$	26.6199	1.33602	0.668008	+	0.744154i	$0.267147\pi$
$398$	11.6596	0.584445
$399$	−14.0028	−0.701019
$400$	10.0715	0.503575
$401$	29.3578	1.46606	0.733030	−	0.680196i	$-0.238106\pi$
$401$	29.3578	1.46606	0.733030	+	0.680196i	$0.238106\pi$
$402$	4.76408	0.237610
$403$	4.43628	0.220987
$404$	−23.8887	−1.18851
$405$	−95.3637	−4.73866
$406$	0.422764	0.0209814
$407$	29.6361	1.46901
$408$	11.2269	0.555813
$409$	19.6346	0.970869	0.485435	−	0.874273i	$-0.338661\pi$
$409$	19.6346	0.970869	0.485435	+	0.874273i	$0.338661\pi$
$410$	−9.56277	−0.472271
$411$	23.2649	1.14757
$412$	6.60242	0.325278
$413$	−9.95068	−0.489641
$414$	25.9664	1.27618
$415$	17.6516	0.866482
$416$	6.90716	0.338651
$417$	37.5973	1.84115
$418$	−1.64920	−0.0806652
$419$	−16.5136	−0.806743	−0.403372	−	0.915036i	$-0.632162\pi$
$419$	−16.5136	−0.806743	−0.403372	+	0.915036i	$0.632162\pi$
$420$	−73.7498	−3.59862
$421$	−17.4047	−0.848251	−0.424125	−	0.905603i	$-0.639418\pi$
$421$	−17.4047	−0.848251	−0.424125	+	0.905603i	$0.639418\pi$
$422$	−0.440540	−0.0214452
$423$	−5.79718	−0.281869
$424$	19.7524	0.959263
$425$	7.03563	0.341278
$426$	11.2401	0.544586
$427$	43.5974	2.10982
$428$	28.5483	1.37994
$429$	−18.6753	−0.901652
$430$	−6.60940	−0.318733
$431$	34.8027	1.67639	0.838194	−	0.545372i	$-0.183611\pi$
$431$	34.8027	1.67639	0.838194	+	0.545372i	$0.183611\pi$
$432$	49.1669	2.36554
$433$	28.1802	1.35426	0.677128	−	0.735865i	$-0.263224\pi$
$433$	28.1802	1.35426	0.677128	+	0.735865i	$0.263224\pi$
$434$	−5.48582	−0.263328
$435$	−2.22440	−0.106652
$436$	17.7328	0.849245
$437$	−7.25048	−0.346838
$438$	23.0022	1.09909
$439$	34.9849	1.66974	0.834869	−	0.550448i	$-0.185543\pi$
$439$	34.9849	1.66974	0.834869	+	0.550448i	$0.185543\pi$
$440$	−18.3054	−0.872676
$441$	86.3191	4.11043
$442$	1.32222	0.0628918
$443$	−2.64371	−0.125606	−0.0628032	−	0.998026i	$-0.520004\pi$
$443$	−2.64371	−0.125606	−0.0628032	+	0.998026i	$0.520004\pi$
$444$	−47.6945	−2.26348
$445$	7.15700	0.339274
$446$	−3.05521	−0.144669
$447$	−37.6062	−1.77871
$448$	15.5787	0.736026
$449$	−35.5126	−1.67595	−0.837973	−	0.545712i	$-0.816259\pi$
$449$	−35.5126	−1.67595	−0.837973	+	0.545712i	$0.816259\pi$
$450$	−12.5537	−0.591787
$451$	−27.8639	−1.31206
$452$	−3.73218	−0.175547
$453$	8.82235	0.414510
$454$	7.99440	0.375196
$455$	−18.3049	−0.858147
$456$	5.59348	0.261939
$457$	22.5157	1.05324	0.526619	−	0.850101i	$-0.323459\pi$
$457$	22.5157	1.05324	0.526619	+	0.850101i	$0.323459\pi$
$458$	−8.57551	−0.400707
$459$	34.3464	1.60315
$460$	−38.1867	−1.78046
$461$	−33.9385	−1.58067	−0.790336	−	0.612674i	$-0.790094\pi$
$461$	−33.9385	−1.58067	−0.790336	+	0.612674i	$0.790094\pi$
$462$	23.0935	1.07441
$463$	−9.33970	−0.434053	−0.217026	−	0.976166i	$-0.569636\pi$
$463$	−9.33970	−0.434053	−0.217026	+	0.976166i	$0.569636\pi$
$464$	0.656901	0.0304959
$465$	28.8641	1.33854
$466$	0.000321340 0	1.48858e−5 0
$467$	−32.5752	−1.50740	−0.753700	−	0.657218i	$-0.771733\pi$
$467$	−32.5752	−1.50740	−0.753700	+	0.657218i	$0.771733\pi$
$468$	21.9534	1.01480
$469$	13.6062	0.628275
$470$	−0.916195	−0.0422609
$471$	−7.62183	−0.351195
$472$	3.97483	0.182956
$473$	−19.2584	−0.885501
$474$	21.8190	1.00218
$475$	3.50531	0.160835
$476$	15.2144	0.697352
$477$	95.7711	4.38506
$478$	8.62457	0.394479
$479$	19.2900	0.881382	0.440691	−	0.897659i	$-0.354734\pi$
$479$	19.2900	0.881382	0.440691	+	0.897659i	$0.354734\pi$
$480$	44.9405	2.05124
$481$	−11.8379	−0.539763
$482$	3.45838	0.157525
$483$	101.527	4.61965
$484$	−5.44392	−0.247451
$485$	−16.1778	−0.734596
$486$	−25.4416	−1.15405
$487$	−26.3555	−1.19428	−0.597140	−	0.802137i	$-0.703696\pi$
$487$	−26.3555	−1.19428	−0.597140	+	0.802137i	$0.703696\pi$
$488$	−17.4151	−0.788345
$489$	−63.7948	−2.88490
$490$	13.6420	0.616283
$491$	−19.6692	−0.887660	−0.443830	−	0.896111i	$-0.646381\pi$
$491$	−19.6692	−0.887660	−0.443830	+	0.896111i	$0.646381\pi$
$492$	44.8424	2.02165
$493$	0.458890	0.0206674
$494$	0.658762	0.0296391
$495$	−88.7551	−3.98925
$496$	−8.52400	−0.382739
$497$	32.1018	1.43996
$498$	8.89530	0.398608
$499$	17.1194	0.766370	0.383185	−	0.923672i	$-0.374827\pi$
$499$	17.1194	0.766370	0.383185	+	0.923672i	$0.374827\pi$
$500$	−7.87219	−0.352055
$501$	−68.8180	−3.07456
$502$	2.61669	0.116788
$503$	−39.7108	−1.77062	−0.885308	−	0.465005i	$-0.846052\pi$
$503$	−39.7108	−1.77062	−0.885308	+	0.465005i	$0.846052\pi$
$504$	−57.2118	−2.54841
$505$	−38.5779	−1.71669
$506$	11.9575	0.531577
$507$	−35.9093	−1.59479
$508$	−3.82722	−0.169806
$509$	−7.22687	−0.320325	−0.160163	−	0.987091i	$-0.551202\pi$
$509$	−7.22687	−0.320325	−0.160163	+	0.987091i	$0.551202\pi$
$510$	8.60288	0.380942
$511$	65.6942	2.90614
$512$	−22.9064	−1.01233
$513$	17.1122	0.755521
$514$	−11.9210	−0.525813
$515$	10.6622	0.469835
$516$	30.9932	1.36440
$517$	−2.66960	−0.117409
$518$	14.6385	0.643181
$519$	0.315269	0.0138388
$520$	7.31196	0.320651
$521$	−19.5942	−0.858438	−0.429219	−	0.903200i	$-0.641211\pi$
$521$	−19.5942	−0.858438	−0.429219	+	0.903200i	$0.641211\pi$
$522$	−0.818800	−0.0358379
$523$	22.7046	0.992803	0.496401	−	0.868093i	$-0.334654\pi$
$523$	22.7046	0.992803	0.496401	+	0.868093i	$0.334654\pi$
$524$	11.5531	0.504698
$525$	−49.0843	−2.14221
$526$	−10.0041	−0.436201
$527$	−5.95460	−0.259386
$528$	35.8833	1.56162
$529$	29.5695	1.28563
$530$	15.1358	0.657457
$531$	19.2723	0.836345
$532$	7.58017	0.328642
$533$	11.1300	0.482094
$534$	3.60668	0.156076
$535$	46.1027	1.99319
$536$	−5.43504	−0.234758
$537$	−11.2738	−0.486499
$538$	−0.455726	−0.0196477
$539$	39.7499	1.71215
$540$	90.1259	3.87840
$541$	29.5684	1.27125	0.635623	−	0.772000i	$-0.280743\pi$
$541$	29.5684	1.27125	0.635623	+	0.772000i	$0.280743\pi$
$542$	−0.984710	−0.0422969
$543$	−45.0170	−1.93187
$544$	−9.27114	−0.397497
$545$	28.6366	1.22666
$546$	−9.22454	−0.394774
$547$	6.37441	0.272550	0.136275	−	0.990671i	$-0.456487\pi$
$547$	6.37441	0.272550	0.136275	+	0.990671i	$0.456487\pi$
$548$	−12.5940	−0.537990
$549$	−84.4385	−3.60375
$550$	−5.78097	−0.246501
$551$	0.228630	0.00973995
$552$	−40.5555	−1.72615
$553$	62.3148	2.64990
$554$	−10.2663	−0.436171
$555$	−77.0219	−3.26940
$556$	−20.3526	−0.863142
$557$	33.2948	1.41075	0.705374	−	0.708836i	$-0.250779\pi$
$557$	33.2948	1.41075	0.705374	+	0.708836i	$0.250779\pi$
$558$	10.6248	0.449785
$559$	7.69261	0.325363
$560$	35.1716	1.48627
$561$	25.0669	1.05833
$562$	−9.23333	−0.389484
$563$	−28.8929	−1.21769	−0.608845	−	0.793289i	$-0.708367\pi$
$563$	−28.8929	−1.21769	−0.608845	+	0.793289i	$0.708367\pi$
$564$	4.29628	0.180906
$565$	−6.02710	−0.253562
$566$	−1.79717	−0.0755408
$567$	−137.252	−5.76403
$568$	−12.8232	−0.538048
$569$	−27.0480	−1.13391	−0.566955	−	0.823749i	$-0.691879\pi$
$569$	−27.0480	−1.13391	−0.566955	+	0.823749i	$0.691879\pi$
$570$	4.28615	0.179527
$571$	5.64000	0.236027	0.118013	−	0.993012i	$-0.462347\pi$
$571$	5.64000	0.236027	0.118013	+	0.993012i	$0.462347\pi$
$572$	10.1095	0.422701
$573$	44.6307	1.86447
$574$	−13.7632	−0.574463
$575$	−25.4152	−1.05989
$576$	−30.1726	−1.25719
$577$	−38.7527	−1.61330	−0.806648	−	0.591032i	$-0.798721\pi$
$577$	−38.7527	−1.61330	−0.806648	+	0.591032i	$0.798721\pi$
$578$	5.71443	0.237689
$579$	−72.9049	−3.02983
$580$	1.20414	0.0499992
$581$	25.4049	1.05397
$582$	−8.15261	−0.337936
$583$	44.1025	1.82654
$584$	−26.2418	−1.08589
$585$	35.4526	1.46578
$586$	7.89493	0.326136
$587$	12.9999	0.536565	0.268282	−	0.963340i	$-0.413544\pi$
$587$	12.9999	0.536565	0.268282	+	0.963340i	$0.413544\pi$
$588$	−63.9710	−2.63812
$589$	−2.96672	−0.122241
$590$	3.04582	0.125394
$591$	9.01201	0.370705
$592$	22.7457	0.934844
$593$	48.1817	1.97859	0.989293	−	0.145941i	$-0.0466208\pi$
$593$	48.1817	1.97859	0.989293	+	0.145941i	$0.0466208\pi$
$594$	−28.2214	−1.15794
$595$	24.5698	1.00726
$596$	20.3574	0.833871
$597$	−88.2949	−3.61367
$598$	−4.77634	−0.195319
$599$	14.6204	0.597373	0.298686	−	0.954351i	$-0.403452\pi$
$599$	14.6204	0.597373	0.298686	+	0.954351i	$0.403452\pi$
$600$	19.6069	0.800448
$601$	−7.80232	−0.318263	−0.159132	−	0.987257i	$-0.550869\pi$
$601$	−7.80232	−0.318263	−0.159132	+	0.987257i	$0.550869\pi$
$602$	−9.51254	−0.387702
$603$	−26.3522	−1.07314
$604$	−4.77582	−0.194325
$605$	−8.79138	−0.357421
$606$	−19.4408	−0.789730
$607$	−2.34298	−0.0950986	−0.0475493	−	0.998869i	$-0.515141\pi$
$607$	−2.34298	−0.0950986	−0.0475493	+	0.998869i	$0.515141\pi$
$608$	−4.61909	−0.187329
$609$	−3.20146	−0.129730
$610$	−13.3448	−0.540314
$611$	1.06635	0.0431399
$612$	−29.4670	−1.19113
$613$	37.6079	1.51897	0.759484	−	0.650526i	$-0.225451\pi$
$613$	37.6079	1.51897	0.759484	+	0.650526i	$0.225451\pi$
$614$	5.13091	0.207067
$615$	72.4159	2.92009
$616$	−26.3460	−1.06151
$617$	22.3167	0.898438	0.449219	−	0.893422i	$-0.351702\pi$
$617$	22.3167	0.898438	0.449219	+	0.893422i	$0.351702\pi$
$618$	5.37311	0.216138
$619$	38.9033	1.56366	0.781829	−	0.623493i	$-0.214287\pi$
$619$	38.9033	1.56366	0.781829	+	0.623493i	$0.214287\pi$
$620$	−15.6250	−0.627516
$621$	−124.072	−4.97882
$622$	−12.4612	−0.499648
$623$	10.3007	0.412688
$624$	−14.3333	−0.573792
$625$	−30.2394	−1.20957
$626$	−10.4207	−0.416494
$627$	12.4889	0.498759
$628$	4.12593	0.164643
$629$	15.8895	0.633554
$630$	−43.8400	−1.74663
$631$	35.3180	1.40599	0.702994	−	0.711196i	$-0.251846\pi$
$631$	35.3180	1.40599	0.702994	+	0.711196i	$0.251846\pi$
$632$	−24.8919	−0.990146
$633$	3.33608	0.132597
$634$	−6.20942	−0.246608
$635$	−6.18058	−0.245269
$636$	−70.9758	−2.81437
$637$	−15.8778	−0.629101
$638$	−0.377057	−0.0149278
$639$	−62.1741	−2.45957
$640$	−31.7106	−1.25347
$641$	6.51631	0.257379	0.128689	−	0.991685i	$-0.458923\pi$
$641$	6.51631	0.257379	0.128689	+	0.991685i	$0.458923\pi$
$642$	23.2329	0.916929
$643$	−40.6573	−1.60337	−0.801684	−	0.597748i	$-0.796062\pi$
$643$	−40.6573	−1.60337	−0.801684	+	0.597748i	$0.796062\pi$
$644$	−54.9599	−2.16572
$645$	50.0510	1.97075
$646$	−0.884224	−0.0347893
$647$	43.3432	1.70400	0.851998	−	0.523545i	$-0.175391\pi$
$647$	43.3432	1.70400	0.851998	+	0.523545i	$0.175391\pi$
$648$	54.8257	2.15376
$649$	8.87486	0.348369
$650$	2.30917	0.0905729
$651$	41.5424	1.62818
$652$	34.5341	1.35246
$653$	−6.38596	−0.249902	−0.124951	−	0.992163i	$-0.539877\pi$
$653$	−6.38596	−0.249902	−0.124951	+	0.992163i	$0.539877\pi$
$654$	14.4311	0.564300
$655$	18.6570	0.728990
$656$	−21.3855	−0.834965
$657$	−127.235	−4.96392
$658$	−1.31863	−0.0514055
$659$	8.52791	0.332200	0.166100	−	0.986109i	$-0.446883\pi$
$659$	8.52791	0.332200	0.166100	+	0.986109i	$0.446883\pi$
$660$	65.7763	2.56034
$661$	22.1656	0.862144	0.431072	−	0.902318i	$-0.358136\pi$
$661$	22.1656	0.862144	0.431072	+	0.902318i	$0.358136\pi$
$662$	11.0267	0.428565
$663$	−10.0128	−0.388865
$664$	−10.1481	−0.393822
$665$	12.2412	0.474694
$666$	−28.3516	−1.09860
$667$	−1.65767	−0.0641854
$668$	37.2534	1.44138
$669$	23.1362	0.894497
$670$	−4.16473	−0.160898
$671$	−38.8838	−1.50109
$672$	64.6804	2.49510
$673$	3.75977	0.144928	0.0724642	−	0.997371i	$-0.476914\pi$
$673$	3.75977	0.144928	0.0724642	+	0.997371i	$0.476914\pi$
$674$	−4.58565	−0.176633
$675$	59.9835	2.30876
$676$	19.4388	0.747648
$677$	10.0204	0.385114	0.192557	−	0.981286i	$-0.438322\pi$
$677$	10.0204	0.385114	0.192557	+	0.981286i	$0.438322\pi$
$678$	−3.03728	−0.116646
$679$	−23.2838	−0.893551
$680$	−9.81448	−0.376368
$681$	−60.5392	−2.31987
$682$	4.89272	0.187352
$683$	−4.86316	−0.186084	−0.0930418	−	0.995662i	$-0.529659\pi$
$683$	−4.86316	−0.186084	−0.0930418	+	0.995662i	$0.529659\pi$
$684$	−14.6811	−0.561347
$685$	−20.3381	−0.777079
$686$	6.69032	0.255438
$687$	64.9398	2.47761
$688$	−14.7808	−0.563513
$689$	−17.6164	−0.671132
$690$	−31.0766	−1.18307
$691$	−37.2086	−1.41548	−0.707741	−	0.706472i	$-0.750285\pi$
$691$	−37.2086	−1.41548	−0.707741	+	0.706472i	$0.750285\pi$
$692$	−0.170665	−0.00648770
$693$	−127.740	−4.85245
$694$	9.97876	0.378788
$695$	−32.8674	−1.24673
$696$	1.27884	0.0484741
$697$	−14.9393	−0.565865
$698$	−7.70576	−0.291667
$699$	−0.00243341	−9.20400e−5 0
$700$	26.5709	1.00428
$701$	−29.4718	−1.11313	−0.556567	−	0.830803i	$-0.687882\pi$
$701$	−29.4718	−1.11313	−0.556567	+	0.830803i	$0.687882\pi$
$702$	11.2728	0.425466
$703$	7.91649	0.298576
$704$	−13.8944	−0.523666
$705$	6.93807	0.261303
$706$	8.74046	0.328952
$707$	−55.5230	−2.08816
$708$	−14.2827	−0.536775
$709$	−26.7054	−1.00294	−0.501472	−	0.865174i	$-0.667208\pi$
$709$	−26.7054	−1.00294	−0.501472	+	0.865174i	$0.667208\pi$
$710$	−9.82608	−0.368766
$711$	−120.690	−4.52623
$712$	−4.11464	−0.154203
$713$	21.5101	0.805561
$714$	12.3816	0.463371
$715$	16.3259	0.610553
$716$	6.10285	0.228074
$717$	−65.3112	−2.43909
$718$	11.3092	0.422055
$719$	−41.7520	−1.55709	−0.778543	−	0.627592i	$-0.784041\pi$
$719$	−41.7520	−1.55709	−0.778543	+	0.627592i	$0.784041\pi$
$720$	−68.1196	−2.53867
$721$	15.3456	0.571499
$722$	−0.440540	−0.0163952
$723$	−26.1893	−0.973990
$724$	24.3691	0.905671
$725$	0.801417	0.0297639
$726$	−4.43031	−0.164424
$727$	36.3123	1.34675	0.673374	−	0.739302i	$-0.264844\pi$
$727$	36.3123	1.34675	0.673374	+	0.739302i	$0.264844\pi$
$728$	10.5237	0.390034
$729$	94.5637	3.50236
$730$	−20.1084	−0.744247
$731$	−10.3254	−0.381899
$732$	62.5772	2.31292
$733$	18.2870	0.675446	0.337723	−	0.941245i	$-0.390343\pi$
$733$	18.2870	0.675446	0.337723	+	0.941245i	$0.390343\pi$
$734$	−5.31249	−0.196088
$735$	−103.307	−3.81053
$736$	33.4907	1.23448
$737$	−12.1351	−0.447004
$738$	26.6562	0.981229
$739$	−21.6082	−0.794872	−0.397436	−	0.917630i	$-0.630100\pi$
$739$	−21.6082	−0.794872	−0.397436	+	0.917630i	$0.630100\pi$
$740$	41.6943	1.53271
$741$	−4.98861	−0.183261
$742$	21.7841	0.799720
$743$	7.05135	0.258689	0.129344	−	0.991600i	$-0.458713\pi$
$743$	7.05135	0.258689	0.129344	+	0.991600i	$0.458713\pi$
$744$	−16.5943	−0.608376
$745$	32.8752	1.20445
$746$	−4.39981	−0.161088
$747$	−49.2037	−1.80027
$748$	−13.5695	−0.496151
$749$	66.3530	2.42449
$750$	−6.40646	−0.233931
$751$	25.7144	0.938333	0.469166	−	0.883110i	$-0.344554\pi$
$751$	25.7144	0.938333	0.469166	+	0.883110i	$0.344554\pi$
$752$	−2.04892	−0.0747163
$753$	−19.8154	−0.722113
$754$	0.150612	0.00548498
$755$	−7.71246	−0.280685
$756$	129.713	4.71762
$757$	−7.92463	−0.288026	−0.144013	−	0.989576i	$-0.546001\pi$
$757$	−7.92463	−0.288026	−0.144013	+	0.989576i	$0.546001\pi$
$758$	8.52174	0.309523
$759$	−90.5507	−3.28678
$760$	−4.88980	−0.177372
$761$	−29.3325	−1.06330	−0.531652	−	0.846963i	$-0.678428\pi$
$761$	−29.3325	−1.06330	−0.531652	+	0.846963i	$0.678428\pi$
$762$	−3.11463	−0.112831
$763$	41.2151	1.49209
$764$	−24.1600	−0.874078
$765$	−47.5862	−1.72048
$766$	−8.41850	−0.304173
$767$	−3.54500	−0.128002
$768$	8.78369	0.316954
$769$	−8.61286	−0.310588	−0.155294	−	0.987868i	$-0.549632\pi$
$769$	−8.61286	−0.310588	−0.155294	+	0.987868i	$0.549632\pi$
$770$	−20.1883	−0.727534
$771$	90.2742	3.25114
$772$	39.4657	1.42040
$773$	−39.0702	−1.40526	−0.702628	−	0.711557i	$-0.747990\pi$
$773$	−39.0702	−1.40526	−0.702628	+	0.711557i	$0.747990\pi$
$774$	18.4237	0.662226
$775$	−10.3993	−0.373552
$776$	9.30080	0.333879
$777$	−110.853	−3.97684
$778$	4.92588	0.176601
$779$	−7.44308	−0.266676
$780$	−26.2739	−0.940755
$781$	−28.6311	−1.02450
$782$	6.41105	0.229259
$783$	3.91235	0.139816
$784$	30.5081	1.08957
$785$	6.66297	0.237812
$786$	9.40198	0.335358
$787$	52.7162	1.87913	0.939565	−	0.342370i	$-0.111230\pi$
$787$	52.7162	1.87913	0.939565	+	0.342370i	$0.111230\pi$
$788$	−4.87848	−0.173789
$789$	75.7583	2.69707
$790$	−19.0740	−0.678624
$791$	−8.67447	−0.308429
$792$	51.0263	1.81314
$793$	15.5319	0.551552
$794$	−11.7272	−0.416181
$795$	−114.619	−4.06511
$796$	47.7968	1.69411
$797$	−38.0374	−1.34736	−0.673678	−	0.739025i	$-0.735286\pi$
$797$	−38.0374	−1.34736	−0.673678	+	0.739025i	$0.735286\pi$
$798$	6.16882	0.218374
$799$	−1.43131	−0.0506361
$800$	−16.1913	−0.572451
$801$	−19.9501	−0.704903
$802$	−12.9333	−0.456691
$803$	−58.5917	−2.06766
$804$	19.5296	0.688754
$805$	−88.7548	−3.12819
$806$	−1.95436	−0.0688394
$807$	3.45108	0.121484
$808$	22.1788	0.780249
$809$	29.3930	1.03340	0.516701	−	0.856166i	$-0.327160\pi$
$809$	29.3930	1.03340	0.516701	+	0.856166i	$0.327160\pi$
$810$	42.0116	1.47614
$811$	29.2050	1.02553	0.512763	−	0.858530i	$-0.328622\pi$
$811$	29.2050	1.02553	0.512763	+	0.858530i	$0.328622\pi$
$812$	1.73305	0.0608182
$813$	7.45691	0.261525
$814$	−13.0559	−0.457609
$815$	55.7691	1.95351
$816$	19.2389	0.673496
$817$	−5.14436	−0.179978
$818$	−8.64984	−0.302435
$819$	51.0249	1.78296
$820$	−39.2010	−1.36896
$821$	20.0912	0.701188	0.350594	−	0.936528i	$-0.385980\pi$
$821$	20.0912	0.701188	0.350594	+	0.936528i	$0.385980\pi$
$822$	−10.2491	−0.357480
$823$	−33.9428	−1.18317	−0.591586	−	0.806242i	$-0.701498\pi$
$823$	−33.9428	−1.18317	−0.591586	+	0.806242i	$0.701498\pi$
$824$	−6.12985	−0.213543
$825$	43.7775	1.52414
$826$	4.38367	0.152528
$827$	−21.5715	−0.750116	−0.375058	−	0.927001i	$-0.622377\pi$
$827$	−21.5715	−0.750116	−0.375058	+	0.927001i	$0.622377\pi$
$828$	106.445	3.69923
$829$	−46.9679	−1.63126	−0.815631	−	0.578572i	$-0.803610\pi$
$829$	−46.9679	−1.63126	−0.815631	+	0.578572i	$0.803610\pi$
$830$	−7.77623	−0.269917
$831$	77.7432	2.69688
$832$	5.55003	0.192413
$833$	21.3120	0.738416
$834$	−16.5631	−0.573534
$835$	60.1604	2.08194
$836$	−6.76064	−0.233822
$837$	−50.7670	−1.75476
$838$	7.27492	0.251308
$839$	−35.9253	−1.24028	−0.620140	−	0.784491i	$-0.712924\pi$
$839$	−35.9253	−1.24028	−0.620140	+	0.784491i	$0.712924\pi$
$840$	68.4710	2.36248
$841$	−28.9477	−0.998198
$842$	7.66745	0.264238
$843$	69.9212	2.40821
$844$	−1.80592	−0.0621625
$845$	31.3918	1.07991
$846$	2.55389	0.0878046
$847$	−12.6529	−0.434760
$848$	33.8487	1.16237
$849$	13.6095	0.467075
$850$	−3.09948	−0.106311
$851$	−57.3984	−1.96759
$852$	46.0771	1.57858
$853$	15.7491	0.539240	0.269620	−	0.962967i	$-0.413102\pi$
$853$	15.7491	0.539240	0.269620	+	0.962967i	$0.413102\pi$
$854$	−19.2064	−0.657229
$855$	−23.7085	−0.810815
$856$	−26.5050	−0.905921
$857$	8.45696	0.288884	0.144442	−	0.989513i	$-0.453861\pi$
$857$	8.45696	0.288884	0.144442	+	0.989513i	$0.453861\pi$
$858$	8.22723	0.280873
$859$	−6.26472	−0.213749	−0.106875	−	0.994272i	$-0.534084\pi$
$859$	−6.26472	−0.213749	−0.106875	+	0.994272i	$0.534084\pi$
$860$	−27.0942	−0.923903
$861$	104.224	3.55195
$862$	−15.3320	−0.522210
$863$	−0.201182	−0.00684831	−0.00342416	−	0.999994i	$-0.501090\pi$
$863$	−0.201182	−0.00684831	−0.00342416	+	0.999994i	$0.501090\pi$
$864$	−79.0427	−2.68909
$865$	−0.275606	−0.00937090
$866$	−12.4145	−0.421863
$867$	−43.2737	−1.46965
$868$	−22.4882	−0.763300
$869$	−55.5777	−1.88534
$870$	0.979940	0.0332231
$871$	4.84729	0.164244
$872$	−16.4635	−0.557525
$873$	45.0956	1.52626
$874$	3.19413	0.108043
$875$	−18.2968	−0.618546
$876$	94.2938	3.18589
$877$	−34.8835	−1.17793	−0.588966	−	0.808158i	$-0.700465\pi$
$877$	−34.8835	−1.17793	−0.588966	+	0.808158i	$0.700465\pi$
$878$	−15.4123	−0.520139
$879$	−59.7859	−2.01653
$880$	−31.3690	−1.05745
$881$	−14.7674	−0.497527	−0.248763	−	0.968564i	$-0.580024\pi$
$881$	−14.7674	−0.497527	−0.248763	+	0.968564i	$0.580024\pi$
$882$	−38.0271	−1.28044
$883$	51.6774	1.73908	0.869541	−	0.493860i	$-0.164415\pi$
$883$	51.6774	1.73908	0.869541	+	0.493860i	$0.164415\pi$
$884$	5.42024	0.182303
$885$	−23.0650	−0.775323
$886$	1.16466	0.0391275
$887$	−4.90273	−0.164618	−0.0823088	−	0.996607i	$-0.526229\pi$
$887$	−4.90273	−0.164618	−0.0823088	+	0.996607i	$0.526229\pi$
$888$	44.2807	1.48596
$889$	−8.89537	−0.298341
$890$	−3.15295	−0.105687
$891$	122.413	4.10098
$892$	−12.5244	−0.419346
$893$	−0.713111	−0.0238633
$894$	16.5670	0.554085
$895$	9.85549	0.329433
$896$	−45.6393	−1.52470
$897$	36.1698	1.20767
$898$	15.6448	0.522072
$899$	−0.678279	−0.0226219
$900$	−51.4619	−1.71540
$901$	23.6456	0.787751
$902$	12.2752	0.408718
$903$	72.0356	2.39719
$904$	3.46505	0.115246
$905$	39.3537	1.30816
$906$	−3.88660	−0.129124
$907$	−14.7968	−0.491318	−0.245659	−	0.969356i	$-0.579004\pi$
$907$	−14.7968	−0.491318	−0.245659	+	0.969356i	$0.579004\pi$
$908$	32.7717	1.08757
$909$	107.536	3.56674
$910$	8.06405	0.267321
$911$	−16.5739	−0.549117	−0.274558	−	0.961570i	$-0.588532\pi$
$911$	−16.5739	−0.549117	−0.274558	+	0.961570i	$0.588532\pi$
$912$	9.58526	0.317400
$913$	−22.6583	−0.749879
$914$	−9.91907	−0.328094
$915$	101.056	3.34081
$916$	−35.1539	−1.16152
$917$	26.8520	0.886732
$918$	−15.1310	−0.499397
$919$	−23.6716	−0.780854	−0.390427	−	0.920634i	$-0.627673\pi$
$919$	−23.6716	−0.780854	−0.390427	+	0.920634i	$0.627673\pi$
$920$	35.4534	1.16886
$921$	−38.8548	−1.28031
$922$	14.9513	0.492393
$923$	11.4365	0.376436
$924$	94.6682	3.11435
$925$	27.7497	0.912406
$926$	4.11452	0.135211
$927$	−29.7210	−0.976166
$928$	−1.05606	−0.0346669
$929$	22.4380	0.736168	0.368084	−	0.929793i	$-0.380014\pi$
$929$	22.4380	0.736168	0.368084	+	0.929793i	$0.380014\pi$
$930$	−12.7158	−0.416967
$931$	10.6181	0.347994
$932$	0.00131728	4.31490e−5 0
$933$	94.3648	3.08937
$934$	14.3507	0.469569
$935$	−21.9134	−0.716645
$936$	−20.3821	−0.666209
$937$	52.6002	1.71837	0.859187	−	0.511662i	$-0.170970\pi$
$937$	52.6002	1.71837	0.859187	+	0.511662i	$0.170970\pi$
$938$	−5.99407	−0.195713
$939$	78.9126	2.57521
$940$	−3.75579	−0.122500
$941$	−31.0389	−1.01184	−0.505920	−	0.862580i	$-0.668847\pi$
$941$	−31.0389	−1.01184	−0.505920	+	0.862580i	$0.668847\pi$
$942$	3.35772	0.109401
$943$	53.9659	1.75737
$944$	6.81146	0.221694
$945$	209.474	6.81418
$946$	8.48409	0.275841
$947$	6.10034	0.198234	0.0991172	−	0.995076i	$-0.468398\pi$
$947$	6.10034	0.198234	0.0991172	+	0.995076i	$0.468398\pi$
$948$	89.4433	2.90498
$949$	23.4040	0.759726
$950$	−1.54423	−0.0501015
$951$	47.0221	1.52480
$952$	−14.1254	−0.457808
$953$	8.52781	0.276243	0.138121	−	0.990415i	$-0.455894\pi$
$953$	8.52781	0.276243	0.138121	+	0.990415i	$0.455894\pi$
$954$	−42.1910	−1.36599
$955$	−39.0160	−1.26253
$956$	35.3550	1.14346
$957$	2.85534	0.0922999
$958$	−8.49801	−0.274558
$959$	−29.2715	−0.945226
$960$	36.1105	1.16546
$961$	−22.1986	−0.716084
$962$	5.21508	0.168141
$963$	−128.511	−4.14122
$964$	14.1771	0.456613
$965$	63.7332	2.05164
$966$	−44.7269	−1.43906
$967$	−26.4340	−0.850062	−0.425031	−	0.905179i	$-0.639737\pi$
$967$	−26.4340	−0.850062	−0.425031	+	0.905179i	$0.639737\pi$
$968$	5.05426	0.162450
$969$	6.69596	0.215105
$970$	7.12697	0.228833
$971$	32.7344	1.05050	0.525248	−	0.850949i	$-0.323973\pi$
$971$	32.7344	1.05050	0.525248	+	0.850949i	$0.323973\pi$
$972$	−104.294	−3.34522
$973$	−47.3042	−1.51650
$974$	11.6106	0.372029
$975$	−17.4866	−0.560020
$976$	−29.8434	−0.955263
$977$	−3.93670	−0.125946	−0.0629730	−	0.998015i	$-0.520058\pi$
$977$	−3.93670	−0.125946	−0.0629730	+	0.998015i	$0.520058\pi$
$978$	28.1042	0.898672
$979$	−9.18701	−0.293618
$980$	55.9232	1.78640
$981$	−79.8246	−2.54860
$982$	8.66510	0.276514
$983$	28.2976	0.902555	0.451277	−	0.892384i	$-0.350969\pi$
$983$	28.2976	0.902555	0.451277	+	0.892384i	$0.350969\pi$
$984$	−41.6327	−1.32720
$985$	−7.87826	−0.251022
$986$	−0.202160	−0.00643808
$987$	9.98557	0.317844
$988$	2.70049	0.0859140
$989$	37.2991	1.18604
$990$	39.1002	1.24269
$991$	61.6616	1.95875	0.979373	−	0.202060i	$-0.0647636\pi$
$991$	61.6616	1.95875	0.979373	+	0.202060i	$0.0647636\pi$
$992$	13.7035	0.435088
$993$	−83.5019	−2.64985
$994$	−14.1421	−0.448561
$995$	77.1870	2.44699
$996$	36.4648	1.15543
$997$	−1.52360	−0.0482530	−0.0241265	−	0.999709i	$-0.507680\pi$
$997$	−1.52360	−0.0482530	−0.0241265	+	0.999709i	$0.507680\pi$
$998$	−7.54179	−0.238731
$999$	135.468	4.28603

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4009.2.a.c.1.34	✓	71

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4009.2.a.c.1.34	✓	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)

\(f(q)\)	\(=\)	\(q-0.440540 q^{2} +3.33608 q^{3} -1.80592 q^{4} -2.91639 q^{5} -1.46968 q^{6} -4.19739 q^{7} +1.67666 q^{8} +8.12943 q^{9} +O(q^{10})\)	\(q-0.440540 q^{2} +3.33608 q^{3} -1.80592 q^{4} -2.91639 q^{5} -1.46968 q^{6} -4.19739 q^{7} +1.67666 q^{8} +8.12943 q^{9} +1.28479 q^{10} +3.74359 q^{11} -6.02471 q^{12} -1.49535 q^{13} +1.84912 q^{14} -9.72930 q^{15} +2.87321 q^{16} +2.00713 q^{17} -3.58134 q^{18} +1.00000 q^{19} +5.26677 q^{20} -14.0028 q^{21} -1.64920 q^{22} -7.25048 q^{23} +5.59348 q^{24} +3.50531 q^{25} +0.658762 q^{26} +17.1122 q^{27} +7.58017 q^{28} +0.228630 q^{29} +4.28615 q^{30} -2.96672 q^{31} -4.61909 q^{32} +12.4889 q^{33} -0.884224 q^{34} +12.2412 q^{35} -14.6811 q^{36} +7.91649 q^{37} -0.440540 q^{38} -4.98861 q^{39} -4.88980 q^{40} -7.44308 q^{41} +6.16882 q^{42} -5.14436 q^{43} -6.76064 q^{44} -23.7085 q^{45} +3.19413 q^{46} -0.713111 q^{47} +9.58526 q^{48} +10.6181 q^{49} -1.54423 q^{50} +6.69596 q^{51} +2.70049 q^{52} +11.7808 q^{53} -7.53860 q^{54} -10.9178 q^{55} -7.03762 q^{56} +3.33608 q^{57} -0.100721 q^{58} +2.37068 q^{59} +17.5704 q^{60} -10.3868 q^{61} +1.30696 q^{62} -34.1224 q^{63} -3.71152 q^{64} +4.36102 q^{65} -5.50187 q^{66} -3.24158 q^{67} -3.62473 q^{68} -24.1882 q^{69} -5.39275 q^{70} -7.64803 q^{71} +13.6303 q^{72} -15.6512 q^{73} -3.48753 q^{74} +11.6940 q^{75} -1.80592 q^{76} -15.7133 q^{77} +2.19768 q^{78} -14.8461 q^{79} -8.37939 q^{80} +32.6993 q^{81} +3.27898 q^{82} -6.05255 q^{83} +25.2881 q^{84} -5.85358 q^{85} +2.26630 q^{86} +0.762726 q^{87} +6.27674 q^{88} -2.45406 q^{89} +10.4446 q^{90} +6.27657 q^{91} +13.0938 q^{92} -9.89720 q^{93} +0.314154 q^{94} -2.91639 q^{95} -15.4097 q^{96} +5.54721 q^{97} -4.67771 q^{98} +30.4332 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9}+O(q^{10}) \) 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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