LMFDB - Embedded newform 619.2.a.a.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [619,2,Mod(1,619)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(619, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("619.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 619 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	619.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:	$4.94273988512$
Analytic rank:	$1$
Dimension:	$21$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Character	$\chi$	$=$	619.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.42511 q^{2} -0.995739 q^{3} +0.0309400 q^{4} +2.82686 q^{5} +1.41904 q^{6} -2.89800 q^{7} +2.80613 q^{8} -2.00850 q^{9} +O(q^{10})$	\(q-1.42511 q^{2} -0.995739 q^{3} +0.0309400 q^{4} +2.82686 q^{5} +1.41904 q^{6} -2.89800 q^{7} +2.80613 q^{8} -2.00850 q^{9} -4.02859 q^{10} +3.29421 q^{11} -0.0308082 q^{12} -5.29605 q^{13} +4.12998 q^{14} -2.81482 q^{15} -4.06092 q^{16} +4.53632 q^{17} +2.86234 q^{18} -2.71161 q^{19} +0.0874631 q^{20} +2.88566 q^{21} -4.69462 q^{22} +4.19598 q^{23} -2.79417 q^{24} +2.99115 q^{25} +7.54746 q^{26} +4.98716 q^{27} -0.0896642 q^{28} -4.17097 q^{29} +4.01143 q^{30} -1.69469 q^{31} +0.175007 q^{32} -3.28018 q^{33} -6.46476 q^{34} -8.19226 q^{35} -0.0621430 q^{36} -10.8272 q^{37} +3.86434 q^{38} +5.27349 q^{39} +7.93254 q^{40} -12.3934 q^{41} -4.11238 q^{42} +8.20151 q^{43} +0.101923 q^{44} -5.67776 q^{45} -5.97973 q^{46} -0.630310 q^{47} +4.04362 q^{48} +1.39843 q^{49} -4.26272 q^{50} -4.51699 q^{51} -0.163860 q^{52} -2.55226 q^{53} -7.10726 q^{54} +9.31229 q^{55} -8.13217 q^{56} +2.70005 q^{57} +5.94410 q^{58} -7.05972 q^{59} -0.0870904 q^{60} -10.6837 q^{61} +2.41513 q^{62} +5.82065 q^{63} +7.87244 q^{64} -14.9712 q^{65} +4.67462 q^{66} -13.1274 q^{67} +0.140354 q^{68} -4.17810 q^{69} +11.6749 q^{70} -13.0265 q^{71} -5.63612 q^{72} -6.99619 q^{73} +15.4300 q^{74} -2.97841 q^{75} -0.0838971 q^{76} -9.54664 q^{77} -7.51530 q^{78} -1.98182 q^{79} -11.4797 q^{80} +1.05959 q^{81} +17.6620 q^{82} +12.5676 q^{83} +0.0892822 q^{84} +12.8236 q^{85} -11.6881 q^{86} +4.15320 q^{87} +9.24398 q^{88} +10.4766 q^{89} +8.09144 q^{90} +15.3480 q^{91} +0.129823 q^{92} +1.68747 q^{93} +0.898261 q^{94} -7.66534 q^{95} -0.174262 q^{96} -7.10553 q^{97} -1.99291 q^{98} -6.61644 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 21 q - 9 q^{2} - 5 q^{3} + 15 q^{4} - 21 q^{5} - 6 q^{6} - 4 q^{7} - 21 q^{8} + 6 q^{9}+O(q^{10}) $ 21 * q - 9 * q^2 - 5 * q^3 + 15 * q^4 - 21 * q^5 - 6 * q^6 - 4 * q^7 - 21 * q^8 + 6 * q^9	\( 21 q - 9 q^{2} - 5 q^{3} + 15 q^{4} - 21 q^{5} - 6 q^{6} - 4 q^{7} - 21 q^{8} + 6 q^{9} + q^{10} - 27 q^{11} - 8 q^{12} - 11 q^{13} - 19 q^{14} - 10 q^{15} + 11 q^{16} - 14 q^{17} - 14 q^{18} - 15 q^{19} - 25 q^{20} - 42 q^{21} + 12 q^{22} - 14 q^{23} - 8 q^{24} + 16 q^{25} - 11 q^{26} - 5 q^{27} + q^{28} - 78 q^{29} + q^{30} - 8 q^{31} - 41 q^{32} - 6 q^{33} + 7 q^{34} - 3 q^{35} - q^{36} - 23 q^{37} + 21 q^{38} - 4 q^{39} + 12 q^{40} - 59 q^{41} + 39 q^{42} + 2 q^{43} - 50 q^{44} - 36 q^{45} - 15 q^{46} - 12 q^{47} + 10 q^{48} + 17 q^{49} - 23 q^{50} - 8 q^{51} + 18 q^{52} - 36 q^{53} - 4 q^{54} + 23 q^{55} - 28 q^{56} - 24 q^{57} + 46 q^{58} - 17 q^{59} + 8 q^{60} - 22 q^{61} + 42 q^{62} - 6 q^{63} + 49 q^{64} - 53 q^{65} + 29 q^{66} + 15 q^{67} - 16 q^{68} - 30 q^{69} + 44 q^{70} - 56 q^{71} + 12 q^{72} - 2 q^{73} - 12 q^{74} + 2 q^{75} - 4 q^{76} - 47 q^{77} + 36 q^{78} + 5 q^{79} + 15 q^{80} - 19 q^{81} + 47 q^{82} - q^{83} - 20 q^{84} - 29 q^{85} - 23 q^{86} + 44 q^{87} + 61 q^{88} - 12 q^{89} + 91 q^{90} + 5 q^{91} + 35 q^{92} - 15 q^{93} + 34 q^{94} - 17 q^{95} + 14 q^{96} + 21 q^{97} + 24 q^{98} - 38 q^{99}+O(q^{100}) \) 21 * q - 9 * q^2 - 5 * q^3 + 15 * q^4 - 21 * q^5 - 6 * q^6 - 4 * q^7 - 21 * q^8 + 6 * q^9 + q^10 - 27 * q^11 - 8 * q^12 - 11 * q^13 - 19 * q^14 - 10 * q^15 + 11 * q^16 - 14 * q^17 - 14 * q^18 - 15 * q^19 - 25 * q^20 - 42 * q^21 + 12 * q^22 - 14 * q^23 - 8 * q^24 + 16 * q^25 - 11 * q^26 - 5 * q^27 + q^28 - 78 * q^29 + q^30 - 8 * q^31 - 41 * q^32 - 6 * q^33 + 7 * q^34 - 3 * q^35 - q^36 - 23 * q^37 + 21 * q^38 - 4 * q^39 + 12 * q^40 - 59 * q^41 + 39 * q^42 + 2 * q^43 - 50 * q^44 - 36 * q^45 - 15 * q^46 - 12 * q^47 + 10 * q^48 + 17 * q^49 - 23 * q^50 - 8 * q^51 + 18 * q^52 - 36 * q^53 - 4 * q^54 + 23 * q^55 - 28 * q^56 - 24 * q^57 + 46 * q^58 - 17 * q^59 + 8 * q^60 - 22 * q^61 + 42 * q^62 - 6 * q^63 + 49 * q^64 - 53 * q^65 + 29 * q^66 + 15 * q^67 - 16 * q^68 - 30 * q^69 + 44 * q^70 - 56 * q^71 + 12 * q^72 - 2 * q^73 - 12 * q^74 + 2 * q^75 - 4 * q^76 - 47 * q^77 + 36 * q^78 + 5 * q^79 + 15 * q^80 - 19 * q^81 + 47 * q^82 - q^83 - 20 * q^84 - 29 * q^85 - 23 * q^86 + 44 * q^87 + 61 * q^88 - 12 * q^89 + 91 * q^90 + 5 * q^91 + 35 * q^92 - 15 * q^93 + 34 * q^94 - 17 * q^95 + 14 * q^96 + 21 * q^97 + 24 * q^98 - 38 * 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 $

$2$	−1.42511	−1.00771	−0.503853	−	0.863790i	$-0.668085\pi$
$2$	−1.42511	−1.00771	−0.503853	+	0.863790i	$0.668085\pi$
$3$	−0.995739	−0.574890	−0.287445	−	0.957797i	$-0.592806\pi$
$3$	−0.995739	−0.574890	−0.287445	+	0.957797i	$0.592806\pi$
$4$	0.0309400	0.0154700
$5$	2.82686	1.26421	0.632106	−	0.774882i	$-0.282191\pi$
$5$	2.82686	1.26421	0.632106	+	0.774882i	$0.282191\pi$
$6$	1.41904	0.579320
$7$	−2.89800	−1.09534	−0.547671	−	0.836694i	$-0.684486\pi$
$7$	−2.89800	−1.09534	−0.547671	+	0.836694i	$0.684486\pi$
$8$	2.80613	0.992116
$9$	−2.00850	−0.669501
$10$	−4.02859	−1.27395
$11$	3.29421	0.993242	0.496621	−	0.867967i	$-0.334574\pi$
$11$	3.29421	0.993242	0.496621	+	0.867967i	$0.334574\pi$
$12$	−0.0308082	−0.00889355
$13$	−5.29605	−1.46886	−0.734430	−	0.678684i	$-0.762551\pi$
$13$	−5.29605	−1.46886	−0.734430	+	0.678684i	$0.762551\pi$
$14$	4.12998	1.10378
$15$	−2.81482	−0.726783
$16$	−4.06092	−1.01523
$17$	4.53632	1.10022	0.550110	−	0.835092i	$-0.314586\pi$
$17$	4.53632	1.10022	0.550110	+	0.835092i	$0.314586\pi$
$18$	2.86234	0.674660
$19$	−2.71161	−0.622085	−0.311043	−	0.950396i	$-0.600678\pi$
$19$	−2.71161	−0.622085	−0.311043	+	0.950396i	$0.600678\pi$
$20$	0.0874631	0.0195573
$21$	2.88566	0.629702
$22$	−4.69462	−1.00090
$23$	4.19598	0.874922	0.437461	−	0.899237i	$-0.355878\pi$
$23$	4.19598	0.874922	0.437461	+	0.899237i	$0.355878\pi$
$24$	−2.79417	−0.570358
$25$	2.99115	0.598231
$26$	7.54746	1.48018
$27$	4.98716	0.959780
$28$	−0.0896642	−0.0169449
$29$	−4.17097	−0.774530	−0.387265	−	0.921968i	$-0.626580\pi$
$29$	−4.17097	−0.774530	−0.387265	+	0.921968i	$0.626580\pi$
$30$	4.01143	0.732383
$31$	−1.69469	−0.304376	−0.152188	−	0.988352i	$-0.548632\pi$
$31$	−1.69469	−0.304376	−0.152188	+	0.988352i	$0.548632\pi$
$32$	0.175007	0.0309372
$33$	−3.28018	−0.571006
$34$	−6.46476	−1.10870
$35$	−8.19226	−1.38474
$36$	−0.0621430	−0.0103572
$37$	−10.8272	−1.77999	−0.889993	−	0.455975i	$-0.849291\pi$
$37$	−10.8272	−1.77999	−0.889993	+	0.455975i	$0.849291\pi$
$38$	3.86434	0.626879
$39$	5.27349	0.844434
$40$	7.93254	1.25424
$41$	−12.3934	−1.93552	−0.967762	−	0.251865i	$-0.918956\pi$
$41$	−12.3934	−1.93552	−0.967762	+	0.251865i	$0.918956\pi$
$42$	−4.11238	−0.634554
$43$	8.20151	1.25072	0.625359	−	0.780337i	$-0.284952\pi$
$43$	8.20151	1.25072	0.625359	+	0.780337i	$0.284952\pi$
$44$	0.101923	0.0153654
$45$	−5.67776	−0.846391
$46$	−5.97973	−0.881663
$47$	−0.630310	−0.0919401	−0.0459701	−	0.998943i	$-0.514638\pi$
$47$	−0.630310	−0.0919401	−0.0459701	+	0.998943i	$0.514638\pi$
$48$	4.04362	0.583646
$49$	1.39843	0.199775
$50$	−4.26272	−0.602840
$51$	−4.51699	−0.632506
$52$	−0.163860	−0.0227233
$53$	−2.55226	−0.350580	−0.175290	−	0.984517i	$-0.556086\pi$
$53$	−2.55226	−0.350580	−0.175290	+	0.984517i	$0.556086\pi$
$54$	−7.10726	−0.967176
$55$	9.31229	1.25567
$56$	−8.13217	−1.08671
$57$	2.70005	0.357631
$58$	5.94410	0.780498
$59$	−7.05972	−0.919098	−0.459549	−	0.888153i	$-0.651989\pi$
$59$	−7.05972	−0.919098	−0.459549	+	0.888153i	$0.651989\pi$
$60$	−0.0870904	−0.0112433
$61$	−10.6837	−1.36791	−0.683957	−	0.729523i	$-0.739742\pi$
$61$	−10.6837	−1.36791	−0.683957	+	0.729523i	$0.739742\pi$
$62$	2.41513	0.306721
$63$	5.82065	0.733333
$64$	7.87244	0.984055
$65$	−14.9712	−1.85695
$66$	4.67462	0.575405
$67$	−13.1274	−1.60376	−0.801882	−	0.597483i	$-0.796168\pi$
$67$	−13.1274	−1.60376	−0.801882	+	0.597483i	$0.796168\pi$
$68$	0.140354	0.0170204
$69$	−4.17810	−0.502984
$70$	11.6749	1.39541
$71$	−13.0265	−1.54596	−0.772981	−	0.634429i	$-0.781235\pi$
$71$	−13.0265	−1.54596	−0.772981	+	0.634429i	$0.781235\pi$
$72$	−5.63612	−0.664223
$73$	−6.99619	−0.818842	−0.409421	−	0.912346i	$-0.634269\pi$
$73$	−6.99619	−0.818842	−0.409421	+	0.912346i	$0.634269\pi$
$74$	15.4300	1.79370
$75$	−2.97841	−0.343917
$76$	−0.0838971	−0.00962365
$77$	−9.54664	−1.08794
$78$	−7.51530	−0.850940
$79$	−1.98182	−0.222972	−0.111486	−	0.993766i	$-0.535561\pi$
$79$	−1.98182	−0.222972	−0.111486	+	0.993766i	$0.535561\pi$
$80$	−11.4797	−1.28347
$81$	1.05959	0.117732
$82$	17.6620	1.95044
$83$	12.5676	1.37947	0.689737	−	0.724060i	$-0.257726\pi$
$83$	12.5676	1.37947	0.689737	+	0.724060i	$0.257726\pi$
$84$	0.0892822	0.00974148
$85$	12.8236	1.39091
$86$	−11.6881	−1.26036
$87$	4.15320	0.445270
$88$	9.24398	0.985412
$89$	10.4766	1.11052	0.555259	−	0.831677i	$-0.312619\pi$
$89$	10.4766	1.11052	0.555259	+	0.831677i	$0.312619\pi$
$90$	8.09144	0.852912
$91$	15.3480	1.60891
$92$	0.129823	0.0135350
$93$	1.68747	0.174983
$94$	0.898261	0.0926485
$95$	−7.66534	−0.786447
$96$	−0.174262	−0.0177855
$97$	−7.10553	−0.721458	−0.360729	−	0.932671i	$-0.617472\pi$
$97$	−7.10553	−0.721458	−0.360729	+	0.932671i	$0.617472\pi$
$98$	−1.99291	−0.201315
$99$	−6.61644	−0.664977
$100$	0.0925462	0.00925462
$101$	5.90837	0.587904	0.293952	−	0.955820i	$-0.405029\pi$
$101$	5.90837	0.587904	0.293952	+	0.955820i	$0.405029\pi$
$102$	6.43722	0.637379
$103$	9.69860	0.955631	0.477816	−	0.878460i	$-0.341429\pi$
$103$	9.69860	0.955631	0.477816	+	0.878460i	$0.341429\pi$
$104$	−14.8614	−1.45728
$105$	8.15736	0.796076
$106$	3.63725	0.353281
$107$	7.93356	0.766966	0.383483	−	0.923548i	$-0.374724\pi$
$107$	7.93356	0.766966	0.383483	+	0.923548i	$0.374724\pi$
$108$	0.154303	0.0148478
$109$	−11.5292	−1.10429	−0.552147	−	0.833747i	$-0.686191\pi$
$109$	−11.5292	−1.10429	−0.552147	+	0.833747i	$0.686191\pi$
$110$	−13.2710	−1.26534
$111$	10.7811	1.02330
$112$	11.7686	1.11203
$113$	8.14887	0.766581	0.383291	−	0.923628i	$-0.374791\pi$
$113$	8.14887	0.766581	0.383291	+	0.923628i	$0.374791\pi$
$114$	−3.84788	−0.360387
$115$	11.8615	1.10609
$116$	−0.129050	−0.0119820
$117$	10.6371	0.983403
$118$	10.0609	0.926180
$119$	−13.1463	−1.20512
$120$	−7.89874	−0.721053
$121$	−0.148163	−0.0134694
$122$	15.2255	1.37845
$123$	12.3406	1.11271
$124$	−0.0524338	−0.00470870
$125$	−5.67874	−0.507922
$126$	−8.29507	−0.738983
$127$	−16.6638	−1.47867	−0.739335	−	0.673338i	$-0.764860\pi$
$127$	−16.6638	−1.47867	−0.739335	+	0.673338i	$0.764860\pi$
$128$	−11.5691	−1.02257
$129$	−8.16657	−0.719026
$130$	21.3356	1.87126
$131$	15.2335	1.33096	0.665480	−	0.746416i	$-0.268227\pi$
$131$	15.2335	1.33096	0.665480	+	0.746416i	$0.268227\pi$
$132$	−0.101489	−0.00883345
$133$	7.85825	0.681397
$134$	18.7080	1.61612
$135$	14.0980	1.21337
$136$	12.7295	1.09155
$137$	21.7696	1.85990	0.929951	−	0.367683i	$-0.119849\pi$
$137$	21.7696	1.85990	0.929951	+	0.367683i	$0.119849\pi$
$138$	5.95425	0.506860
$139$	−9.73848	−0.826007	−0.413004	−	0.910729i	$-0.635520\pi$
$139$	−9.73848	−0.826007	−0.413004	+	0.910729i	$0.635520\pi$
$140$	−0.253468	−0.0214220
$141$	0.627624	0.0528555
$142$	18.5642	1.55787
$143$	−17.4463	−1.45893
$144$	8.15637	0.679698
$145$	−11.7908	−0.979170
$146$	9.97035	0.825152
$147$	−1.39247	−0.114849
$148$	−0.334994	−0.0275364
$149$	−23.9092	−1.95872	−0.979358	−	0.202136i	$-0.935212\pi$
$149$	−23.9092	−1.95872	−0.979358	+	0.202136i	$0.935212\pi$
$150$	4.24456	0.346567
$151$	2.26424	0.184261	0.0921307	−	0.995747i	$-0.470632\pi$
$151$	2.26424	0.184261	0.0921307	+	0.995747i	$0.470632\pi$
$152$	−7.60912	−0.617181
$153$	−9.11122	−0.736598
$154$	13.6050	1.09632
$155$	−4.79067	−0.384796
$156$	0.163162	0.0130634
$157$	4.17308	0.333048	0.166524	−	0.986037i	$-0.446746\pi$
$157$	4.17308	0.333048	0.166524	+	0.986037i	$0.446746\pi$
$158$	2.82431	0.224690
$159$	2.54138	0.201545
$160$	0.494722	0.0391112
$161$	−12.1600	−0.958339
$162$	−1.51004	−0.118640
$163$	6.76963	0.530239	0.265119	−	0.964216i	$-0.414589\pi$
$163$	6.76963	0.530239	0.265119	+	0.964216i	$0.414589\pi$
$164$	−0.383452	−0.0299425
$165$	−9.27261	−0.721872
$166$	−17.9102	−1.39010
$167$	−6.43038	−0.497598	−0.248799	−	0.968555i	$-0.580036\pi$
$167$	−6.43038	−0.497598	−0.248799	+	0.968555i	$0.580036\pi$
$168$	8.09752	0.624737
$169$	15.0482	1.15755
$170$	−18.2750	−1.40163
$171$	5.44627	0.416487
$172$	0.253754	0.0193486
$173$	17.7149	1.34684	0.673419	−	0.739261i	$-0.264825\pi$
$173$	17.7149	1.34684	0.673419	+	0.739261i	$0.264825\pi$
$174$	−5.91877	−0.448701
$175$	−8.66837	−0.655267
$176$	−13.3775	−1.00837
$177$	7.02964	0.528380
$178$	−14.9303	−1.11908
$179$	9.06181	0.677312	0.338656	−	0.940910i	$-0.390028\pi$
$179$	9.06181	0.677312	0.338656	+	0.940910i	$0.390028\pi$
$180$	−0.175670	−0.0130937
$181$	21.9954	1.63491	0.817454	−	0.575994i	$-0.195385\pi$
$181$	21.9954	1.63491	0.817454	+	0.575994i	$0.195385\pi$
$182$	−21.8726	−1.62130
$183$	10.6382	0.786400
$184$	11.7745	0.868024
$185$	−30.6071	−2.25028
$186$	−2.40484	−0.176331
$187$	14.9436	1.09278
$188$	−0.0195018	−0.00142231
$189$	−14.4528	−1.05129
$190$	10.9240	0.792507
$191$	26.1713	1.89369	0.946845	−	0.321690i	$-0.104251\pi$
$191$	26.1713	1.89369	0.946845	+	0.321690i	$0.104251\pi$
$192$	−7.83890	−0.565724
$193$	−9.81005	−0.706143	−0.353071	−	0.935596i	$-0.614863\pi$
$193$	−9.81005	−0.706143	−0.353071	+	0.935596i	$0.614863\pi$
$194$	10.1262	0.727017
$195$	14.9074	1.06754
$196$	0.0432673	0.00309052
$197$	−13.1625	−0.937789	−0.468894	−	0.883254i	$-0.655347\pi$
$197$	−13.1625	−0.937789	−0.468894	+	0.883254i	$0.655347\pi$
$198$	9.42915	0.670101
$199$	−20.0017	−1.41788	−0.708940	−	0.705269i	$-0.750826\pi$
$199$	−20.0017	−1.41788	−0.708940	+	0.705269i	$0.750826\pi$
$200$	8.39356	0.593514
$201$	13.0714	0.921988
$202$	−8.42008	−0.592434
$203$	12.0875	0.848376
$204$	−0.139756	−0.00978486
$205$	−35.0345	−2.44691
$206$	−13.8216	−0.962995
$207$	−8.42763	−0.585761
$208$	21.5069	1.49123
$209$	−8.93261	−0.617882
$210$	−11.6251	−0.802210
$211$	15.5763	1.07232	0.536160	−	0.844116i	$-0.319874\pi$
$211$	15.5763	1.07232	0.536160	+	0.844116i	$0.319874\pi$
$212$	−0.0789668	−0.00542346
$213$	12.9710	0.888759
$214$	−11.3062	−0.772876
$215$	23.1845	1.58117
$216$	13.9946	0.952213
$217$	4.91123	0.333396
$218$	16.4303	1.11280
$219$	6.96638	0.470745
$220$	0.288122	0.0194252
$221$	−24.0246	−1.61607
$222$	−15.3643	−1.03118
$223$	−10.5580	−0.707016	−0.353508	−	0.935432i	$-0.615011\pi$
$223$	−10.5580	−0.707016	−0.353508	+	0.935432i	$0.615011\pi$
$224$	−0.507172	−0.0338868
$225$	−6.00774	−0.400516
$226$	−11.6130	−0.772488
$227$	−4.02639	−0.267241	−0.133620	−	0.991033i	$-0.542660\pi$
$227$	−4.02639	−0.267241	−0.133620	+	0.991033i	$0.542660\pi$
$228$	0.0835396	0.00553255
$229$	24.5125	1.61983	0.809915	−	0.586547i	$-0.199513\pi$
$229$	24.5125	1.61983	0.809915	+	0.586547i	$0.199513\pi$
$230$	−16.9039	−1.11461
$231$	9.50597	0.625447
$232$	−11.7043	−0.768424
$233$	1.99919	0.130971	0.0654855	−	0.997854i	$-0.479140\pi$
$233$	1.99919	0.130971	0.0654855	+	0.997854i	$0.479140\pi$
$234$	−15.1591	−0.990981
$235$	−1.78180	−0.116232
$236$	−0.218428	−0.0142184
$237$	1.97337	0.128184
$238$	18.7349	1.21440
$239$	−20.6066	−1.33293	−0.666464	−	0.745538i	$-0.732193\pi$
$239$	−20.6066	−1.33293	−0.666464	+	0.745538i	$0.732193\pi$
$240$	11.4308	0.737852
$241$	−19.8887	−1.28114	−0.640572	−	0.767898i	$-0.721303\pi$
$241$	−19.8887	−1.28114	−0.640572	+	0.767898i	$0.721303\pi$
$242$	0.211149	0.0135732
$243$	−16.0166	−1.02746
$244$	−0.330555	−0.0211616
$245$	3.95316	0.252558
$246$	−17.5867	−1.12129
$247$	14.3608	0.913757
$248$	−4.75553	−0.301976
$249$	−12.5141	−0.793047
$250$	8.09283	0.511835
$251$	−5.67598	−0.358265	−0.179132	−	0.983825i	$-0.557329\pi$
$251$	−5.67598	−0.358265	−0.179132	+	0.983825i	$0.557329\pi$
$252$	0.180091	0.0113447
$253$	13.8224	0.869009
$254$	23.7477	1.49006
$255$	−12.7689	−0.799621
$256$	0.742383	0.0463989
$257$	11.6733	0.728159	0.364080	−	0.931368i	$-0.381384\pi$
$257$	11.6733	0.728159	0.364080	+	0.931368i	$0.381384\pi$
$258$	11.6383	0.724566
$259$	31.3774	1.94969
$260$	−0.463209	−0.0287270
$261$	8.37741	0.518549
$262$	−21.7095	−1.34122
$263$	12.3393	0.760874	0.380437	−	0.924807i	$-0.375774\pi$
$263$	12.3393	0.760874	0.380437	+	0.924807i	$0.375774\pi$
$264$	−9.20460	−0.566504
$265$	−7.21488	−0.443207
$266$	−11.1989	−0.686647
$267$	−10.4320	−0.638427
$268$	−0.406160	−0.0248102
$269$	−18.0664	−1.10153	−0.550765	−	0.834660i	$-0.685664\pi$
$269$	−18.0664	−1.10153	−0.550765	+	0.834660i	$0.685664\pi$
$270$	−20.0912	−1.22271
$271$	0.807429	0.0490478	0.0245239	−	0.999699i	$-0.492193\pi$
$271$	0.807429	0.0490478	0.0245239	+	0.999699i	$0.492193\pi$
$272$	−18.4217	−1.11698
$273$	−15.2826	−0.924944
$274$	−31.0241	−1.87423
$275$	9.85349	0.594188
$276$	−0.129270	−0.00778116
$277$	10.3264	0.620451	0.310226	−	0.950663i	$-0.399595\pi$
$277$	10.3264	0.620451	0.310226	+	0.950663i	$0.399595\pi$
$278$	13.8784	0.832372
$279$	3.40380	0.203780
$280$	−22.9885	−1.37383
$281$	−13.3878	−0.798649	−0.399324	−	0.916810i	$-0.630755\pi$
$281$	−13.3878	−0.798649	−0.399324	+	0.916810i	$0.630755\pi$
$282$	−0.894434	−0.0532628
$283$	−10.7254	−0.637560	−0.318780	−	0.947829i	$-0.603273\pi$
$283$	−10.7254	−0.637560	−0.318780	+	0.947829i	$0.603273\pi$
$284$	−0.403040	−0.0239160
$285$	7.63268	0.452121
$286$	24.8629	1.47018
$287$	35.9161	2.12006
$288$	−0.351503	−0.0207125
$289$	3.57821	0.210483
$290$	16.8031	0.986715
$291$	7.07526	0.414759
$292$	−0.216462	−0.0126675
$293$	13.7379	0.802578	0.401289	−	0.915951i	$-0.368562\pi$
$293$	13.7379	0.802578	0.401289	+	0.915951i	$0.368562\pi$
$294$	1.98442	0.115734
$295$	−19.9569	−1.16193
$296$	−30.3826	−1.76595
$297$	16.4288	0.953294
$298$	34.0732	1.97381
$299$	−22.2221	−1.28514
$300$	−0.0921519	−0.00532039
$301$	−23.7680	−1.36996
$302$	−3.22679	−0.185681
$303$	−5.88319	−0.337981
$304$	11.0116	0.631560
$305$	−30.2015	−1.72933
$306$	12.9845	0.742274
$307$	−2.31749	−0.132266	−0.0661331	−	0.997811i	$-0.521066\pi$
$307$	−2.31749	−0.132266	−0.0661331	+	0.997811i	$0.521066\pi$
$308$	−0.295373	−0.0168304
$309$	−9.65728	−0.549383
$310$	6.82723	0.387761
$311$	8.92477	0.506077	0.253039	−	0.967456i	$-0.418570\pi$
$311$	8.92477	0.506077	0.253039	+	0.967456i	$0.418570\pi$
$312$	14.7981	0.837776
$313$	4.26754	0.241216	0.120608	−	0.992700i	$-0.461516\pi$
$313$	4.26754	0.241216	0.120608	+	0.992700i	$0.461516\pi$
$314$	−5.94710	−0.335614
$315$	16.4542	0.927088
$316$	−0.0613174	−0.00344937
$317$	−13.1699	−0.739695	−0.369848	−	0.929092i	$-0.620590\pi$
$317$	−13.1699	−0.739695	−0.369848	+	0.929092i	$0.620590\pi$
$318$	−3.62175	−0.203098
$319$	−13.7401	−0.769296
$320$	22.2543	1.24405
$321$	−7.89976	−0.440922
$322$	17.3293	0.965723
$323$	−12.3007	−0.684431
$324$	0.0327838	0.00182132
$325$	−15.8413	−0.878717
$326$	−9.64748	−0.534324
$327$	11.4800	0.634848
$328$	−34.7775	−1.92027
$329$	1.82664	0.100706
$330$	13.2145	0.727434
$331$	10.2212	0.561808	0.280904	−	0.959736i	$-0.409366\pi$
$331$	10.2212	0.561808	0.280904	+	0.959736i	$0.409366\pi$
$332$	0.388842	0.0213405
$333$	21.7465	1.19170
$334$	9.16400	0.501432
$335$	−37.1093	−2.02750
$336$	−11.7184	−0.639293
$337$	1.71905	0.0936424	0.0468212	−	0.998903i	$-0.485091\pi$
$337$	1.71905	0.0936424	0.0468212	+	0.998903i	$0.485091\pi$
$338$	−21.4453	−1.16647
$339$	−8.11415	−0.440700
$340$	0.396761	0.0215174
$341$	−5.58268	−0.302319
$342$	−7.76154	−0.419696
$343$	16.2334	0.876520
$344$	23.0145	1.24086
$345$	−11.8109	−0.635878
$346$	−25.2457	−1.35722
$347$	−23.4244	−1.25749	−0.628743	−	0.777613i	$-0.716430\pi$
$347$	−23.4244	−1.25749	−0.628743	+	0.777613i	$0.716430\pi$
$348$	0.128500	0.00688832
$349$	8.41763	0.450585	0.225293	−	0.974291i	$-0.427666\pi$
$349$	8.41763	0.450585	0.225293	+	0.974291i	$0.427666\pi$
$350$	12.3534	0.660316
$351$	−26.4123	−1.40978
$352$	0.576511	0.0307282
$353$	24.8054	1.32026	0.660129	−	0.751152i	$-0.270502\pi$
$353$	24.8054	1.32026	0.660129	+	0.751152i	$0.270502\pi$
$354$	−10.0180	−0.532452
$355$	−36.8241	−1.95442
$356$	0.324146	0.0171797
$357$	13.0903	0.692810
$358$	−12.9141	−0.682531
$359$	15.3148	0.808282	0.404141	−	0.914697i	$-0.367571\pi$
$359$	15.3148	0.808282	0.404141	+	0.914697i	$0.367571\pi$
$360$	−15.9325	−0.839718
$361$	−11.6472	−0.613010
$362$	−31.3459	−1.64751
$363$	0.147532	0.00774343
$364$	0.474866	0.0248897
$365$	−19.7773	−1.03519
$366$	−15.1606	−0.792460
$367$	−10.3959	−0.542659	−0.271330	−	0.962487i	$-0.587463\pi$
$367$	−10.3959	−0.542659	−0.271330	+	0.962487i	$0.587463\pi$
$368$	−17.0395	−0.888247
$369$	24.8922	1.29584
$370$	43.6185	2.26762
$371$	7.39645	0.384005
$372$	0.0522104	0.00270698
$373$	−0.118665	−0.00614424	−0.00307212	−	0.999995i	$-0.500978\pi$
$373$	−0.118665	−0.00614424	−0.00307212	+	0.999995i	$0.500978\pi$
$374$	−21.2963	−1.10121
$375$	5.65454	0.291999
$376$	−1.76873	−0.0912153
$377$	22.0897	1.13768
$378$	20.5969	1.05939
$379$	8.07056	0.414557	0.207278	−	0.978282i	$-0.433539\pi$
$379$	8.07056	0.414557	0.207278	+	0.978282i	$0.433539\pi$
$380$	−0.237165	−0.0121663
$381$	16.5928	0.850073
$382$	−37.2970	−1.90828
$383$	1.16566	0.0595625	0.0297812	−	0.999556i	$-0.490519\pi$
$383$	1.16566	0.0595625	0.0297812	+	0.999556i	$0.490519\pi$
$384$	11.5198	0.587868
$385$	−26.9870	−1.37539
$386$	13.9804	0.711584
$387$	−16.4728	−0.837357
$388$	−0.219845	−0.0111609
$389$	−4.49125	−0.227715	−0.113858	−	0.993497i	$-0.536321\pi$
$389$	−4.49125	−0.227715	−0.113858	+	0.993497i	$0.536321\pi$
$390$	−21.2447	−1.07577
$391$	19.0343	0.962606
$392$	3.92416	0.198200
$393$	−15.1686	−0.765156
$394$	18.7580	0.945015
$395$	−5.60232	−0.281884
$396$	−0.204712	−0.0102872
$397$	4.18358	0.209968	0.104984	−	0.994474i	$-0.466521\pi$
$397$	4.18358	0.209968	0.104984	+	0.994474i	$0.466521\pi$
$398$	28.5046	1.42880
$399$	−7.82477	−0.391728
$400$	−12.1468	−0.607342
$401$	22.6311	1.13014	0.565071	−	0.825042i	$-0.308849\pi$
$401$	22.6311	1.13014	0.565071	+	0.825042i	$0.308849\pi$
$402$	−18.6282	−0.929092
$403$	8.97519	0.447086
$404$	0.182805	0.00909488
$405$	2.99532	0.148839
$406$	−17.2260	−0.854913
$407$	−35.6672	−1.76796
$408$	−12.6753	−0.627519
$409$	−13.9650	−0.690527	−0.345264	−	0.938506i	$-0.612210\pi$
$409$	−13.9650	−0.690527	−0.345264	+	0.938506i	$0.612210\pi$
$410$	49.9280	2.46577
$411$	−21.6769	−1.06924
$412$	0.300074	0.0147836
$413$	20.4591	1.00673
$414$	12.0103	0.590274
$415$	35.5269	1.74395
$416$	−0.926848	−0.0454424
$417$	9.69699	0.474864
$418$	12.7300	0.622643
$419$	−29.8407	−1.45781	−0.728906	−	0.684613i	$-0.759971\pi$
$419$	−29.8407	−1.45781	−0.728906	+	0.684613i	$0.759971\pi$
$420$	0.252388	0.0123153
$421$	19.8299	0.966451	0.483226	−	0.875496i	$-0.339465\pi$
$421$	19.8299	0.966451	0.483226	+	0.875496i	$0.339465\pi$
$422$	−22.1980	−1.08058
$423$	1.26598	0.0615540
$424$	−7.16196	−0.347816
$425$	13.5688	0.658185
$426$	−18.4851	−0.895607
$427$	30.9615	1.49833
$428$	0.245464	0.0118650
$429$	17.3720	0.838728
$430$	−33.0405	−1.59336
$431$	−3.39783	−0.163667	−0.0818337	−	0.996646i	$-0.526078\pi$
$431$	−3.39783	−0.163667	−0.0818337	+	0.996646i	$0.526078\pi$
$432$	−20.2525	−0.974398
$433$	−26.3348	−1.26557	−0.632786	−	0.774327i	$-0.718089\pi$
$433$	−26.3348	−1.26557	−0.632786	+	0.774327i	$0.718089\pi$
$434$	−6.99905	−0.335965
$435$	11.7405	0.562916
$436$	−0.356712	−0.0170834
$437$	−11.3778	−0.544276
$438$	−9.92787	−0.474372
$439$	−8.72302	−0.416327	−0.208164	−	0.978094i	$-0.566749\pi$
$439$	−8.72302	−0.416327	−0.208164	+	0.978094i	$0.566749\pi$
$440$	26.1315	1.24577
$441$	−2.80874	−0.133750
$442$	34.2377	1.62852
$443$	6.33581	0.301024	0.150512	−	0.988608i	$-0.451908\pi$
$443$	6.33581	0.301024	0.150512	+	0.988608i	$0.451908\pi$
$444$	0.333567	0.0158304
$445$	29.6159	1.40393
$446$	15.0463	0.712464
$447$	23.8073	1.12605
$448$	−22.8144	−1.07788
$449$	−7.85304	−0.370608	−0.185304	−	0.982681i	$-0.559327\pi$
$449$	−7.85304	−0.370608	−0.185304	+	0.982681i	$0.559327\pi$
$450$	8.56169	0.403602
$451$	−40.8265	−1.92245
$452$	0.252126	0.0118590
$453$	−2.25459	−0.105930
$454$	5.73805	0.269300
$455$	43.3866	2.03400
$456$	7.57670	0.354811
$457$	16.4077	0.767519	0.383759	−	0.923433i	$-0.374629\pi$
$457$	16.4077	0.767519	0.383759	+	0.923433i	$0.374629\pi$
$458$	−34.9330	−1.63231
$459$	22.6234	1.05597
$460$	0.366993	0.0171111
$461$	25.3001	1.17834	0.589171	−	0.808009i	$-0.299455\pi$
$461$	25.3001	1.17834	0.589171	+	0.808009i	$0.299455\pi$
$462$	−13.5471	−0.630266
$463$	−12.4730	−0.579669	−0.289834	−	0.957077i	$-0.593600\pi$
$463$	−12.4730	−0.579669	−0.289834	+	0.957077i	$0.593600\pi$
$464$	16.9380	0.786327
$465$	4.77026	0.221215
$466$	−2.84906	−0.131980
$467$	32.8959	1.52224	0.761120	−	0.648612i	$-0.224650\pi$
$467$	32.8959	1.52224	0.761120	+	0.648612i	$0.224650\pi$
$468$	0.329113	0.0152132
$469$	38.0432	1.75667
$470$	2.53926	0.117127
$471$	−4.15530	−0.191466
$472$	−19.8105	−0.911852
$473$	27.0175	1.24227
$474$	−2.81227	−0.129172
$475$	−8.11083	−0.372150
$476$	−0.406746	−0.0186431
$477$	5.12622	0.234713
$478$	29.3666	1.34320
$479$	26.8619	1.22735	0.613675	−	0.789559i	$-0.289691\pi$
$479$	26.8619	1.22735	0.613675	+	0.789559i	$0.289691\pi$
$480$	−0.492614	−0.0224846
$481$	57.3416	2.61455
$482$	28.3436	1.29102
$483$	12.1082	0.550940
$484$	−0.00458417	−0.000208371 0
$485$	−20.0864	−0.912075
$486$	22.8254	1.03538
$487$	38.9069	1.76304	0.881521	−	0.472145i	$-0.156520\pi$
$487$	38.9069	1.76304	0.881521	+	0.472145i	$0.156520\pi$
$488$	−29.9799	−1.35713
$489$	−6.74079	−0.304829
$490$	−5.63369	−0.254504
$491$	−15.7477	−0.710686	−0.355343	−	0.934736i	$-0.615636\pi$
$491$	−15.7477	−0.710686	−0.355343	+	0.934736i	$0.615636\pi$
$492$	0.381818	0.0172137
$493$	−18.9209	−0.852153
$494$	−20.4657	−0.920797
$495$	−18.7038	−0.840671
$496$	6.88202	0.309012
$497$	37.7509	1.69336
$498$	17.8339	0.799157
$499$	−32.4985	−1.45483	−0.727416	−	0.686196i	$-0.759279\pi$
$499$	−32.4985	−1.45483	−0.727416	+	0.686196i	$0.759279\pi$
$500$	−0.175700	−0.00785754
$501$	6.40298	0.286064
$502$	8.08889	0.361025
$503$	−21.9745	−0.979792	−0.489896	−	0.871781i	$-0.662965\pi$
$503$	−21.9745	−0.979792	−0.489896	+	0.871781i	$0.662965\pi$
$504$	16.3335	0.727551
$505$	16.7021	0.743236
$506$	−19.6985	−0.875705
$507$	−14.9840	−0.665465
$508$	−0.515576	−0.0228750
$509$	3.98980	0.176845	0.0884223	−	0.996083i	$-0.471817\pi$
$509$	3.98980	0.176845	0.0884223	+	0.996083i	$0.471817\pi$
$510$	18.1971	0.805782
$511$	20.2750	0.896913
$512$	22.0802	0.975818
$513$	−13.5232	−0.597065
$514$	−16.6357	−0.733770
$515$	27.4166	1.20812
$516$	−0.252673	−0.0111233
$517$	−2.07637	−0.0913188
$518$	−44.7162	−1.96472
$519$	−17.6394	−0.774284
$520$	−42.0111	−1.84231
$521$	−4.03718	−0.176872	−0.0884360	−	0.996082i	$-0.528187\pi$
$521$	−4.03718	−0.176872	−0.0884360	+	0.996082i	$0.528187\pi$
$522$	−11.9387	−0.522544
$523$	0.711792	0.0311245	0.0155623	−	0.999879i	$-0.495046\pi$
$523$	0.711792	0.0311245	0.0155623	+	0.999879i	$0.495046\pi$
$524$	0.471325	0.0205899
$525$	8.63144	0.376707
$526$	−17.5849	−0.766737
$527$	−7.68768	−0.334881
$528$	13.3205	0.579702
$529$	−5.39377	−0.234512
$530$	10.2820	0.446622
$531$	14.1795	0.615337
$532$	0.243134	0.0105412
$533$	65.6361	2.84302
$534$	14.8667	0.643346
$535$	22.4271	0.969608
$536$	−36.8371	−1.59112
$537$	−9.02321	−0.389380
$538$	25.7467	1.11002
$539$	4.60671	0.198425
$540$	0.436193	0.0187707
$541$	−37.4738	−1.61112	−0.805562	−	0.592511i	$-0.798137\pi$
$541$	−37.4738	−1.61112	−0.805562	+	0.592511i	$0.798137\pi$
$542$	−1.15068	−0.0494257
$543$	−21.9017	−0.939893
$544$	0.793889	0.0340377
$545$	−32.5914	−1.39606
$546$	21.7794	0.932071
$547$	24.5254	1.04863	0.524316	−	0.851524i	$-0.324321\pi$
$547$	24.5254	1.04863	0.524316	+	0.851524i	$0.324321\pi$
$548$	0.673551	0.0287727
$549$	21.4583	0.915819
$550$	−14.0423	−0.598766
$551$	11.3100	0.481824
$552$	−11.7243	−0.499019
$553$	5.74331	0.244230
$554$	−14.7162	−0.625232
$555$	30.4767	1.29366
$556$	−0.301308	−0.0127783
$557$	−24.6781	−1.04565	−0.522823	−	0.852442i	$-0.675121\pi$
$557$	−24.6781	−1.04565	−0.522823	+	0.852442i	$0.675121\pi$
$558$	−4.85079	−0.205350
$559$	−43.4356	−1.83713
$560$	33.2681	1.40584
$561$	−14.8799	−0.628232
$562$	19.0791	0.804803
$563$	40.9376	1.72531	0.862656	−	0.505790i	$-0.168799\pi$
$563$	40.9376	1.72531	0.862656	+	0.505790i	$0.168799\pi$
$564$	0.0194187	0.000817674 0
$565$	23.0357	0.969121
$566$	15.2849	0.642472
$567$	−3.07070	−0.128957
$568$	−36.5540	−1.53377
$569$	33.3049	1.39622	0.698108	−	0.715993i	$-0.254026\pi$
$569$	33.3049	1.39622	0.698108	+	0.715993i	$0.254026\pi$
$570$	−10.8774	−0.455605
$571$	−23.8723	−0.999026	−0.499513	−	0.866306i	$-0.666488\pi$
$571$	−23.8723	−0.999026	−0.499513	+	0.866306i	$0.666488\pi$
$572$	−0.539789	−0.0225697
$573$	−26.0598	−1.08866
$574$	−51.1845	−2.13640
$575$	12.5508	0.523405
$576$	−15.8118	−0.658826
$577$	−11.0786	−0.461208	−0.230604	−	0.973048i	$-0.574070\pi$
$577$	−11.0786	−0.461208	−0.230604	+	0.973048i	$0.574070\pi$
$578$	−5.09935	−0.212105
$579$	9.76825	0.405955
$580$	−0.364806	−0.0151477
$581$	−36.4210	−1.51100
$582$	−10.0830	−0.417955
$583$	−8.40768	−0.348210
$584$	−19.6322	−0.812387
$585$	30.0697	1.24323
$586$	−19.5781	−0.808762
$587$	−7.88613	−0.325495	−0.162748	−	0.986668i	$-0.552036\pi$
$587$	−7.88613	−0.325495	−0.162748	+	0.986668i	$0.552036\pi$
$588$	−0.0430829	−0.00177671
$589$	4.59535	0.189348
$590$	28.4407	1.17089
$591$	13.1064	0.539126
$592$	43.9685	1.80710
$593$	−11.5238	−0.473224	−0.236612	−	0.971604i	$-0.576037\pi$
$593$	−11.5238	−0.473224	−0.236612	+	0.971604i	$0.576037\pi$
$594$	−23.4128	−0.960640
$595$	−37.1627	−1.52352
$596$	−0.739749	−0.0303013
$597$	19.9164	0.815125
$598$	31.6690	1.29504
$599$	−6.99725	−0.285900	−0.142950	−	0.989730i	$-0.545659\pi$
$599$	−6.99725	−0.285900	−0.142950	+	0.989730i	$0.545659\pi$
$600$	−8.35780	−0.341206
$601$	−6.91194	−0.281944	−0.140972	−	0.990014i	$-0.545023\pi$
$601$	−6.91194	−0.281944	−0.140972	+	0.990014i	$0.545023\pi$
$602$	33.8720	1.38052
$603$	26.3664	1.07372
$604$	0.0700556	0.00285052
$605$	−0.418838	−0.0170282
$606$	8.38420	0.340585
$607$	4.92306	0.199821	0.0999105	−	0.994996i	$-0.468144\pi$
$607$	4.92306	0.199821	0.0999105	+	0.994996i	$0.468144\pi$
$608$	−0.474551	−0.0192456
$609$	−12.0360	−0.487723
$610$	43.0404	1.74266
$611$	3.33815	0.135047
$612$	−0.281901	−0.0113952
$613$	−28.7819	−1.16249	−0.581246	−	0.813728i	$-0.697434\pi$
$613$	−28.7819	−1.16249	−0.581246	+	0.813728i	$0.697434\pi$
$614$	3.30268	0.133285
$615$	34.8852	1.40671
$616$	−26.7891	−1.07936
$617$	35.1742	1.41606	0.708029	−	0.706183i	$-0.249584\pi$
$617$	35.1742	1.41606	0.708029	+	0.706183i	$0.249584\pi$
$618$	13.7627	0.553617
$619$	−1.00000	−0.0401934
$619$	−1.00000	−0.0401934
$620$	−0.148223	−0.00595279
$621$	20.9260	0.839733
$622$	−12.7188	−0.509977
$623$	−30.3613	−1.21640
$624$	−21.4152	−0.857295
$625$	−31.0088	−1.24035
$626$	−6.08172	−0.243074
$627$	8.89455	0.355214
$628$	0.129115	0.00515225
$629$	−49.1158	−1.95838
$630$	−23.4490	−0.934231
$631$	−19.4331	−0.773619	−0.386809	−	0.922160i	$-0.626423\pi$
$631$	−19.4331	−0.773619	−0.386809	+	0.922160i	$0.626423\pi$
$632$	−5.56123	−0.221214
$633$	−15.5100	−0.616467
$634$	18.7686	0.745395
$635$	−47.1061	−1.86935
$636$	0.0786303	0.00311790
$637$	−7.40614	−0.293442
$638$	19.5811	0.775224
$639$	26.1638	1.03502
$640$	−32.7043	−1.29275
$641$	7.05041	0.278474	0.139237	−	0.990259i	$-0.455535\pi$
$641$	7.05041	0.278474	0.139237	+	0.990259i	$0.455535\pi$
$642$	11.2580	0.444319
$643$	−21.7735	−0.858662	−0.429331	−	0.903147i	$-0.641251\pi$
$643$	−21.7735	−0.858662	−0.429331	+	0.903147i	$0.641251\pi$
$644$	−0.376229	−0.0148255
$645$	−23.0858	−0.909001
$646$	17.5299	0.689704
$647$	19.0364	0.748398	0.374199	−	0.927348i	$-0.377918\pi$
$647$	19.0364	0.748398	0.374199	+	0.927348i	$0.377918\pi$
$648$	2.97335	0.116804
$649$	−23.2562	−0.912887
$650$	22.5756	0.885488
$651$	−4.89031	−0.191666
$652$	0.209452	0.00820279
$653$	−4.76460	−0.186453	−0.0932265	−	0.995645i	$-0.529718\pi$
$653$	−4.76460	−0.186453	−0.0932265	+	0.995645i	$0.529718\pi$
$654$	−16.3603	−0.639740
$655$	43.0631	1.68261
$656$	50.3287	1.96500
$657$	14.0519	0.548216
$658$	−2.60316	−0.101482
$659$	−13.9179	−0.542164	−0.271082	−	0.962556i	$-0.587381\pi$
$659$	−13.9179	−0.542164	−0.271082	+	0.962556i	$0.587381\pi$
$660$	−0.286894	−0.0111673
$661$	21.5193	0.837003	0.418501	−	0.908216i	$-0.362556\pi$
$661$	21.5193	0.837003	0.418501	+	0.908216i	$0.362556\pi$
$662$	−14.5663	−0.566137
$663$	23.9222	0.929063
$664$	35.2663	1.36860
$665$	22.2142	0.861429
$666$	−30.9912	−1.20088
$667$	−17.5013	−0.677653
$668$	−0.198956	−0.00769783
$669$	10.5130	0.406457
$670$	52.8848	2.04312
$671$	−35.1945	−1.35867
$672$	0.505011	0.0194812
$673$	31.5578	1.21646	0.608231	−	0.793760i	$-0.291879\pi$
$673$	31.5578	1.21646	0.608231	+	0.793760i	$0.291879\pi$
$674$	−2.44983	−0.0943639
$675$	14.9174	0.574170
$676$	0.465590	0.0179073
$677$	−14.0561	−0.540219	−0.270110	−	0.962830i	$-0.587060\pi$
$677$	−14.0561	−0.540219	−0.270110	+	0.962830i	$0.587060\pi$
$678$	11.5636	0.444096
$679$	20.5919	0.790243
$680$	35.9845	1.37994
$681$	4.00923	0.153634
$682$	7.95594	0.304649
$683$	23.0995	0.883880	0.441940	−	0.897045i	$-0.354290\pi$
$683$	23.0995	0.883880	0.441940	+	0.897045i	$0.354290\pi$
$684$	0.168507	0.00644305
$685$	61.5397	2.35131
$686$	−23.1344	−0.883274
$687$	−24.4080	−0.931225
$688$	−33.3057	−1.26977
$689$	13.5169	0.514952
$690$	16.8319	0.640778
$691$	12.7900	0.486554	0.243277	−	0.969957i	$-0.421778\pi$
$691$	12.7900	0.486554	0.243277	+	0.969957i	$0.421778\pi$
$692$	0.548098	0.0208356
$693$	19.1745	0.728377
$694$	33.3823	1.26718
$695$	−27.5294	−1.04425
$696$	11.6544	0.441760
$697$	−56.2205	−2.12950
$698$	−11.9961	−0.454057
$699$	−1.99067	−0.0752940
$700$	−0.268199	−0.0101370
$701$	−20.9473	−0.791168	−0.395584	−	0.918430i	$-0.629458\pi$
$701$	−20.9473	−0.791168	−0.395584	+	0.918430i	$0.629458\pi$
$702$	37.6404	1.42065
$703$	29.3592	1.10730
$704$	25.9335	0.977405
$705$	1.77421	0.0668205
$706$	−35.3504	−1.33043
$707$	−17.1225	−0.643957
$708$	0.217497	0.00817404
$709$	26.7680	1.00529	0.502646	−	0.864492i	$-0.332360\pi$
$709$	26.7680	1.00529	0.502646	+	0.864492i	$0.332360\pi$
$710$	52.4785	1.96948
$711$	3.98048	0.149280
$712$	29.3987	1.10176
$713$	−7.11090	−0.266305
$714$	−18.6551	−0.698149
$715$	−49.3183	−1.84440
$716$	0.280372	0.0104780
$717$	20.5188	0.766287
$718$	−21.8252	−0.814510
$719$	−27.9520	−1.04243	−0.521217	−	0.853424i	$-0.674522\pi$
$719$	−27.9520	−1.04243	−0.521217	+	0.853424i	$0.674522\pi$
$720$	23.0570	0.859282
$721$	−28.1066	−1.04674
$722$	16.5985	0.617733
$723$	19.8040	0.736518
$724$	0.680538	0.0252920
$725$	−12.4760	−0.463348
$726$	−0.210250	−0.00780309
$727$	−25.7442	−0.954800	−0.477400	−	0.878686i	$-0.658421\pi$
$727$	−25.7442	−0.954800	−0.477400	+	0.878686i	$0.658421\pi$
$728$	43.0684	1.59622
$729$	12.7696	0.472946
$730$	28.1848	1.04317
$731$	37.2047	1.37606
$732$	0.329146	0.0121656
$733$	0.0489534	0.00180813	0.000904067	−	1.00000i	$-0.499712\pi$
$733$	0.0489534	0.00180813	0.000904067		1.00000i	$0.499712\pi$
$734$	14.8152	0.546840
$735$	−3.93632	−0.145193
$736$	0.734327	0.0270676
$737$	−43.2443	−1.59293
$738$	−35.4741	−1.30582
$739$	−35.1466	−1.29289	−0.646445	−	0.762961i	$-0.723745\pi$
$739$	−35.1466	−1.29289	−0.646445	+	0.762961i	$0.723745\pi$
$740$	−0.946983	−0.0348118
$741$	−14.2996	−0.525310
$742$	−10.5408	−0.386964
$743$	9.83355	0.360758	0.180379	−	0.983597i	$-0.442268\pi$
$743$	9.83355	0.360758	0.180379	+	0.983597i	$0.442268\pi$
$744$	4.73527	0.173603
$745$	−67.5879	−2.47623
$746$	0.169111	0.00619158
$747$	−25.2421	−0.923559
$748$	0.462355	0.0169054
$749$	−22.9915	−0.840091
$750$	−8.05835	−0.294249
$751$	6.56348	0.239505	0.119752	−	0.992804i	$-0.461790\pi$
$751$	6.56348	0.239505	0.119752	+	0.992804i	$0.461790\pi$
$752$	2.55964	0.0933404
$753$	5.65179	0.205963
$754$	−31.4802	−1.14644
$755$	6.40070	0.232945
$756$	−0.447170	−0.0162634
$757$	−34.1021	−1.23946	−0.619731	−	0.784814i	$-0.712758\pi$
$757$	−34.1021	−1.23946	−0.619731	+	0.784814i	$0.712758\pi$
$758$	−11.5014	−0.417751
$759$	−13.7636	−0.499585
$760$	−21.5099	−0.780247
$761$	−46.9148	−1.70066	−0.850330	−	0.526250i	$-0.823597\pi$
$761$	−46.9148	−1.70066	−0.850330	+	0.526250i	$0.823597\pi$
$762$	−23.6465	−0.856623
$763$	33.4116	1.20958
$764$	0.809740	0.0292954
$765$	−25.7562	−0.931216
$766$	−1.66120	−0.0600214
$767$	37.3887	1.35003
$768$	−0.739220	−0.0266743
$769$	−22.3593	−0.806296	−0.403148	−	0.915135i	$-0.632084\pi$
$769$	−22.3593	−0.806296	−0.403148	+	0.915135i	$0.632084\pi$
$770$	38.4595	1.38598
$771$	−11.6235	−0.418612
$772$	−0.303523	−0.0109240
$773$	−33.5391	−1.20632	−0.603159	−	0.797621i	$-0.706092\pi$
$773$	−33.5391	−1.20632	−0.603159	+	0.797621i	$0.706092\pi$
$774$	23.4755	0.843809
$775$	−5.06909	−0.182087
$776$	−19.9390	−0.715770
$777$	−31.2437	−1.12086
$778$	6.40053	0.229470
$779$	33.6061	1.20406
$780$	0.461235	0.0165149
$781$	−42.9121	−1.53552
$782$	−27.1260	−0.970023
$783$	−20.8013	−0.743379
$784$	−5.67890	−0.202818
$785$	11.7967	0.421043
$786$	21.6170	0.771052
$787$	−30.3661	−1.08243	−0.541217	−	0.840883i	$-0.682036\pi$
$787$	−30.3661	−1.08243	−0.541217	+	0.840883i	$0.682036\pi$
$788$	−0.407247	−0.0145076
$789$	−12.2867	−0.437419
$790$	7.98393	0.284056
$791$	−23.6155	−0.839669
$792$	−18.5666	−0.659734
$793$	56.5816	2.00927
$794$	−5.96207	−0.211586
$795$	7.18414	0.254795
$796$	−0.618851	−0.0219346
$797$	−41.3506	−1.46471	−0.732356	−	0.680922i	$-0.761579\pi$
$797$	−41.3506	−1.46471	−0.732356	+	0.680922i	$0.761579\pi$
$798$	11.1512	0.394747
$799$	−2.85929	−0.101154
$800$	0.523474	0.0185076
$801$	−21.0423	−0.743493
$802$	−32.2518	−1.13885
$803$	−23.0469	−0.813309
$804$	0.404430	0.0142631
$805$	−34.3745	−1.21154
$806$	−12.7906	−0.450531
$807$	17.9895	0.633259
$808$	16.5796	0.583270
$809$	−19.3052	−0.678736	−0.339368	−	0.940654i	$-0.610213\pi$
$809$	−19.3052	−0.678736	−0.339368	+	0.940654i	$0.610213\pi$
$810$	−4.26867	−0.149986
$811$	−46.8302	−1.64443	−0.822215	−	0.569176i	$-0.807262\pi$
$811$	−46.8302	−1.64443	−0.822215	+	0.569176i	$0.807262\pi$
$812$	0.373987	0.0131244
$813$	−0.803989	−0.0281971
$814$	50.8297	1.78158
$815$	19.1368	0.670334
$816$	18.3432	0.642139
$817$	−22.2393	−0.778054
$818$	19.9017	0.695848
$819$	−30.8265	−1.07716
$820$	−1.08397	−0.0378537
$821$	11.2555	0.392820	0.196410	−	0.980522i	$-0.437072\pi$
$821$	11.2555	0.392820	0.196410	+	0.980522i	$0.437072\pi$
$822$	30.8919	1.07748
$823$	36.2736	1.26442	0.632210	−	0.774797i	$-0.282148\pi$
$823$	36.2736	1.26442	0.632210	+	0.774797i	$0.282148\pi$
$824$	27.2155	0.948097
$825$	−9.81151	−0.341593
$826$	−29.1565	−1.01448
$827$	35.1579	1.22256	0.611280	−	0.791414i	$-0.290655\pi$
$827$	35.1579	1.22256	0.611280	+	0.791414i	$0.290655\pi$
$828$	−0.260751	−0.00906171
$829$	24.2001	0.840505	0.420253	−	0.907407i	$-0.361941\pi$
$829$	24.2001	0.840505	0.420253	+	0.907407i	$0.361941\pi$
$830$	−50.6298	−1.75738
$831$	−10.2824	−0.356691
$832$	−41.6929	−1.44544
$833$	6.34371	0.219797
$834$	−13.8193	−0.478523
$835$	−18.1778	−0.629068
$836$	−0.276375	−0.00955862
$837$	−8.45172	−0.292134
$838$	42.5263	1.46905
$839$	52.9069	1.82655	0.913274	−	0.407345i	$-0.133545\pi$
$839$	52.9069	1.82655	0.913274	+	0.407345i	$0.133545\pi$
$840$	22.8906	0.789800
$841$	−11.6030	−0.400103
$842$	−28.2598	−0.973898
$843$	13.3308	0.459136
$844$	0.481932	0.0165888
$845$	42.5391	1.46339
$846$	−1.80416	−0.0620283
$847$	0.429378	0.0147536
$848$	10.3645	0.355919
$849$	10.6797	0.366527
$850$	−19.3371	−0.663257
$851$	−45.4308	−1.55735
$852$	0.401323	0.0137491
$853$	−34.6798	−1.18741	−0.593706	−	0.804682i	$-0.702336\pi$
$853$	−34.6798	−1.18741	−0.593706	+	0.804682i	$0.702336\pi$
$854$	−44.1236	−1.50988
$855$	15.3959	0.526527
$856$	22.2626	0.760920
$857$	0.810262	0.0276780	0.0138390	−	0.999904i	$-0.495595\pi$
$857$	0.810262	0.0276780	0.0138390	+	0.999904i	$0.495595\pi$
$858$	−24.7570	−0.845190
$859$	39.1228	1.33485	0.667426	−	0.744676i	$-0.267396\pi$
$859$	39.1228	1.33485	0.667426	+	0.744676i	$0.267396\pi$
$860$	0.717329	0.0244607
$861$	−35.7631	−1.21880
$862$	4.84228	0.164929
$863$	9.76014	0.332239	0.166120	−	0.986106i	$-0.446876\pi$
$863$	9.76014	0.332239	0.166120	+	0.986106i	$0.446876\pi$
$864$	0.872790	0.0296929
$865$	50.0775	1.70269
$866$	37.5301	1.27532
$867$	−3.56297	−0.121005
$868$	0.151953	0.00515763
$869$	−6.52853	−0.221465
$870$	−16.7316	−0.567253
$871$	69.5232	2.35570
$872$	−32.3523	−1.09559
$873$	14.2715	0.483017
$874$	16.2147	0.548470
$875$	16.4570	0.556348
$876$	0.215540	0.00728241
$877$	4.79212	0.161819	0.0809093	−	0.996721i	$-0.474218\pi$
$877$	4.79212	0.161819	0.0809093	+	0.996721i	$0.474218\pi$
$878$	12.4313	0.419535
$879$	−13.6794	−0.461394
$880$	−37.8165	−1.27479
$881$	20.6764	0.696607	0.348303	−	0.937382i	$-0.386758\pi$
$881$	20.6764	0.696607	0.348303	+	0.937382i	$0.386758\pi$
$882$	4.00277	0.134780
$883$	−5.82393	−0.195991	−0.0979955	−	0.995187i	$-0.531243\pi$
$883$	−5.82393	−0.195991	−0.0979955	+	0.995187i	$0.531243\pi$
$884$	−0.743320	−0.0250006
$885$	19.8718	0.667985
$886$	−9.02923	−0.303343
$887$	32.9137	1.10513	0.552567	−	0.833469i	$-0.313648\pi$
$887$	32.9137	1.10513	0.552567	+	0.833469i	$0.313648\pi$
$888$	30.2531	1.01523
$889$	48.2916	1.61965
$890$	−42.2060	−1.41475
$891$	3.49052	0.116937
$892$	−0.326664	−0.0109375
$893$	1.70915	0.0571946
$894$	−33.9280	−1.13472
$895$	25.6165	0.856265
$896$	33.5273	1.12007
$897$	22.1274	0.738814
$898$	11.1915	0.373464
$899$	7.06853	0.235749
$900$	−0.185879	−0.00619598
$901$	−11.5779	−0.385714
$902$	58.1823	1.93726
$903$	23.6667	0.787580
$904$	22.8668	0.760538
$905$	62.1781	2.06687
$906$	3.21305	0.106746
$907$	−0.383458	−0.0127325	−0.00636626	−	0.999980i	$-0.502026\pi$
$907$	−0.383458	−0.0127325	−0.00636626	+	0.999980i	$0.502026\pi$
$908$	−0.124576	−0.00413421
$909$	−11.8670	−0.393603
$910$	−61.8307	−2.04967
$911$	0.928045	0.0307475	0.0153738	−	0.999882i	$-0.495106\pi$
$911$	0.928045	0.0307475	0.0153738	+	0.999882i	$0.495106\pi$
$912$	−10.9647	−0.363078
$913$	41.4004	1.37015
$914$	−23.3827	−0.773433
$915$	30.0728	0.994176
$916$	0.758416	0.0250588
$917$	−44.1468	−1.45786
$918$	−32.2408	−1.06411
$919$	38.5179	1.27059	0.635294	−	0.772270i	$-0.280879\pi$
$919$	38.5179	1.27059	0.635294	+	0.772270i	$0.280879\pi$
$920$	33.2848	1.09737
$921$	2.30762	0.0760386
$922$	−36.0554	−1.18742
$923$	68.9890	2.27080
$924$	0.294114	0.00967565
$925$	−32.3859	−1.06484
$926$	17.7754	0.584135
$927$	−19.4797	−0.639796
$928$	−0.729951	−0.0239618
$929$	−48.7038	−1.59792	−0.798960	−	0.601384i	$-0.794616\pi$
$929$	−48.7038	−1.59792	−0.798960	+	0.601384i	$0.794616\pi$
$930$	−6.79815	−0.222920
$931$	−3.79198	−0.124277
$932$	0.0618548	0.00202612
$933$	−8.88675	−0.290939
$934$	−46.8802	−1.53397
$935$	42.2435	1.38151
$936$	29.8492	0.975650
$937$	16.3327	0.533565	0.266782	−	0.963757i	$-0.414039\pi$
$937$	16.3327	0.533565	0.266782	+	0.963757i	$0.414039\pi$
$938$	−54.2157	−1.77021
$939$	−4.24936	−0.138673
$940$	−0.0551288	−0.00179810
$941$	−47.9485	−1.56307	−0.781537	−	0.623858i	$-0.785564\pi$
$941$	−47.9485	−1.56307	−0.781537	+	0.623858i	$0.785564\pi$
$942$	5.92176	0.192941
$943$	−52.0025	−1.69343
$944$	28.6690	0.933096
$945$	−40.8561	−1.32905
$946$	−38.5029	−1.25184
$947$	−39.8868	−1.29615	−0.648074	−	0.761578i	$-0.724425\pi$
$947$	−39.8868	−1.29615	−0.648074	+	0.761578i	$0.724425\pi$
$948$	0.0610561	0.00198301
$949$	37.0522	1.20276
$950$	11.5588	0.375018
$951$	13.1138	0.425244
$952$	−36.8901	−1.19562
$953$	−34.3873	−1.11391	−0.556957	−	0.830541i	$-0.688031\pi$
$953$	−34.3873	−1.11391	−0.556957	+	0.830541i	$0.688031\pi$
$954$	−7.30542	−0.236522
$955$	73.9827	2.39402
$956$	−0.637566	−0.0206204
$957$	13.6815	0.442261
$958$	−38.2811	−1.23681
$959$	−63.0884	−2.03723
$960$	−22.1595	−0.715195
$961$	−28.1280	−0.907355
$962$	−81.7181	−2.63470
$963$	−15.9346	−0.513485
$964$	−0.615356	−0.0198193
$965$	−27.7317	−0.892714
$966$	−17.2555	−0.555185
$967$	25.0929	0.806932	0.403466	−	0.914995i	$-0.367805\pi$
$967$	25.0929	0.806932	0.403466	+	0.914995i	$0.367805\pi$
$968$	−0.415765	−0.0133632
$969$	12.2483	0.393473
$970$	28.6253	0.919103
$971$	47.5585	1.52623	0.763113	−	0.646265i	$-0.223670\pi$
$971$	47.5585	1.52623	0.763113	+	0.646265i	$0.223670\pi$
$972$	−0.495552	−0.0158948
$973$	28.2222	0.904761
$974$	−55.4467	−1.77663
$975$	15.7738	0.505166
$976$	43.3859	1.38875
$977$	8.11115	0.259499	0.129749	−	0.991547i	$-0.458583\pi$
$977$	8.11115	0.259499	0.129749	+	0.991547i	$0.458583\pi$
$978$	9.60637	0.307178
$979$	34.5122	1.10301
$980$	0.122311	0.00390707
$981$	23.1564	0.739326
$982$	22.4423	0.716162
$983$	−1.96320	−0.0626163	−0.0313082	−	0.999510i	$-0.509967\pi$
$983$	−1.96320	−0.0626163	−0.0313082	+	0.999510i	$0.509967\pi$
$984$	34.6293	1.10394
$985$	−37.2086	−1.18556
$986$	26.9643	0.858720
$987$	−1.81886	−0.0578949
$988$	0.444323	0.0141358
$989$	34.4133	1.09428
$990$	26.6549	0.847149
$991$	30.5433	0.970240	0.485120	−	0.874447i	$-0.338776\pi$
$991$	30.5433	0.970240	0.485120	+	0.874447i	$0.338776\pi$
$992$	−0.296584	−0.00941655
$993$	−10.1776	−0.322978
$994$	−53.7992	−1.70641
$995$	−56.5419	−1.79250
$996$	−0.387185	−0.0122684
$997$	−47.6927	−1.51044	−0.755222	−	0.655469i	$-0.772471\pi$
$997$	−47.6927	−1.51044	−0.755222	+	0.655469i	$0.772471\pi$
$998$	46.3140	1.46604
$999$	−53.9972	−1.70839

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	619.2.a.a.1.7	✓	21
3.2	odd	2		5571.2.a.e.1.15		21
4.3	odd	2		9904.2.a.j.1.13		21

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
619.2.a.a.1.7	✓	21	1.1	even	1	trivial
5571.2.a.e.1.15		21	3.2	odd	2
9904.2.a.j.1.13		21	4.3	odd	2

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:	\( N \)	\(=\)	\( 619 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	619.a (trivial)

\(f(q)\)	\(=\)	\(q-1.42511 q^{2} -0.995739 q^{3} +0.0309400 q^{4} +2.82686 q^{5} +1.41904 q^{6} -2.89800 q^{7} +2.80613 q^{8} -2.00850 q^{9} +O(q^{10})\)	\(q-1.42511 q^{2} -0.995739 q^{3} +0.0309400 q^{4} +2.82686 q^{5} +1.41904 q^{6} -2.89800 q^{7} +2.80613 q^{8} -2.00850 q^{9} -4.02859 q^{10} +3.29421 q^{11} -0.0308082 q^{12} -5.29605 q^{13} +4.12998 q^{14} -2.81482 q^{15} -4.06092 q^{16} +4.53632 q^{17} +2.86234 q^{18} -2.71161 q^{19} +0.0874631 q^{20} +2.88566 q^{21} -4.69462 q^{22} +4.19598 q^{23} -2.79417 q^{24} +2.99115 q^{25} +7.54746 q^{26} +4.98716 q^{27} -0.0896642 q^{28} -4.17097 q^{29} +4.01143 q^{30} -1.69469 q^{31} +0.175007 q^{32} -3.28018 q^{33} -6.46476 q^{34} -8.19226 q^{35} -0.0621430 q^{36} -10.8272 q^{37} +3.86434 q^{38} +5.27349 q^{39} +7.93254 q^{40} -12.3934 q^{41} -4.11238 q^{42} +8.20151 q^{43} +0.101923 q^{44} -5.67776 q^{45} -5.97973 q^{46} -0.630310 q^{47} +4.04362 q^{48} +1.39843 q^{49} -4.26272 q^{50} -4.51699 q^{51} -0.163860 q^{52} -2.55226 q^{53} -7.10726 q^{54} +9.31229 q^{55} -8.13217 q^{56} +2.70005 q^{57} +5.94410 q^{58} -7.05972 q^{59} -0.0870904 q^{60} -10.6837 q^{61} +2.41513 q^{62} +5.82065 q^{63} +7.87244 q^{64} -14.9712 q^{65} +4.67462 q^{66} -13.1274 q^{67} +0.140354 q^{68} -4.17810 q^{69} +11.6749 q^{70} -13.0265 q^{71} -5.63612 q^{72} -6.99619 q^{73} +15.4300 q^{74} -2.97841 q^{75} -0.0838971 q^{76} -9.54664 q^{77} -7.51530 q^{78} -1.98182 q^{79} -11.4797 q^{80} +1.05959 q^{81} +17.6620 q^{82} +12.5676 q^{83} +0.0892822 q^{84} +12.8236 q^{85} -11.6881 q^{86} +4.15320 q^{87} +9.24398 q^{88} +10.4766 q^{89} +8.09144 q^{90} +15.3480 q^{91} +0.129823 q^{92} +1.68747 q^{93} +0.898261 q^{94} -7.66534 q^{95} -0.174262 q^{96} -7.10553 q^{97} -1.99291 q^{98} -6.61644 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q - 9 q^{2} - 5 q^{3} + 15 q^{4} - 21 q^{5} - 6 q^{6} - 4 q^{7} - 21 q^{8} + 6 q^{9}+O(q^{10}) \) 21 * q - 9 * q^2 - 5 * q^3 + 15 * q^4 - 21 * q^5 - 6 * q^6 - 4 * q^7 - 21 * q^8 + 6 * q^9	\( 21 q - 9 q^{2} - 5 q^{3} + 15 q^{4} - 21 q^{5} - 6 q^{6} - 4 q^{7} - 21 q^{8} + 6 q^{9} + q^{10} - 27 q^{11} - 8 q^{12} - 11 q^{13} - 19 q^{14} - 10 q^{15} + 11 q^{16} - 14 q^{17} - 14 q^{18} - 15 q^{19} - 25 q^{20} - 42 q^{21} + 12 q^{22} - 14 q^{23} - 8 q^{24} + 16 q^{25} - 11 q^{26} - 5 q^{27} + q^{28} - 78 q^{29} + q^{30} - 8 q^{31} - 41 q^{32} - 6 q^{33} + 7 q^{34} - 3 q^{35} - q^{36} - 23 q^{37} + 21 q^{38} - 4 q^{39} + 12 q^{40} - 59 q^{41} + 39 q^{42} + 2 q^{43} - 50 q^{44} - 36 q^{45} - 15 q^{46} - 12 q^{47} + 10 q^{48} + 17 q^{49} - 23 q^{50} - 8 q^{51} + 18 q^{52} - 36 q^{53} - 4 q^{54} + 23 q^{55} - 28 q^{56} - 24 q^{57} + 46 q^{58} - 17 q^{59} + 8 q^{60} - 22 q^{61} + 42 q^{62} - 6 q^{63} + 49 q^{64} - 53 q^{65} + 29 q^{66} + 15 q^{67} - 16 q^{68} - 30 q^{69} + 44 q^{70} - 56 q^{71} + 12 q^{72} - 2 q^{73} - 12 q^{74} + 2 q^{75} - 4 q^{76} - 47 q^{77} + 36 q^{78} + 5 q^{79} + 15 q^{80} - 19 q^{81} + 47 q^{82} - q^{83} - 20 q^{84} - 29 q^{85} - 23 q^{86} + 44 q^{87} + 61 q^{88} - 12 q^{89} + 91 q^{90} + 5 q^{91} + 35 q^{92} - 15 q^{93} + 34 q^{94} - 17 q^{95} + 14 q^{96} + 21 q^{97} + 24 q^{98} - 38 q^{99}+O(q^{100}) \) 21 * q - 9 * q^2 - 5 * q^3 + 15 * q^4 - 21 * q^5 - 6 * q^6 - 4 * q^7 - 21 * q^8 + 6 * q^9 + q^10 - 27 * q^11 - 8 * q^12 - 11 * q^13 - 19 * q^14 - 10 * q^15 + 11 * q^16 - 14 * q^17 - 14 * q^18 - 15 * q^19 - 25 * q^20 - 42 * q^21 + 12 * q^22 - 14 * q^23 - 8 * q^24 + 16 * q^25 - 11 * q^26 - 5 * q^27 + q^28 - 78 * q^29 + q^30 - 8 * q^31 - 41 * q^32 - 6 * q^33 + 7 * q^34 - 3 * q^35 - q^36 - 23 * q^37 + 21 * q^38 - 4 * q^39 + 12 * q^40 - 59 * q^41 + 39 * q^42 + 2 * q^43 - 50 * q^44 - 36 * q^45 - 15 * q^46 - 12 * q^47 + 10 * q^48 + 17 * q^49 - 23 * q^50 - 8 * q^51 + 18 * q^52 - 36 * q^53 - 4 * q^54 + 23 * q^55 - 28 * q^56 - 24 * q^57 + 46 * q^58 - 17 * q^59 + 8 * q^60 - 22 * q^61 + 42 * q^62 - 6 * q^63 + 49 * q^64 - 53 * q^65 + 29 * q^66 + 15 * q^67 - 16 * q^68 - 30 * q^69 + 44 * q^70 - 56 * q^71 + 12 * q^72 - 2 * q^73 - 12 * q^74 + 2 * q^75 - 4 * q^76 - 47 * q^77 + 36 * q^78 + 5 * q^79 + 15 * q^80 - 19 * q^81 + 47 * q^82 - q^83 - 20 * q^84 - 29 * q^85 - 23 * q^86 + 44 * q^87 + 61 * q^88 - 12 * q^89 + 91 * q^90 + 5 * q^91 + 35 * q^92 - 15 * q^93 + 34 * q^94 - 17 * q^95 + 14 * q^96 + 21 * q^97 + 24 * q^98 - 38 * q^99

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