LMFDB - Embedded newform 3997.2.a.e.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3997,2,Mod(1,3997)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3997, 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("3997.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3997 = 7 \cdot 571 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3997.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:	$31.9162056879$
Analytic rank:	$1$
Dimension:	$73$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Character	$\chi$	$=$	3997.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.14277 q^{2} -1.83467 q^{3} +2.59148 q^{4} +4.00463 q^{5} +3.93129 q^{6} -1.00000 q^{7} -1.26740 q^{8} +0.366032 q^{9} +O(q^{10})$	\(q-2.14277 q^{2} -1.83467 q^{3} +2.59148 q^{4} +4.00463 q^{5} +3.93129 q^{6} -1.00000 q^{7} -1.26740 q^{8} +0.366032 q^{9} -8.58101 q^{10} -4.06484 q^{11} -4.75452 q^{12} +1.11140 q^{13} +2.14277 q^{14} -7.34719 q^{15} -2.46720 q^{16} -4.91667 q^{17} -0.784324 q^{18} +0.337966 q^{19} +10.3779 q^{20} +1.83467 q^{21} +8.71003 q^{22} +0.307343 q^{23} +2.32527 q^{24} +11.0370 q^{25} -2.38148 q^{26} +4.83247 q^{27} -2.59148 q^{28} -0.334140 q^{29} +15.7434 q^{30} +6.95104 q^{31} +7.82145 q^{32} +7.45766 q^{33} +10.5353 q^{34} -4.00463 q^{35} +0.948565 q^{36} +2.79010 q^{37} -0.724184 q^{38} -2.03906 q^{39} -5.07548 q^{40} +2.81024 q^{41} -3.93129 q^{42} -11.3588 q^{43} -10.5339 q^{44} +1.46582 q^{45} -0.658567 q^{46} +2.57566 q^{47} +4.52650 q^{48} +1.00000 q^{49} -23.6499 q^{50} +9.02049 q^{51} +2.88017 q^{52} -3.22706 q^{53} -10.3549 q^{54} -16.2782 q^{55} +1.26740 q^{56} -0.620058 q^{57} +0.715987 q^{58} -1.51780 q^{59} -19.0401 q^{60} +7.39953 q^{61} -14.8945 q^{62} -0.366032 q^{63} -11.8252 q^{64} +4.45075 q^{65} -15.9801 q^{66} -6.54849 q^{67} -12.7414 q^{68} -0.563875 q^{69} +8.58101 q^{70} -7.60957 q^{71} -0.463911 q^{72} -5.55325 q^{73} -5.97855 q^{74} -20.2494 q^{75} +0.875831 q^{76} +4.06484 q^{77} +4.36924 q^{78} +4.91528 q^{79} -9.88021 q^{80} -9.96412 q^{81} -6.02171 q^{82} +7.20462 q^{83} +4.75452 q^{84} -19.6894 q^{85} +24.3393 q^{86} +0.613038 q^{87} +5.15179 q^{88} +8.63203 q^{89} -3.14093 q^{90} -1.11140 q^{91} +0.796474 q^{92} -12.7529 q^{93} -5.51905 q^{94} +1.35343 q^{95} -14.3498 q^{96} +4.68486 q^{97} -2.14277 q^{98} -1.48786 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 73 q - 12 q^{2} - 18 q^{3} + 74 q^{4} - 22 q^{5} - 7 q^{6} - 73 q^{7} - 39 q^{8} + 81 q^{9}+O(q^{10}) $ 73 * q - 12 * q^2 - 18 * q^3 + 74 * q^4 - 22 * q^5 - 7 * q^6 - 73 * q^7 - 39 * q^8 + 81 * q^9	\( 73 q - 12 q^{2} - 18 q^{3} + 74 q^{4} - 22 q^{5} - 7 q^{6} - 73 q^{7} - 39 q^{8} + 81 q^{9} - 8 q^{10} - 24 q^{11} - 33 q^{12} - 24 q^{13} + 12 q^{14} - 9 q^{15} + 76 q^{16} - 31 q^{17} - 42 q^{18} - 31 q^{19} - 54 q^{20} + 18 q^{21} - 4 q^{22} - 30 q^{23} - 6 q^{24} + 71 q^{25} - 22 q^{26} - 78 q^{27} - 74 q^{28} + 39 q^{29} - 13 q^{30} - 30 q^{31} - 100 q^{32} - 51 q^{33} - 10 q^{34} + 22 q^{35} + 97 q^{36} - 22 q^{37} - 61 q^{38} - 4 q^{39} - 15 q^{40} - 49 q^{41} + 7 q^{42} - 50 q^{43} - 21 q^{44} - 86 q^{45} + 17 q^{46} - 73 q^{47} - 70 q^{48} + 73 q^{49} - 50 q^{50} - 47 q^{51} - 82 q^{52} + 4 q^{53} - 8 q^{54} - 34 q^{55} + 39 q^{56} - 27 q^{57} - 27 q^{58} - 118 q^{59} - 24 q^{60} - 25 q^{61} - 61 q^{62} - 81 q^{63} + 77 q^{64} - 32 q^{65} - 46 q^{66} - 71 q^{67} - 92 q^{68} - 21 q^{69} + 8 q^{70} + 2 q^{71} - 141 q^{72} - 54 q^{73} - 12 q^{74} - 66 q^{75} - 36 q^{76} + 24 q^{77} - 28 q^{78} - 6 q^{79} - 133 q^{80} + 101 q^{81} - 38 q^{82} - 214 q^{83} + 33 q^{84} - 21 q^{85} - 34 q^{86} - 115 q^{87} + 3 q^{88} - 60 q^{89} - 67 q^{90} + 24 q^{91} - 99 q^{92} - 15 q^{93} - 11 q^{94} + q^{95} - 43 q^{96} - 48 q^{97} - 12 q^{98} - 55 q^{99}+O(q^{100}) \) 73 * q - 12 * q^2 - 18 * q^3 + 74 * q^4 - 22 * q^5 - 7 * q^6 - 73 * q^7 - 39 * q^8 + 81 * q^9 - 8 * q^10 - 24 * q^11 - 33 * q^12 - 24 * q^13 + 12 * q^14 - 9 * q^15 + 76 * q^16 - 31 * q^17 - 42 * q^18 - 31 * q^19 - 54 * q^20 + 18 * q^21 - 4 * q^22 - 30 * q^23 - 6 * q^24 + 71 * q^25 - 22 * q^26 - 78 * q^27 - 74 * q^28 + 39 * q^29 - 13 * q^30 - 30 * q^31 - 100 * q^32 - 51 * q^33 - 10 * q^34 + 22 * q^35 + 97 * q^36 - 22 * q^37 - 61 * q^38 - 4 * q^39 - 15 * q^40 - 49 * q^41 + 7 * q^42 - 50 * q^43 - 21 * q^44 - 86 * q^45 + 17 * q^46 - 73 * q^47 - 70 * q^48 + 73 * q^49 - 50 * q^50 - 47 * q^51 - 82 * q^52 + 4 * q^53 - 8 * q^54 - 34 * q^55 + 39 * q^56 - 27 * q^57 - 27 * q^58 - 118 * q^59 - 24 * q^60 - 25 * q^61 - 61 * q^62 - 81 * q^63 + 77 * q^64 - 32 * q^65 - 46 * q^66 - 71 * q^67 - 92 * q^68 - 21 * q^69 + 8 * q^70 + 2 * q^71 - 141 * q^72 - 54 * q^73 - 12 * q^74 - 66 * q^75 - 36 * q^76 + 24 * q^77 - 28 * q^78 - 6 * q^79 - 133 * q^80 + 101 * q^81 - 38 * q^82 - 214 * q^83 + 33 * q^84 - 21 * q^85 - 34 * q^86 - 115 * q^87 + 3 * q^88 - 60 * q^89 - 67 * q^90 + 24 * q^91 - 99 * q^92 - 15 * q^93 - 11 * q^94 + q^95 - 43 * q^96 - 48 * q^97 - 12 * q^98 - 55 * 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 $

$2$	−2.14277	−1.51517	−0.757585	−	0.652737i	$-0.773621\pi$
$2$	−2.14277	−1.51517	−0.757585	+	0.652737i	$0.773621\pi$
$3$	−1.83467	−1.05925	−0.529625	−	0.848232i	$-0.677667\pi$
$3$	−1.83467	−1.05925	−0.529625	+	0.848232i	$0.677667\pi$
$4$	2.59148	1.29574
$5$	4.00463	1.79092	0.895462	−	0.445138i	$-0.146845\pi$
$5$	4.00463	1.79092	0.895462	+	0.445138i	$0.146845\pi$
$6$	3.93129	1.60494
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.26740	−0.448095
$9$	0.366032	0.122011
$10$	−8.58101	−2.71355
$11$	−4.06484	−1.22560	−0.612798	−	0.790240i	$-0.709956\pi$
$11$	−4.06484	−1.22560	−0.612798	+	0.790240i	$0.709956\pi$
$12$	−4.75452	−1.37251
$13$	1.11140	0.308247	0.154124	−	0.988052i	$-0.450745\pi$
$13$	1.11140	0.308247	0.154124	+	0.988052i	$0.450745\pi$
$14$	2.14277	0.572680
$15$	−7.34719	−1.89704
$16$	−2.46720	−0.616799
$17$	−4.91667	−1.19247	−0.596234	−	0.802811i	$-0.703337\pi$
$17$	−4.91667	−1.19247	−0.596234	+	0.802811i	$0.703337\pi$
$18$	−0.784324	−0.184867
$19$	0.337966	0.0775347	0.0387673	−	0.999248i	$-0.487657\pi$
$19$	0.337966	0.0775347	0.0387673	+	0.999248i	$0.487657\pi$
$20$	10.3779	2.32057
$21$	1.83467	0.400359
$22$	8.71003	1.85698
$23$	0.307343	0.0640855	0.0320428	−	0.999486i	$-0.489799\pi$
$23$	0.307343	0.0640855	0.0320428	+	0.999486i	$0.489799\pi$
$24$	2.32527	0.474645
$25$	11.0370	2.20741
$26$	−2.38148	−0.467047
$27$	4.83247	0.930010
$28$	−2.59148	−0.489743
$29$	−0.334140	−0.0620483	−0.0310241	−	0.999519i	$-0.509877\pi$
$29$	−0.334140	−0.0620483	−0.0310241	+	0.999519i	$0.509877\pi$
$30$	15.7434	2.87433
$31$	6.95104	1.24844	0.624222	−	0.781247i	$-0.285416\pi$
$31$	6.95104	1.24844	0.624222	+	0.781247i	$0.285416\pi$
$32$	7.82145	1.38265
$33$	7.45766	1.29821
$34$	10.5353	1.80679
$35$	−4.00463	−0.676906
$36$	0.948565	0.158094
$37$	2.79010	0.458689	0.229345	−	0.973345i	$-0.426342\pi$
$37$	2.79010	0.458689	0.229345	+	0.973345i	$0.426342\pi$
$38$	−0.724184	−0.117478
$39$	−2.03906	−0.326511
$40$	−5.07548	−0.802504
$41$	2.81024	0.438886	0.219443	−	0.975625i	$-0.429576\pi$
$41$	2.81024	0.438886	0.219443	+	0.975625i	$0.429576\pi$
$42$	−3.93129	−0.606612
$43$	−11.3588	−1.73220	−0.866100	−	0.499870i	$-0.833381\pi$
$43$	−11.3588	−1.73220	−0.866100	+	0.499870i	$0.833381\pi$
$44$	−10.5339	−1.58805
$45$	1.46582	0.218512
$46$	−0.658567	−0.0971004
$47$	2.57566	0.375698	0.187849	−	0.982198i	$-0.439848\pi$
$47$	2.57566	0.375698	0.187849	+	0.982198i	$0.439848\pi$
$48$	4.52650	0.653345
$49$	1.00000	0.142857
$50$	−23.6499	−3.34460
$51$	9.02049	1.26312
$52$	2.88017	0.399408
$53$	−3.22706	−0.443271	−0.221636	−	0.975130i	$-0.571140\pi$
$53$	−3.22706	−0.443271	−0.221636	+	0.975130i	$0.571140\pi$
$54$	−10.3549	−1.40912
$55$	−16.2782	−2.19495
$56$	1.26740	0.169364
$57$	−0.620058	−0.0821286
$58$	0.715987	0.0940136
$59$	−1.51780	−0.197601	−0.0988006	−	0.995107i	$-0.531501\pi$
$59$	−1.51780	−0.197601	−0.0988006	+	0.995107i	$0.531501\pi$
$60$	−19.0401	−2.45806
$61$	7.39953	0.947412	0.473706	−	0.880683i	$-0.342916\pi$
$61$	7.39953	0.947412	0.473706	+	0.880683i	$0.342916\pi$
$62$	−14.8945	−1.89160
$63$	−0.366032	−0.0461157
$64$	−11.8252	−1.47815
$65$	4.45075	0.552047
$66$	−15.9801	−1.96701
$67$	−6.54849	−0.800026	−0.400013	−	0.916510i	$-0.630994\pi$
$67$	−6.54849	−0.800026	−0.400013	+	0.916510i	$0.630994\pi$
$68$	−12.7414	−1.54513
$69$	−0.563875	−0.0678826
$70$	8.58101	1.02563
$71$	−7.60957	−0.903090	−0.451545	−	0.892248i	$-0.649127\pi$
$71$	−7.60957	−0.903090	−0.451545	+	0.892248i	$0.649127\pi$
$72$	−0.463911	−0.0546724
$73$	−5.55325	−0.649959	−0.324979	−	0.945721i	$-0.605357\pi$
$73$	−5.55325	−0.649959	−0.324979	+	0.945721i	$0.605357\pi$
$74$	−5.97855	−0.694992
$75$	−20.2494	−2.33820
$76$	0.875831	0.100465
$77$	4.06484	0.463232
$78$	4.36924	0.494719
$79$	4.91528	0.553012	0.276506	−	0.961012i	$-0.410823\pi$
$79$	4.91528	0.553012	0.276506	+	0.961012i	$0.410823\pi$
$80$	−9.88021	−1.10464
$81$	−9.96412	−1.10712
$82$	−6.02171	−0.664987
$83$	7.20462	0.790810	0.395405	−	0.918507i	$-0.370604\pi$
$83$	7.20462	0.790810	0.395405	+	0.918507i	$0.370604\pi$
$84$	4.75452	0.518761
$85$	−19.6894	−2.13562
$86$	24.3393	2.62458
$87$	0.613038	0.0657246
$88$	5.15179	0.549183
$89$	8.63203	0.914993	0.457497	−	0.889211i	$-0.348746\pi$
$89$	8.63203	0.914993	0.457497	+	0.889211i	$0.348746\pi$
$90$	−3.14093	−0.331083
$91$	−1.11140	−0.116506
$92$	0.796474	0.0830381
$93$	−12.7529	−1.32241
$94$	−5.51905	−0.569246
$95$	1.35343	0.138859
$96$	−14.3498	−1.46457
$97$	4.68486	0.475675	0.237838	−	0.971305i	$-0.423561\pi$
$97$	4.68486	0.475675	0.237838	+	0.971305i	$0.423561\pi$
$98$	−2.14277	−0.216453
$99$	−1.48786	−0.149536
$100$	28.6023	2.86023
$101$	−7.06455	−0.702949	−0.351475	−	0.936197i	$-0.614320\pi$
$101$	−7.06455	−0.702949	−0.351475	+	0.936197i	$0.614320\pi$
$102$	−19.3289	−1.91384
$103$	10.3508	1.01990	0.509948	−	0.860205i	$-0.329665\pi$
$103$	10.3508	1.01990	0.509948	+	0.860205i	$0.329665\pi$
$104$	−1.40859	−0.138124
$105$	7.34719	0.717012
$106$	6.91486	0.671631
$107$	−4.67490	−0.451939	−0.225970	−	0.974134i	$-0.572555\pi$
$107$	−4.67490	−0.451939	−0.225970	+	0.974134i	$0.572555\pi$
$108$	12.5233	1.20505
$109$	−0.219551	−0.0210291	−0.0105146	−	0.999945i	$-0.503347\pi$
$109$	−0.219551	−0.0210291	−0.0105146	+	0.999945i	$0.503347\pi$
$110$	34.8804	3.32572
$111$	−5.11892	−0.485866
$112$	2.46720	0.233128
$113$	−0.997076	−0.0937970	−0.0468985	−	0.998900i	$-0.514934\pi$
$113$	−0.997076	−0.0937970	−0.0468985	+	0.998900i	$0.514934\pi$
$114$	1.32864	0.124439
$115$	1.23080	0.114772
$116$	−0.865917	−0.0803984
$117$	0.406809	0.0376095
$118$	3.25231	0.299399
$119$	4.91667	0.450710
$120$	9.31186	0.850052
$121$	5.52292	0.502084
$122$	−15.8555	−1.43549
$123$	−5.15588	−0.464890
$124$	18.0135	1.61766
$125$	24.1761	2.16238
$126$	0.784324	0.0698732
$127$	−3.20677	−0.284555	−0.142277	−	0.989827i	$-0.545442\pi$
$127$	−3.20677	−0.284555	−0.142277	+	0.989827i	$0.545442\pi$
$128$	9.69583	0.856999
$129$	20.8397	1.83483
$130$	−9.53694	−0.836445
$131$	−16.3518	−1.42866	−0.714331	−	0.699808i	$-0.753269\pi$
$131$	−16.3518	−1.42866	−0.714331	+	0.699808i	$0.753269\pi$
$132$	19.3264	1.68214
$133$	−0.337966	−0.0293054
$134$	14.0319	1.21217
$135$	19.3523	1.66558
$136$	6.23141	0.534339
$137$	8.16144	0.697279	0.348639	−	0.937257i	$-0.386644\pi$
$137$	8.16144	0.697279	0.348639	+	0.937257i	$0.386644\pi$
$138$	1.20826	0.102854
$139$	−18.4245	−1.56274	−0.781371	−	0.624066i	$-0.785479\pi$
$139$	−18.4245	−1.56274	−0.781371	+	0.624066i	$0.785479\pi$
$140$	−10.3779	−0.877093
$141$	−4.72549	−0.397958
$142$	16.3056	1.36833
$143$	−4.51767	−0.377786
$144$	−0.903074	−0.0752561
$145$	−1.33811	−0.111124
$146$	11.8994	0.984798
$147$	−1.83467	−0.151321
$148$	7.23048	0.594341
$149$	18.7121	1.53296	0.766479	−	0.642269i	$-0.222007\pi$
$149$	18.7121	1.53296	0.766479	+	0.642269i	$0.222007\pi$
$150$	43.3899	3.54277
$151$	−11.1434	−0.906836	−0.453418	−	0.891298i	$-0.649795\pi$
$151$	−11.1434	−0.906836	−0.453418	+	0.891298i	$0.649795\pi$
$152$	−0.428339	−0.0347429
$153$	−1.79966	−0.145494
$154$	−8.71003	−0.701874
$155$	27.8363	2.23587
$156$	−5.28418	−0.423073
$157$	−11.8784	−0.948003	−0.474001	−	0.880524i	$-0.657191\pi$
$157$	−11.8784	−0.948003	−0.474001	+	0.880524i	$0.657191\pi$
$158$	−10.5323	−0.837908
$159$	5.92061	0.469535
$160$	31.3220	2.47622
$161$	−0.307343	−0.0242220
$162$	21.3508	1.67748
$163$	−19.2017	−1.50399	−0.751996	−	0.659167i	$-0.770909\pi$
$163$	−19.2017	−1.50399	−0.751996	+	0.659167i	$0.770909\pi$
$164$	7.28268	0.568682
$165$	29.8652	2.32500
$166$	−15.4379	−1.19821
$167$	−10.2728	−0.794932	−0.397466	−	0.917617i	$-0.630110\pi$
$167$	−10.2728	−0.794932	−0.397466	+	0.917617i	$0.630110\pi$
$168$	−2.32527	−0.179399
$169$	−11.7648	−0.904984
$170$	42.1900	3.23582
$171$	0.123706	0.00946007
$172$	−29.4361	−2.24448
$173$	−6.46907	−0.491834	−0.245917	−	0.969291i	$-0.579089\pi$
$173$	−6.46907	−0.491834	−0.245917	+	0.969291i	$0.579089\pi$
$174$	−1.31360	−0.0995840
$175$	−11.0370	−0.834322
$176$	10.0288	0.755946
$177$	2.78468	0.209309
$178$	−18.4965	−1.38637
$179$	−7.41112	−0.553933	−0.276967	−	0.960880i	$-0.589329\pi$
$179$	−7.41112	−0.553933	−0.276967	+	0.960880i	$0.589329\pi$
$180$	3.79865	0.283135
$181$	18.9956	1.41193	0.705964	−	0.708247i	$-0.250514\pi$
$181$	18.9956	1.41193	0.705964	+	0.708247i	$0.250514\pi$
$182$	2.38148	0.176527
$183$	−13.5757	−1.00355
$184$	−0.389528	−0.0287164
$185$	11.1733	0.821477
$186$	27.3266	2.00368
$187$	19.9855	1.46148
$188$	6.67476	0.486807
$189$	−4.83247	−0.351511
$190$	−2.90009	−0.210395
$191$	6.01475	0.435212	0.217606	−	0.976037i	$-0.430175\pi$
$191$	6.01475	0.435212	0.217606	+	0.976037i	$0.430175\pi$
$192$	21.6954	1.56573
$193$	2.30469	0.165895	0.0829476	−	0.996554i	$-0.473567\pi$
$193$	2.30469	0.165895	0.0829476	+	0.996554i	$0.473567\pi$
$194$	−10.0386	−0.720729
$195$	−8.16568	−0.584756
$196$	2.59148	0.185106
$197$	−10.3756	−0.739232	−0.369616	−	0.929185i	$-0.620511\pi$
$197$	−10.3756	−0.739232	−0.369616	+	0.929185i	$0.620511\pi$
$198$	3.18815	0.226572
$199$	−13.6096	−0.964756	−0.482378	−	0.875963i	$-0.660227\pi$
$199$	−13.6096	−0.964756	−0.482378	+	0.875963i	$0.660227\pi$
$200$	−13.9884	−0.989129
$201$	12.0144	0.847427
$202$	15.1377	1.06509
$203$	0.334140	0.0234520
$204$	23.3764	1.63668
$205$	11.2540	0.786011
$206$	−22.1794	−1.54532
$207$	0.112498	0.00781912
$208$	−2.74205	−0.190127
$209$	−1.37378	−0.0950262
$210$	−15.7434	−1.08640
$211$	−4.14094	−0.285074	−0.142537	−	0.989789i	$-0.545526\pi$
$211$	−4.14094	−0.285074	−0.142537	+	0.989789i	$0.545526\pi$
$212$	−8.36286	−0.574364
$213$	13.9611	0.956598
$214$	10.0172	0.684765
$215$	−45.4878	−3.10224
$216$	−6.12470	−0.416733
$217$	−6.95104	−0.471868
$218$	0.470447	0.0318627
$219$	10.1884	0.688469
$220$	−42.1845	−2.84408
$221$	−5.46439	−0.367575
$222$	10.9687	0.736170
$223$	11.9176	0.798065	0.399032	−	0.916937i	$-0.369346\pi$
$223$	11.9176	0.798065	0.399032	+	0.916937i	$0.369346\pi$
$224$	−7.82145	−0.522593
$225$	4.03991	0.269328
$226$	2.13651	0.142118
$227$	−15.7276	−1.04388	−0.521939	−	0.852983i	$-0.674791\pi$
$227$	−15.7276	−1.04388	−0.521939	+	0.852983i	$0.674791\pi$
$228$	−1.60687	−0.106417
$229$	22.6419	1.49622	0.748109	−	0.663576i	$-0.230962\pi$
$229$	22.6419	1.49622	0.748109	+	0.663576i	$0.230962\pi$
$230$	−2.63732	−0.173899
$231$	−7.45766	−0.490678
$232$	0.423490	0.0278035
$233$	−16.2494	−1.06453	−0.532265	−	0.846578i	$-0.678659\pi$
$233$	−16.2494	−1.06453	−0.532265	+	0.846578i	$0.678659\pi$
$234$	−0.871699	−0.0569847
$235$	10.3145	0.672847
$236$	−3.93336	−0.256040
$237$	−9.01795	−0.585778
$238$	−10.5353	−0.682903
$239$	−15.8278	−1.02381	−0.511907	−	0.859041i	$-0.671061\pi$
$239$	−15.8278	−1.02381	−0.511907	+	0.859041i	$0.671061\pi$
$240$	18.1270	1.17009
$241$	24.9919	1.60987	0.804934	−	0.593364i	$-0.202201\pi$
$241$	24.9919	1.60987	0.804934	+	0.593364i	$0.202201\pi$
$242$	−11.8344	−0.760743
$243$	3.78349	0.242711
$244$	19.1757	1.22760
$245$	4.00463	0.255846
$246$	11.0479	0.704387
$247$	0.375616	0.0238999
$248$	−8.80978	−0.559422
$249$	−13.2181	−0.837666
$250$	−51.8039	−3.27637
$251$	6.45976	0.407736	0.203868	−	0.978998i	$-0.434649\pi$
$251$	6.45976	0.407736	0.203868	+	0.978998i	$0.434649\pi$
$252$	−0.948565	−0.0597540
$253$	−1.24930	−0.0785429
$254$	6.87138	0.431148
$255$	36.1237	2.26215
$256$	2.87444	0.179652
$257$	−12.9697	−0.809026	−0.404513	−	0.914532i	$-0.632559\pi$
$257$	−12.9697	−0.809026	−0.404513	+	0.914532i	$0.632559\pi$
$258$	−44.6548	−2.78008
$259$	−2.79010	−0.173368
$260$	11.5340	0.715309
$261$	−0.122306	−0.00757055
$262$	35.0382	2.16467
$263$	−2.11179	−0.130218	−0.0651091	−	0.997878i	$-0.520740\pi$
$263$	−2.11179	−0.130218	−0.0651091	+	0.997878i	$0.520740\pi$
$264$	−9.45187	−0.581722
$265$	−12.9232	−0.793865
$266$	0.724184	0.0444026
$267$	−15.8370	−0.969206
$268$	−16.9703	−1.03662
$269$	12.1095	0.738329	0.369165	−	0.929364i	$-0.379644\pi$
$269$	12.1095	0.738329	0.369165	+	0.929364i	$0.379644\pi$
$270$	−41.4675	−2.52363
$271$	13.2963	0.807694	0.403847	−	0.914827i	$-0.367673\pi$
$271$	13.2963	0.807694	0.403847	+	0.914827i	$0.367673\pi$
$272$	12.1304	0.735513
$273$	2.03906	0.123410
$274$	−17.4881	−1.05650
$275$	−44.8638	−2.70539
$276$	−1.46127	−0.0879581
$277$	4.97400	0.298858	0.149429	−	0.988772i	$-0.452256\pi$
$277$	4.97400	0.298858	0.149429	+	0.988772i	$0.452256\pi$
$278$	39.4795	2.36782
$279$	2.54431	0.152324
$280$	5.07548	0.303318
$281$	−2.23415	−0.133278	−0.0666390	−	0.997777i	$-0.521228\pi$
$281$	−2.23415	−0.133278	−0.0666390	+	0.997777i	$0.521228\pi$
$282$	10.1257	0.602974
$283$	9.23065	0.548705	0.274352	−	0.961629i	$-0.411537\pi$
$283$	9.23065	0.548705	0.274352	+	0.961629i	$0.411537\pi$
$284$	−19.7200	−1.17017
$285$	−2.48310	−0.147086
$286$	9.68034	0.572410
$287$	−2.81024	−0.165883
$288$	2.86290	0.168698
$289$	7.17363	0.421978
$290$	2.86726	0.168371
$291$	−8.59520	−0.503859
$292$	−14.3911	−0.842177
$293$	−21.0293	−1.22854	−0.614272	−	0.789095i	$-0.710550\pi$
$293$	−21.0293	−1.22854	−0.614272	+	0.789095i	$0.710550\pi$
$294$	3.93129	0.229278
$295$	−6.07824	−0.353889
$296$	−3.53618	−0.205536
$297$	−19.6432	−1.13982
$298$	−40.0959	−2.32269
$299$	0.341582	0.0197542
$300$	−52.4759	−3.02969
$301$	11.3588	0.654710
$302$	23.8778	1.37401
$303$	12.9612	0.744599
$304$	−0.833828	−0.0478233
$305$	29.6323	1.69674
$306$	3.85626	0.220448
$307$	9.29783	0.530655	0.265328	−	0.964158i	$-0.414520\pi$
$307$	9.29783	0.530655	0.265328	+	0.964158i	$0.414520\pi$
$308$	10.5339	0.600227
$309$	−18.9904	−1.08032
$310$	−59.6470	−3.38772
$311$	20.2591	1.14879	0.574395	−	0.818578i	$-0.305237\pi$
$311$	20.2591	1.14879	0.574395	+	0.818578i	$0.305237\pi$
$312$	2.58431	0.146308
$313$	15.2971	0.864643	0.432322	−	0.901719i	$-0.357695\pi$
$313$	15.2971	0.864643	0.432322	+	0.901719i	$0.357695\pi$
$314$	25.4528	1.43638
$315$	−1.46582	−0.0825898
$316$	12.7378	0.716560
$317$	−4.73396	−0.265886	−0.132943	−	0.991124i	$-0.542443\pi$
$317$	−4.73396	−0.265886	−0.132943	+	0.991124i	$0.542443\pi$
$318$	−12.6865	−0.711425
$319$	1.35823	0.0760461
$320$	−47.3556	−2.64726
$321$	8.57691	0.478717
$322$	0.658567	0.0367005
$323$	−1.66167	−0.0924576
$324$	−25.8218	−1.43454
$325$	12.2666	0.680428
$326$	41.1449	2.27880
$327$	0.402804	0.0222751
$328$	−3.56171	−0.196663
$329$	−2.57566	−0.142001
$330$	−63.9943	−3.52277
$331$	−11.4639	−0.630113	−0.315056	−	0.949073i	$-0.602023\pi$
$331$	−11.4639	−0.630113	−0.315056	+	0.949073i	$0.602023\pi$
$332$	18.6706	1.02468
$333$	1.02127	0.0559650
$334$	22.0123	1.20446
$335$	−26.2243	−1.43279
$336$	−4.52650	−0.246941
$337$	−22.3861	−1.21945	−0.609723	−	0.792614i	$-0.708719\pi$
$337$	−22.3861	−1.21945	−0.609723	+	0.792614i	$0.708719\pi$
$338$	25.2093	1.37120
$339$	1.82931	0.0993545
$340$	−51.0247	−2.76720
$341$	−28.2549	−1.53009
$342$	−0.265075	−0.0143336
$343$	−1.00000	−0.0539949
$344$	14.3962	0.776191
$345$	−2.25811	−0.121573
$346$	13.8618	0.745213
$347$	−32.7887	−1.76019	−0.880096	−	0.474796i	$-0.842522\pi$
$347$	−32.7887	−1.76019	−0.880096	+	0.474796i	$0.842522\pi$
$348$	1.58868	0.0851620
$349$	−25.0816	−1.34259	−0.671294	−	0.741191i	$-0.734261\pi$
$349$	−25.0816	−1.34259	−0.671294	+	0.741191i	$0.734261\pi$
$350$	23.6499	1.26414
$351$	5.37082	0.286673
$352$	−31.7930	−1.69457
$353$	−8.74627	−0.465517	−0.232758	−	0.972535i	$-0.574775\pi$
$353$	−8.74627	−0.465517	−0.232758	+	0.972535i	$0.574775\pi$
$354$	−5.96693	−0.317139
$355$	−30.4735	−1.61737
$356$	22.3697	1.18559
$357$	−9.02049	−0.477415
$358$	15.8804	0.839303
$359$	10.0553	0.530698	0.265349	−	0.964152i	$-0.414513\pi$
$359$	10.0553	0.530698	0.265349	+	0.964152i	$0.414513\pi$
$360$	−1.85779	−0.0979141
$361$	−18.8858	−0.993988
$362$	−40.7032	−2.13931
$363$	−10.1328	−0.531833
$364$	−2.88017	−0.150962
$365$	−22.2387	−1.16403
$366$	29.0897	1.52054
$367$	−26.9824	−1.40847	−0.704235	−	0.709967i	$-0.748710\pi$
$367$	−26.9824	−1.40847	−0.704235	+	0.709967i	$0.748710\pi$
$368$	−0.758277	−0.0395279
$369$	1.02864	0.0535488
$370$	−23.9419	−1.24468
$371$	3.22706	0.167541
$372$	−33.0489	−1.71350
$373$	−11.8673	−0.614468	−0.307234	−	0.951634i	$-0.599403\pi$
$373$	−11.8673	−0.614468	−0.307234	+	0.951634i	$0.599403\pi$
$374$	−42.8243	−2.21439
$375$	−44.3553	−2.29050
$376$	−3.26440	−0.168348
$377$	−0.371364	−0.0191262
$378$	10.3549	0.532599
$379$	8.69868	0.446821	0.223411	−	0.974724i	$-0.428281\pi$
$379$	8.69868	0.446821	0.223411	+	0.974724i	$0.428281\pi$
$380$	3.50738	0.179925
$381$	5.88338	0.301414
$382$	−12.8882	−0.659420
$383$	−8.20690	−0.419353	−0.209677	−	0.977771i	$-0.567241\pi$
$383$	−8.20690	−0.419353	−0.209677	+	0.977771i	$0.567241\pi$
$384$	−17.7887	−0.907776
$385$	16.2782	0.829612
$386$	−4.93843	−0.251359
$387$	−4.15769	−0.211347
$388$	12.1407	0.616351
$389$	−35.6631	−1.80819	−0.904095	−	0.427331i	$-0.859454\pi$
$389$	−35.6631	−1.80819	−0.904095	+	0.427331i	$0.859454\pi$
$390$	17.4972	0.886005
$391$	−1.51111	−0.0764199
$392$	−1.26740	−0.0640136
$393$	30.0002	1.51331
$394$	22.2326	1.12006
$395$	19.6839	0.990403
$396$	−3.85576	−0.193759
$397$	−7.84157	−0.393557	−0.196779	−	0.980448i	$-0.563048\pi$
$397$	−7.84157	−0.393557	−0.196779	+	0.980448i	$0.563048\pi$
$398$	29.1622	1.46177
$399$	0.620058	0.0310417
$400$	−27.2306	−1.36153
$401$	−4.93905	−0.246644	−0.123322	−	0.992367i	$-0.539355\pi$
$401$	−4.93905	−0.246644	−0.123322	+	0.992367i	$0.539355\pi$
$402$	−25.7440	−1.28400
$403$	7.72540	0.384829
$404$	−18.3076	−0.910839
$405$	−39.9026	−1.98278
$406$	−0.715987	−0.0355338
$407$	−11.3413	−0.562167
$408$	−11.4326	−0.565998
$409$	18.7729	0.928260	0.464130	−	0.885767i	$-0.346367\pi$
$409$	18.7729	0.928260	0.464130	+	0.885767i	$0.346367\pi$
$410$	−24.1147	−1.19094
$411$	−14.9736	−0.738593
$412$	26.8239	1.32152
$413$	1.51780	0.0746863
$414$	−0.241057	−0.0118473
$415$	28.8518	1.41628
$416$	8.69277	0.426198
$417$	33.8029	1.65534
$418$	2.94369	0.143981
$419$	−19.5514	−0.955149	−0.477575	−	0.878591i	$-0.658484\pi$
$419$	−19.5514	−0.955149	−0.477575	+	0.878591i	$0.658484\pi$
$420$	19.0401	0.929061
$421$	−4.06128	−0.197935	−0.0989674	−	0.995091i	$-0.531554\pi$
$421$	−4.06128	−0.197935	−0.0989674	+	0.995091i	$0.531554\pi$
$422$	8.87309	0.431935
$423$	0.942773	0.0458392
$424$	4.08999	0.198628
$425$	−54.2655	−2.63226
$426$	−29.9155	−1.44941
$427$	−7.39953	−0.358088
$428$	−12.1149	−0.585595
$429$	8.28845	0.400170
$430$	97.4700	4.70042
$431$	−2.28656	−0.110140	−0.0550698	−	0.998483i	$-0.517538\pi$
$431$	−2.28656	−0.110140	−0.0550698	+	0.998483i	$0.517538\pi$
$432$	−11.9227	−0.573630
$433$	15.4497	0.742467	0.371233	−	0.928540i	$-0.378935\pi$
$433$	15.4497	0.742467	0.371233	+	0.928540i	$0.378935\pi$
$434$	14.8945	0.714959
$435$	2.45499	0.117708
$436$	−0.568961	−0.0272483
$437$	0.103872	0.00496885
$438$	−21.8315	−1.04315
$439$	16.4247	0.783910	0.391955	−	0.919984i	$-0.371799\pi$
$439$	16.4247	0.783910	0.391955	+	0.919984i	$0.371799\pi$
$440$	20.6310	0.983545
$441$	0.366032	0.0174301
$442$	11.7090	0.556938
$443$	−19.4730	−0.925189	−0.462595	−	0.886570i	$-0.653081\pi$
$443$	−19.4730	−0.925189	−0.462595	+	0.886570i	$0.653081\pi$
$444$	−13.2656	−0.629556
$445$	34.5681	1.63868
$446$	−25.5368	−1.20920
$447$	−34.3307	−1.62379
$448$	11.8252	0.558688
$449$	−38.2186	−1.80365	−0.901824	−	0.432105i	$-0.857771\pi$
$449$	−38.2186	−1.80365	−0.901824	+	0.432105i	$0.857771\pi$
$450$	−8.65662	−0.408077
$451$	−11.4232	−0.537897
$452$	−2.58390	−0.121536
$453$	20.4445	0.960566
$454$	33.7007	1.58165
$455$	−4.45075	−0.208654
$456$	0.785863	0.0368014
$457$	7.82860	0.366206	0.183103	−	0.983094i	$-0.441386\pi$
$457$	7.82860	0.366206	0.183103	+	0.983094i	$0.441386\pi$
$458$	−48.5164	−2.26702
$459$	−23.7597	−1.10901
$460$	3.18958	0.148715
$461$	5.63596	0.262493	0.131246	−	0.991350i	$-0.458102\pi$
$461$	5.63596	0.262493	0.131246	+	0.991350i	$0.458102\pi$
$462$	15.9801	0.743460
$463$	−22.3620	−1.03925	−0.519625	−	0.854394i	$-0.673928\pi$
$463$	−22.3620	−1.03925	−0.519625	+	0.854394i	$0.673928\pi$
$464$	0.824389	0.0382713
$465$	−51.0706	−2.36834
$466$	34.8187	1.61294
$467$	−30.3325	−1.40362	−0.701811	−	0.712363i	$-0.747625\pi$
$467$	−30.3325	−1.40362	−0.701811	+	0.712363i	$0.747625\pi$
$468$	1.05424	0.0487321
$469$	6.54849	0.302381
$470$	−22.1017	−1.01948
$471$	21.7931	1.00417
$472$	1.92367	0.0885441
$473$	46.1717	2.12298
$474$	19.3234	0.887554
$475$	3.73014	0.171151
$476$	12.7414	0.584003
$477$	−1.18121	−0.0540838
$478$	33.9154	1.55125
$479$	37.6022	1.71809	0.859043	−	0.511903i	$-0.171059\pi$
$479$	37.6022	1.71809	0.859043	+	0.511903i	$0.171059\pi$
$480$	−57.4657	−2.62294
$481$	3.10092	0.141390
$482$	−53.5519	−2.43922
$483$	0.563875	0.0256572
$484$	14.3125	0.650570
$485$	18.7611	0.851899
$486$	−8.10717	−0.367749
$487$	−40.9402	−1.85518	−0.927589	−	0.373603i	$-0.878122\pi$
$487$	−40.9402	−1.85518	−0.927589	+	0.373603i	$0.878122\pi$
$488$	−9.37819	−0.424531
$489$	35.2289	1.59310
$490$	−8.58101	−0.387651
$491$	−2.84681	−0.128475	−0.0642375	−	0.997935i	$-0.520462\pi$
$491$	−2.84681	−0.128475	−0.0642375	+	0.997935i	$0.520462\pi$
$492$	−13.3613	−0.602376
$493$	1.64286	0.0739905
$494$	−0.804859	−0.0362123
$495$	−5.95834	−0.267807
$496$	−17.1496	−0.770039
$497$	7.60957	0.341336
$498$	28.3235	1.26921
$499$	19.1986	0.859448	0.429724	−	0.902960i	$-0.358611\pi$
$499$	19.1986	0.859448	0.429724	+	0.902960i	$0.358611\pi$
$500$	62.6519	2.80188
$501$	18.8472	0.842032
$502$	−13.8418	−0.617790
$503$	−24.4444	−1.08992	−0.544961	−	0.838461i	$-0.683455\pi$
$503$	−24.4444	−1.08992	−0.544961	+	0.838461i	$0.683455\pi$
$504$	0.463911	0.0206642
$505$	−28.2909	−1.25893
$506$	2.67697	0.119006
$507$	21.5846	0.958604
$508$	−8.31027	−0.368708
$509$	39.3259	1.74309	0.871544	−	0.490317i	$-0.163119\pi$
$509$	39.3259	1.74309	0.871544	+	0.490317i	$0.163119\pi$
$510$	−77.4049	−3.42755
$511$	5.55325	0.245661
$512$	−25.5509	−1.12920
$513$	1.63321	0.0721080
$514$	27.7911	1.22581
$515$	41.4511	1.82656
$516$	54.0057	2.37747
$517$	−10.4696	−0.460454
$518$	5.97855	0.262682
$519$	11.8686	0.520976
$520$	−5.64090	−0.247370
$521$	−17.4424	−0.764164	−0.382082	−	0.924128i	$-0.624793\pi$
$521$	−17.4424	−0.764164	−0.382082	+	0.924128i	$0.624793\pi$
$522$	0.262074	0.0114707
$523$	−1.30370	−0.0570067	−0.0285034	−	0.999594i	$-0.509074\pi$
$523$	−1.30370	−0.0570067	−0.0285034	+	0.999594i	$0.509074\pi$
$524$	−42.3753	−1.85117
$525$	20.2494	0.883756
$526$	4.52508	0.197303
$527$	−34.1760	−1.48873
$528$	−18.3995	−0.800736
$529$	−22.9055	−0.995893
$530$	27.6914	1.20284
$531$	−0.555565	−0.0241095
$532$	−0.875831	−0.0379721
$533$	3.12331	0.135285
$534$	33.9350	1.46851
$535$	−18.7212	−0.809389
$536$	8.29959	0.358488
$537$	13.5970	0.586754
$538$	−25.9479	−1.11869
$539$	−4.06484	−0.175085
$540$	50.1510	2.15815
$541$	18.6436	0.801549	0.400774	−	0.916177i	$-0.368741\pi$
$541$	18.6436	0.801549	0.400774	+	0.916177i	$0.368741\pi$
$542$	−28.4910	−1.22379
$543$	−34.8507	−1.49559
$544$	−38.4555	−1.64877
$545$	−0.879218	−0.0376616
$546$	−4.36924	−0.186986
$547$	23.0255	0.984498	0.492249	−	0.870454i	$-0.336175\pi$
$547$	23.0255	0.984498	0.492249	+	0.870454i	$0.336175\pi$
$548$	21.1502	0.903492
$549$	2.70847	0.115594
$550$	96.1330	4.09913
$551$	−0.112928	−0.00481089
$552$	0.714658	0.0304178
$553$	−4.91528	−0.209019
$554$	−10.6581	−0.452821
$555$	−20.4994	−0.870150
$556$	−47.7466	−2.02491
$557$	4.65131	0.197082	0.0985412	−	0.995133i	$-0.468582\pi$
$557$	4.65131	0.197082	0.0985412	+	0.995133i	$0.468582\pi$
$558$	−5.45187	−0.230796
$559$	−12.6242	−0.533946
$560$	9.88021	0.417515
$561$	−36.6668	−1.54808
$562$	4.78727	0.201939
$563$	−30.0434	−1.26618	−0.633090	−	0.774078i	$-0.718214\pi$
$563$	−30.0434	−1.26618	−0.633090	+	0.774078i	$0.718214\pi$
$564$	−12.2460	−0.515650
$565$	−3.99292	−0.167983
$566$	−19.7792	−0.831381
$567$	9.96412	0.418454
$568$	9.64440	0.404670
$569$	−32.1811	−1.34910	−0.674550	−	0.738229i	$-0.735663\pi$
$569$	−32.1811	−1.34910	−0.674550	+	0.738229i	$0.735663\pi$
$570$	5.32072	0.222860
$571$	−1.00000	−0.0418487
$571$	−1.00000	−0.0418487
$572$	−11.7074	−0.489513
$573$	−11.0351	−0.460998
$574$	6.02171	0.251341
$575$	3.39216	0.141463
$576$	−4.32841	−0.180350
$577$	3.57387	0.148782	0.0743912	−	0.997229i	$-0.476299\pi$
$577$	3.57387	0.148782	0.0743912	+	0.997229i	$0.476299\pi$
$578$	−15.3715	−0.639369
$579$	−4.22836	−0.175725
$580$	−3.46767	−0.143987
$581$	−7.20462	−0.298898
$582$	18.4176	0.763432
$583$	13.1175	0.543271
$584$	7.03821	0.291243
$585$	1.62912	0.0673557
$586$	45.0610	1.86145
$587$	−26.6819	−1.10128	−0.550641	−	0.834742i	$-0.685616\pi$
$587$	−26.6819	−1.10128	−0.550641	+	0.834742i	$0.685616\pi$
$588$	−4.75452	−0.196073
$589$	2.34922	0.0967977
$590$	13.0243	0.536202
$591$	19.0359	0.783032
$592$	−6.88372	−0.282919
$593$	11.8756	0.487671	0.243835	−	0.969817i	$-0.421594\pi$
$593$	11.8756	0.487671	0.243835	+	0.969817i	$0.421594\pi$
$594$	42.0910	1.72701
$595$	19.6894	0.807188
$596$	48.4921	1.98631
$597$	24.9691	1.02192
$598$	−0.731932	−0.0299309
$599$	−28.0695	−1.14689	−0.573445	−	0.819244i	$-0.694393\pi$
$599$	−28.0695	−1.14689	−0.573445	+	0.819244i	$0.694393\pi$
$600$	25.6642	1.04773
$601$	25.3089	1.03237	0.516186	−	0.856477i	$-0.327351\pi$
$601$	25.3089	1.03237	0.516186	+	0.856477i	$0.327351\pi$
$602$	−24.3393	−0.991997
$603$	−2.39696	−0.0976117
$604$	−28.8778	−1.17502
$605$	22.1173	0.899194
$606$	−27.7728	−1.12819
$607$	4.36352	0.177110	0.0885549	−	0.996071i	$-0.471775\pi$
$607$	4.36352	0.177110	0.0885549	+	0.996071i	$0.471775\pi$
$608$	2.64338	0.107203
$609$	−0.613038	−0.0248416
$610$	−63.4954	−2.57085
$611$	2.86259	0.115808
$612$	−4.66378	−0.188522
$613$	21.8557	0.882745	0.441373	−	0.897324i	$-0.354492\pi$
$613$	21.8557	0.882745	0.441373	+	0.897324i	$0.354492\pi$
$614$	−19.9232	−0.804033
$615$	−20.6474	−0.832583
$616$	−5.15179	−0.207572
$617$	−9.30924	−0.374776	−0.187388	−	0.982286i	$-0.560002\pi$
$617$	−9.30924	−0.374776	−0.187388	+	0.982286i	$0.560002\pi$
$618$	40.6921	1.63688
$619$	44.6630	1.79516	0.897578	−	0.440856i	$-0.145325\pi$
$619$	44.6630	1.79516	0.897578	+	0.440856i	$0.145325\pi$
$620$	72.1373	2.89710
$621$	1.48523	0.0596002
$622$	−43.4107	−1.74061
$623$	−8.63203	−0.345835
$624$	5.03076	0.201392
$625$	41.6311	1.66525
$626$	−32.7782	−1.31008
$627$	2.52043	0.100656
$628$	−30.7827	−1.22836
$629$	−13.7180	−0.546972
$630$	3.14093	0.125138
$631$	23.9994	0.955400	0.477700	−	0.878523i	$-0.341471\pi$
$631$	23.9994	0.955400	0.477700	+	0.878523i	$0.341471\pi$
$632$	−6.22965	−0.247802
$633$	7.59728	0.301965
$634$	10.1438	0.402862
$635$	−12.8419	−0.509616
$636$	15.3431	0.608395
$637$	1.11140	0.0440353
$638$	−2.91037	−0.115223
$639$	−2.78535	−0.110187
$640$	38.8282	1.53482
$641$	46.0890	1.82041	0.910203	−	0.414163i	$-0.135926\pi$
$641$	46.0890	1.82041	0.910203	+	0.414163i	$0.135926\pi$
$642$	−18.3784	−0.725337
$643$	16.4694	0.649489	0.324745	−	0.945802i	$-0.394722\pi$
$643$	16.4694	0.649489	0.324745	+	0.945802i	$0.394722\pi$
$644$	−0.796474	−0.0313855
$645$	83.4553	3.28605
$646$	3.56057	0.140089
$647$	−14.7442	−0.579654	−0.289827	−	0.957079i	$-0.593598\pi$
$647$	−14.7442	−0.579654	−0.289827	+	0.957079i	$0.593598\pi$
$648$	12.6286	0.496097
$649$	6.16963	0.242179
$650$	−26.2845	−1.03096
$651$	12.7529	0.499826
$652$	−49.7608	−1.94878
$653$	33.4874	1.31046	0.655232	−	0.755428i	$-0.272571\pi$
$653$	33.4874	1.31046	0.655232	+	0.755428i	$0.272571\pi$
$654$	−0.863118	−0.0337506
$655$	−65.4828	−2.55863
$656$	−6.93342	−0.270705
$657$	−2.03267	−0.0793020
$658$	5.51905	0.215155
$659$	−12.0786	−0.470515	−0.235257	−	0.971933i	$-0.575593\pi$
$659$	−12.0786	−0.470515	−0.235257	+	0.971933i	$0.575593\pi$
$660$	77.3949	3.01259
$661$	−30.0260	−1.16788	−0.583938	−	0.811798i	$-0.698489\pi$
$661$	−30.0260	−1.16788	−0.583938	+	0.811798i	$0.698489\pi$
$662$	24.5645	0.954728
$663$	10.0254	0.389354
$664$	−9.13117	−0.354358
$665$	−1.35343	−0.0524837
$666$	−2.18834	−0.0847965
$667$	−0.102696	−0.00397639
$668$	−26.6217	−1.03002
$669$	−21.8650	−0.845350
$670$	56.1927	2.17091
$671$	−30.0779	−1.16114
$672$	14.3498	0.553556
$673$	−28.8170	−1.11081	−0.555406	−	0.831579i	$-0.687437\pi$
$673$	−28.8170	−1.11081	−0.555406	+	0.831579i	$0.687437\pi$
$674$	47.9683	1.84767
$675$	53.3362	2.05291
$676$	−30.4882	−1.17262
$677$	−23.1077	−0.888101	−0.444051	−	0.896002i	$-0.646459\pi$
$677$	−23.1077	−0.888101	−0.444051	+	0.896002i	$0.646459\pi$
$678$	−3.91980	−0.150539
$679$	−4.68486	−0.179788
$680$	24.9545	0.956960
$681$	28.8550	1.10573
$682$	60.5438	2.31834
$683$	−19.3026	−0.738592	−0.369296	−	0.929312i	$-0.620401\pi$
$683$	−19.3026	−0.738592	−0.369296	+	0.929312i	$0.620401\pi$
$684$	0.320582	0.0122578
$685$	32.6835	1.24877
$686$	2.14277	0.0818115
$687$	−41.5405	−1.58487
$688$	28.0244	1.06842
$689$	−3.58656	−0.136637
$690$	4.83862	0.184203
$691$	35.1693	1.33790	0.668951	−	0.743307i	$-0.266744\pi$
$691$	35.1693	1.33790	0.668951	+	0.743307i	$0.266744\pi$
$692$	−16.7645	−0.637289
$693$	1.48786	0.0565192
$694$	70.2588	2.66699
$695$	−73.7831	−2.79875
$696$	−0.776967	−0.0294509
$697$	−13.8170	−0.523357
$698$	53.7442	2.03425
$699$	29.8123	1.12760
$700$	−28.6023	−1.08106
$701$	11.2224	0.423863	0.211932	−	0.977284i	$-0.432025\pi$
$701$	11.2224	0.423863	0.211932	+	0.977284i	$0.432025\pi$
$702$	−11.5084	−0.434358
$703$	0.942957	0.0355643
$704$	48.0676	1.81161
$705$	−18.9238	−0.712713
$706$	18.7413	0.705337
$707$	7.06455	0.265690
$708$	7.21643	0.271210
$709$	9.94106	0.373344	0.186672	−	0.982422i	$-0.440230\pi$
$709$	9.94106	0.373344	0.186672	+	0.982422i	$0.440230\pi$
$710$	65.2978	2.45058
$711$	1.79915	0.0674735
$712$	−10.9403	−0.410004
$713$	2.13636	0.0800072
$714$	19.3289	0.723365
$715$	−18.0916	−0.676587
$716$	−19.2058	−0.717753
$717$	29.0389	1.08448
$718$	−21.5462	−0.804098
$719$	12.3937	0.462206	0.231103	−	0.972929i	$-0.425766\pi$
$719$	12.3937	0.462206	0.231103	+	0.972929i	$0.425766\pi$
$720$	−3.61647	−0.134778
$721$	−10.3508	−0.385484
$722$	40.4679	1.50606
$723$	−45.8520	−1.70525
$724$	49.2266	1.82949
$725$	−3.68792	−0.136966
$726$	21.7122	0.805817
$727$	10.6248	0.394052	0.197026	−	0.980398i	$-0.436872\pi$
$727$	10.6248	0.394052	0.197026	+	0.980398i	$0.436872\pi$
$728$	1.40859	0.0522060
$729$	22.9509	0.850032
$730$	47.6525	1.76370
$731$	55.8475	2.06559
$732$	−35.1812	−1.30033
$733$	−16.8911	−0.623887	−0.311943	−	0.950101i	$-0.600980\pi$
$733$	−16.8911	−0.623887	−0.311943	+	0.950101i	$0.600980\pi$
$734$	57.8172	2.13407
$735$	−7.34719	−0.271005
$736$	2.40387	0.0886079
$737$	26.6186	0.980508
$738$	−2.20414	−0.0811355
$739$	1.82307	0.0670626	0.0335313	−	0.999438i	$-0.489325\pi$
$739$	1.82307	0.0670626	0.0335313	+	0.999438i	$0.489325\pi$
$740$	28.9554	1.06442
$741$	−0.689133	−0.0253159
$742$	−6.91486	−0.253853
$743$	2.85597	0.104775	0.0523877	−	0.998627i	$-0.483317\pi$
$743$	2.85597	0.104775	0.0523877	+	0.998627i	$0.483317\pi$
$744$	16.1631	0.592567
$745$	74.9351	2.74541
$746$	25.4290	0.931023
$747$	2.63712	0.0964873
$748$	51.7919	1.89370
$749$	4.67490	0.170817
$750$	95.0434	3.47049
$751$	13.2311	0.482811	0.241405	−	0.970424i	$-0.422392\pi$
$751$	13.2311	0.482811	0.241405	+	0.970424i	$0.422392\pi$
$752$	−6.35465	−0.231730
$753$	−11.8516	−0.431895
$754$	0.795748	0.0289794
$755$	−44.6251	−1.62407
$756$	−12.5233	−0.455466
$757$	38.5435	1.40089	0.700444	−	0.713707i	$-0.252985\pi$
$757$	38.5435	1.40089	0.700444	+	0.713707i	$0.252985\pi$
$758$	−18.6393	−0.677010
$759$	2.29206	0.0831966
$760$	−1.71534	−0.0622219
$761$	−31.5541	−1.14384	−0.571918	−	0.820311i	$-0.693801\pi$
$761$	−31.5541	−1.14384	−0.571918	+	0.820311i	$0.693801\pi$
$762$	−12.6067	−0.456694
$763$	0.219551	0.00794826
$764$	15.5871	0.563921
$765$	−7.20697	−0.260568
$766$	17.5855	0.635391
$767$	−1.68689	−0.0609100
$768$	−5.27365	−0.190297
$769$	−13.4934	−0.486583	−0.243291	−	0.969953i	$-0.578227\pi$
$769$	−13.4934	−0.486583	−0.243291	+	0.969953i	$0.578227\pi$
$770$	−34.8804	−1.25700
$771$	23.7951	0.856961
$772$	5.97255	0.214957
$773$	27.3113	0.982319	0.491159	−	0.871070i	$-0.336573\pi$
$773$	27.3113	0.982319	0.491159	+	0.871070i	$0.336573\pi$
$774$	8.90898	0.320227
$775$	76.7190	2.75583
$776$	−5.93761	−0.213148
$777$	5.11892	0.183640
$778$	76.4179	2.73972
$779$	0.949766	0.0340289
$780$	−21.1612	−0.757691
$781$	30.9317	1.10682
$782$	3.23796	0.115789
$783$	−1.61472	−0.0577055
$784$	−2.46720	−0.0881142
$785$	−47.5687	−1.69780
$786$	−64.2837	−2.29292
$787$	−32.1979	−1.14773	−0.573866	−	0.818949i	$-0.694557\pi$
$787$	−32.1979	−1.14773	−0.573866	+	0.818949i	$0.694557\pi$
$788$	−26.8882	−0.957852
$789$	3.87444	0.137934
$790$	−42.1781	−1.50063
$791$	0.997076	0.0354519
$792$	1.88572	0.0670062
$793$	8.22384	0.292037
$794$	16.8027	0.596306
$795$	23.7098	0.840901
$796$	−35.2689	−1.25007
$797$	6.06590	0.214865	0.107433	−	0.994212i	$-0.465737\pi$
$797$	6.06590	0.214865	0.107433	+	0.994212i	$0.465737\pi$
$798$	−1.32864	−0.0470334
$799$	−12.6636	−0.448008
$800$	86.3257	3.05207
$801$	3.15960	0.111639
$802$	10.5833	0.373708
$803$	22.5731	0.796587
$804$	31.1349	1.09804
$805$	−1.23080	−0.0433798
$806$	−16.5538	−0.583082
$807$	−22.2170	−0.782075
$808$	8.95364	0.314988
$809$	33.9853	1.19486	0.597429	−	0.801922i	$-0.296189\pi$
$809$	33.9853	1.19486	0.597429	+	0.801922i	$0.296189\pi$
$810$	85.5022	3.00424
$811$	25.0729	0.880430	0.440215	−	0.897893i	$-0.354902\pi$
$811$	25.0729	0.880430	0.440215	+	0.897893i	$0.354902\pi$
$812$	0.865917	0.0303877
$813$	−24.3944	−0.855550
$814$	24.3018	0.851779
$815$	−76.8956	−2.69354
$816$	−22.2553	−0.779092
$817$	−3.83889	−0.134306
$818$	−40.2261	−1.40647
$819$	−0.406809	−0.0142150
$820$	29.1644	1.01847
$821$	−41.5562	−1.45032	−0.725161	−	0.688580i	$-0.758235\pi$
$821$	−41.5562	−1.45032	−0.725161	+	0.688580i	$0.758235\pi$
$822$	32.0850	1.11909
$823$	−16.0997	−0.561199	−0.280599	−	0.959825i	$-0.590533\pi$
$823$	−16.0997	−0.561199	−0.280599	+	0.959825i	$0.590533\pi$
$824$	−13.1187	−0.457010
$825$	82.3105	2.86568
$826$	−3.25231	−0.113162
$827$	−18.5725	−0.645831	−0.322915	−	0.946428i	$-0.604663\pi$
$827$	−18.5725	−0.645831	−0.322915	+	0.946428i	$0.604663\pi$
$828$	0.291535	0.0101315
$829$	−1.09863	−0.0381571	−0.0190785	−	0.999818i	$-0.506073\pi$
$829$	−1.09863	−0.0381571	−0.0190785	+	0.999818i	$0.506073\pi$
$830$	−61.8229	−2.14591
$831$	−9.12567	−0.316566
$832$	−13.1425	−0.455636
$833$	−4.91667	−0.170352
$834$	−72.4320	−2.50811
$835$	−41.1387	−1.42366
$836$	−3.56011	−0.123129
$837$	33.5907	1.16107
$838$	41.8943	1.44721
$839$	32.8893	1.13546	0.567732	−	0.823213i	$-0.307821\pi$
$839$	32.8893	1.13546	0.567732	+	0.823213i	$0.307821\pi$
$840$	−9.31186	−0.321290
$841$	−28.8884	−0.996150
$842$	8.70241	0.299905
$843$	4.09893	0.141175
$844$	−10.7312	−0.369382
$845$	−47.1136	−1.62076
$846$	−2.02015	−0.0694542
$847$	−5.52292	−0.189770
$848$	7.96180	0.273409
$849$	−16.9352	−0.581216
$850$	116.279	3.98832
$851$	0.857518	0.0293953
$852$	36.1799	1.23950
$853$	−57.5004	−1.96878	−0.984389	−	0.176008i	$-0.943682\pi$
$853$	−57.5004	−1.96878	−0.984389	+	0.176008i	$0.943682\pi$
$854$	15.8555	0.542564
$855$	0.495398	0.0169423
$856$	5.92498	0.202512
$857$	−44.9442	−1.53527	−0.767633	−	0.640890i	$-0.778565\pi$
$857$	−44.9442	−1.53527	−0.767633	+	0.640890i	$0.778565\pi$
$858$	−17.7603	−0.606326
$859$	−1.66267	−0.0567297	−0.0283648	−	0.999598i	$-0.509030\pi$
$859$	−1.66267	−0.0567297	−0.0283648	+	0.999598i	$0.509030\pi$
$860$	−117.881	−4.01969
$861$	5.15588	0.175712
$862$	4.89958	0.166880
$863$	−38.7370	−1.31862	−0.659312	−	0.751870i	$-0.729152\pi$
$863$	−38.7370	−1.31862	−0.659312	+	0.751870i	$0.729152\pi$
$864$	37.7970	1.28588
$865$	−25.9062	−0.880838
$866$	−33.1053	−1.12496
$867$	−13.1613	−0.446980
$868$	−18.0135	−0.611417
$869$	−19.9798	−0.677769
$870$	−5.26049	−0.178347
$871$	−7.27800	−0.246606
$872$	0.278259	0.00942305
$873$	1.71481	0.0580375
$874$	−0.222573	−0.00752865
$875$	−24.1761	−0.817302
$876$	26.4030	0.892076
$877$	−12.7902	−0.431894	−0.215947	−	0.976405i	$-0.569284\pi$
$877$	−12.7902	−0.431894	−0.215947	+	0.976405i	$0.569284\pi$
$878$	−35.1945	−1.18776
$879$	38.5819	1.30133
$880$	40.1615	1.35384
$881$	−47.3232	−1.59436	−0.797179	−	0.603743i	$-0.793675\pi$
$881$	−47.3232	−1.59436	−0.797179	+	0.603743i	$0.793675\pi$
$882$	−0.784324	−0.0264096
$883$	−27.9844	−0.941751	−0.470875	−	0.882200i	$-0.656062\pi$
$883$	−27.9844	−0.941751	−0.470875	+	0.882200i	$0.656062\pi$
$884$	−14.1609	−0.476281
$885$	11.1516	0.374857
$886$	41.7262	1.40182
$887$	−13.5759	−0.455833	−0.227917	−	0.973681i	$-0.573191\pi$
$887$	−13.5759	−0.455833	−0.227917	+	0.973681i	$0.573191\pi$
$888$	6.48774	0.217714
$889$	3.20677	0.107552
$890$	−74.0715	−2.48288
$891$	40.5025	1.35689
$892$	30.8843	1.03408
$893$	0.870484	0.0291296
$894$	73.5629	2.46031
$895$	−29.6788	−0.992053
$896$	−9.69583	−0.323915
$897$	−0.626691	−0.0209246
$898$	81.8938	2.73283
$899$	−2.32262	−0.0774638
$900$	10.4694	0.348978
$901$	15.8664	0.528586
$902$	24.4773	0.815005
$903$	−20.8397	−0.693502
$904$	1.26370	0.0420300
$905$	76.0701	2.52866
$906$	−43.8079	−1.45542
$907$	50.6853	1.68298	0.841489	−	0.540274i	$-0.181679\pi$
$907$	50.6853	1.68298	0.841489	+	0.540274i	$0.181679\pi$
$908$	−40.7577	−1.35259
$909$	−2.58585	−0.0857674
$910$	9.53694	0.316147
$911$	−17.5384	−0.581074	−0.290537	−	0.956864i	$-0.593834\pi$
$911$	−17.5384	−0.581074	−0.290537	+	0.956864i	$0.593834\pi$
$912$	1.52980	0.0506569
$913$	−29.2856	−0.969213
$914$	−16.7749	−0.554865
$915$	−54.3657	−1.79728
$916$	58.6759	1.93871
$917$	16.3518	0.539984
$918$	50.9116	1.68033
$919$	−39.3246	−1.29720	−0.648599	−	0.761130i	$-0.724645\pi$
$919$	−39.3246	−1.29720	−0.648599	+	0.761130i	$0.724645\pi$
$920$	−1.55992	−0.0514289
$921$	−17.0585	−0.562097
$922$	−12.0766	−0.397721
$923$	−8.45729	−0.278375
$924$	−19.3264	−0.635791
$925$	30.7944	1.01251
$926$	47.9167	1.57464
$927$	3.78873	0.124438
$928$	−2.61346	−0.0857911
$929$	7.63452	0.250480	0.125240	−	0.992126i	$-0.460030\pi$
$929$	7.63452	0.250480	0.125240	+	0.992126i	$0.460030\pi$
$930$	109.433	3.58844
$931$	0.337966	0.0110764
$932$	−42.1099	−1.37935
$933$	−37.1689	−1.21686
$934$	64.9958	2.12673
$935$	80.0344	2.61740
$936$	−0.515591	−0.0168526
$937$	−57.5351	−1.87959	−0.939794	−	0.341740i	$-0.888984\pi$
$937$	−57.5351	−1.87959	−0.939794	+	0.341740i	$0.888984\pi$
$938$	−14.0319	−0.458159
$939$	−28.0652	−0.915874
$940$	26.7299	0.871834
$941$	−29.6955	−0.968045	−0.484023	−	0.875056i	$-0.660825\pi$
$941$	−29.6955	−0.968045	−0.484023	+	0.875056i	$0.660825\pi$
$942$	−46.6976	−1.52149
$943$	0.863709	0.0281262
$944$	3.74472	0.121880
$945$	−19.3523	−0.629529
$946$	−98.9355	−3.21667
$947$	−32.6994	−1.06259	−0.531294	−	0.847187i	$-0.678294\pi$
$947$	−32.6994	−1.06259	−0.531294	+	0.847187i	$0.678294\pi$
$948$	−23.3698	−0.759016
$949$	−6.17189	−0.200348
$950$	−7.99285	−0.259322
$951$	8.68528	0.281640
$952$	−6.23141	−0.201961
$953$	5.39764	0.174847	0.0874234	−	0.996171i	$-0.472137\pi$
$953$	5.39764	0.174847	0.0874234	+	0.996171i	$0.472137\pi$
$954$	2.53106	0.0819462
$955$	24.0868	0.779431
$956$	−41.0174	−1.32660
$957$	−2.49190	−0.0805518
$958$	−80.5729	−2.60319
$959$	−8.16144	−0.263547
$960$	86.8820	2.80411
$961$	17.3170	0.558613
$962$	−6.64456	−0.214229
$963$	−1.71116	−0.0551414
$964$	64.7659	2.08597
$965$	9.22943	0.297106
$966$	−1.20826	−0.0388750
$967$	26.9253	0.865859	0.432930	−	0.901428i	$-0.357480\pi$
$967$	26.9253	0.865859	0.432930	+	0.901428i	$0.357480\pi$
$968$	−6.99978	−0.224981
$969$	3.04862	0.0979357
$970$	−40.2008	−1.29077
$971$	−15.4070	−0.494433	−0.247217	−	0.968960i	$-0.579516\pi$
$971$	−15.4070	−0.494433	−0.247217	+	0.968960i	$0.579516\pi$
$972$	9.80484	0.314490
$973$	18.4245	0.590661
$974$	87.7256	2.81091
$975$	−22.5052	−0.720743
$976$	−18.2561	−0.584363
$977$	36.8513	1.17898	0.589489	−	0.807777i	$-0.299329\pi$
$977$	36.8513	1.17898	0.589489	+	0.807777i	$0.299329\pi$
$978$	−75.4875	−2.41382
$979$	−35.0878	−1.12141
$980$	10.3779	0.331510
$981$	−0.0803626	−0.00256578
$982$	6.10008	0.194661
$983$	−19.9237	−0.635468	−0.317734	−	0.948180i	$-0.602922\pi$
$983$	−19.9237	−0.635468	−0.317734	+	0.948180i	$0.602922\pi$
$984$	6.53458	0.208315
$985$	−41.5505	−1.32391
$986$	−3.52027	−0.112108
$987$	4.72549	0.150414
$988$	0.973400	0.0309680
$989$	−3.49105	−0.111009
$990$	12.7674	0.405773
$991$	14.2862	0.453816	0.226908	−	0.973916i	$-0.427138\pi$
$991$	14.2862	0.453816	0.226908	+	0.973916i	$0.427138\pi$
$992$	54.3672	1.72616
$993$	21.0325	0.667447
$994$	−16.3056	−0.517182
$995$	−54.5012	−1.72780
$996$	−34.2545	−1.08540
$997$	−44.3203	−1.40364	−0.701819	−	0.712356i	$-0.747628\pi$
$997$	−44.3203	−1.40364	−0.701819	+	0.712356i	$0.747628\pi$
$998$	−41.1383	−1.30221
$999$	13.4831	0.426586

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3997.2.a.e.1.13	✓	73

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3997.2.a.e.1.13	✓	73	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:	\( N \)	\(=\)	\( 3997 = 7 \cdot 571 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3997.a (trivial)

\(f(q)\)	\(=\)	\(q-2.14277 q^{2} -1.83467 q^{3} +2.59148 q^{4} +4.00463 q^{5} +3.93129 q^{6} -1.00000 q^{7} -1.26740 q^{8} +0.366032 q^{9} +O(q^{10})\)	\(q-2.14277 q^{2} -1.83467 q^{3} +2.59148 q^{4} +4.00463 q^{5} +3.93129 q^{6} -1.00000 q^{7} -1.26740 q^{8} +0.366032 q^{9} -8.58101 q^{10} -4.06484 q^{11} -4.75452 q^{12} +1.11140 q^{13} +2.14277 q^{14} -7.34719 q^{15} -2.46720 q^{16} -4.91667 q^{17} -0.784324 q^{18} +0.337966 q^{19} +10.3779 q^{20} +1.83467 q^{21} +8.71003 q^{22} +0.307343 q^{23} +2.32527 q^{24} +11.0370 q^{25} -2.38148 q^{26} +4.83247 q^{27} -2.59148 q^{28} -0.334140 q^{29} +15.7434 q^{30} +6.95104 q^{31} +7.82145 q^{32} +7.45766 q^{33} +10.5353 q^{34} -4.00463 q^{35} +0.948565 q^{36} +2.79010 q^{37} -0.724184 q^{38} -2.03906 q^{39} -5.07548 q^{40} +2.81024 q^{41} -3.93129 q^{42} -11.3588 q^{43} -10.5339 q^{44} +1.46582 q^{45} -0.658567 q^{46} +2.57566 q^{47} +4.52650 q^{48} +1.00000 q^{49} -23.6499 q^{50} +9.02049 q^{51} +2.88017 q^{52} -3.22706 q^{53} -10.3549 q^{54} -16.2782 q^{55} +1.26740 q^{56} -0.620058 q^{57} +0.715987 q^{58} -1.51780 q^{59} -19.0401 q^{60} +7.39953 q^{61} -14.8945 q^{62} -0.366032 q^{63} -11.8252 q^{64} +4.45075 q^{65} -15.9801 q^{66} -6.54849 q^{67} -12.7414 q^{68} -0.563875 q^{69} +8.58101 q^{70} -7.60957 q^{71} -0.463911 q^{72} -5.55325 q^{73} -5.97855 q^{74} -20.2494 q^{75} +0.875831 q^{76} +4.06484 q^{77} +4.36924 q^{78} +4.91528 q^{79} -9.88021 q^{80} -9.96412 q^{81} -6.02171 q^{82} +7.20462 q^{83} +4.75452 q^{84} -19.6894 q^{85} +24.3393 q^{86} +0.613038 q^{87} +5.15179 q^{88} +8.63203 q^{89} -3.14093 q^{90} -1.11140 q^{91} +0.796474 q^{92} -12.7529 q^{93} -5.51905 q^{94} +1.35343 q^{95} -14.3498 q^{96} +4.68486 q^{97} -2.14277 q^{98} -1.48786 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 73 q - 12 q^{2} - 18 q^{3} + 74 q^{4} - 22 q^{5} - 7 q^{6} - 73 q^{7} - 39 q^{8} + 81 q^{9}+O(q^{10}) \) 73 * q - 12 * q^2 - 18 * q^3 + 74 * q^4 - 22 * q^5 - 7 * q^6 - 73 * q^7 - 39 * q^8 + 81 * q^9	\( 73 q - 12 q^{2} - 18 q^{3} + 74 q^{4} - 22 q^{5} - 7 q^{6} - 73 q^{7} - 39 q^{8} + 81 q^{9} - 8 q^{10} - 24 q^{11} - 33 q^{12} - 24 q^{13} + 12 q^{14} - 9 q^{15} + 76 q^{16} - 31 q^{17} - 42 q^{18} - 31 q^{19} - 54 q^{20} + 18 q^{21} - 4 q^{22} - 30 q^{23} - 6 q^{24} + 71 q^{25} - 22 q^{26} - 78 q^{27} - 74 q^{28} + 39 q^{29} - 13 q^{30} - 30 q^{31} - 100 q^{32} - 51 q^{33} - 10 q^{34} + 22 q^{35} + 97 q^{36} - 22 q^{37} - 61 q^{38} - 4 q^{39} - 15 q^{40} - 49 q^{41} + 7 q^{42} - 50 q^{43} - 21 q^{44} - 86 q^{45} + 17 q^{46} - 73 q^{47} - 70 q^{48} + 73 q^{49} - 50 q^{50} - 47 q^{51} - 82 q^{52} + 4 q^{53} - 8 q^{54} - 34 q^{55} + 39 q^{56} - 27 q^{57} - 27 q^{58} - 118 q^{59} - 24 q^{60} - 25 q^{61} - 61 q^{62} - 81 q^{63} + 77 q^{64} - 32 q^{65} - 46 q^{66} - 71 q^{67} - 92 q^{68} - 21 q^{69} + 8 q^{70} + 2 q^{71} - 141 q^{72} - 54 q^{73} - 12 q^{74} - 66 q^{75} - 36 q^{76} + 24 q^{77} - 28 q^{78} - 6 q^{79} - 133 q^{80} + 101 q^{81} - 38 q^{82} - 214 q^{83} + 33 q^{84} - 21 q^{85} - 34 q^{86} - 115 q^{87} + 3 q^{88} - 60 q^{89} - 67 q^{90} + 24 q^{91} - 99 q^{92} - 15 q^{93} - 11 q^{94} + q^{95} - 43 q^{96} - 48 q^{97} - 12 q^{98} - 55 q^{99}+O(q^{100}) \) 73 * q - 12 * q^2 - 18 * q^3 + 74 * q^4 - 22 * q^5 - 7 * q^6 - 73 * q^7 - 39 * q^8 + 81 * q^9 - 8 * q^10 - 24 * q^11 - 33 * q^12 - 24 * q^13 + 12 * q^14 - 9 * q^15 + 76 * q^16 - 31 * q^17 - 42 * q^18 - 31 * q^19 - 54 * q^20 + 18 * q^21 - 4 * q^22 - 30 * q^23 - 6 * q^24 + 71 * q^25 - 22 * q^26 - 78 * q^27 - 74 * q^28 + 39 * q^29 - 13 * q^30 - 30 * q^31 - 100 * q^32 - 51 * q^33 - 10 * q^34 + 22 * q^35 + 97 * q^36 - 22 * q^37 - 61 * q^38 - 4 * q^39 - 15 * q^40 - 49 * q^41 + 7 * q^42 - 50 * q^43 - 21 * q^44 - 86 * q^45 + 17 * q^46 - 73 * q^47 - 70 * q^48 + 73 * q^49 - 50 * q^50 - 47 * q^51 - 82 * q^52 + 4 * q^53 - 8 * q^54 - 34 * q^55 + 39 * q^56 - 27 * q^57 - 27 * q^58 - 118 * q^59 - 24 * q^60 - 25 * q^61 - 61 * q^62 - 81 * q^63 + 77 * q^64 - 32 * q^65 - 46 * q^66 - 71 * q^67 - 92 * q^68 - 21 * q^69 + 8 * q^70 + 2 * q^71 - 141 * q^72 - 54 * q^73 - 12 * q^74 - 66 * q^75 - 36 * q^76 + 24 * q^77 - 28 * q^78 - 6 * q^79 - 133 * q^80 + 101 * q^81 - 38 * q^82 - 214 * q^83 + 33 * q^84 - 21 * q^85 - 34 * q^86 - 115 * q^87 + 3 * q^88 - 60 * q^89 - 67 * q^90 + 24 * q^91 - 99 * q^92 - 15 * q^93 - 11 * q^94 + q^95 - 43 * q^96 - 48 * q^97 - 12 * q^98 - 55 * q^99

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