LMFDB - Embedded newform 4007.2.a.a.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4007,2,Mod(1,4007)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4007, 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("4007.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4007 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4007.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:	$31.9960560899$
Analytic rank:	$1$
Dimension:	$139$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Character	$\chi$	$=$	4007.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.64078 q^{2} +1.79793 q^{3} +4.97369 q^{4} -0.236310 q^{5} -4.74793 q^{6} -1.11879 q^{7} -7.85286 q^{8} +0.232555 q^{9} +O(q^{10})$	\(q-2.64078 q^{2} +1.79793 q^{3} +4.97369 q^{4} -0.236310 q^{5} -4.74793 q^{6} -1.11879 q^{7} -7.85286 q^{8} +0.232555 q^{9} +0.624041 q^{10} -0.498274 q^{11} +8.94236 q^{12} +4.19458 q^{13} +2.95447 q^{14} -0.424868 q^{15} +10.7902 q^{16} +4.73229 q^{17} -0.614124 q^{18} -0.737817 q^{19} -1.17533 q^{20} -2.01151 q^{21} +1.31583 q^{22} +6.86740 q^{23} -14.1189 q^{24} -4.94416 q^{25} -11.0770 q^{26} -4.97567 q^{27} -5.56452 q^{28} -10.2071 q^{29} +1.12198 q^{30} -8.97308 q^{31} -12.7889 q^{32} -0.895862 q^{33} -12.4969 q^{34} +0.264381 q^{35} +1.15666 q^{36} -1.41700 q^{37} +1.94841 q^{38} +7.54157 q^{39} +1.85571 q^{40} +5.50467 q^{41} +5.31194 q^{42} -12.5845 q^{43} -2.47826 q^{44} -0.0549549 q^{45} -18.1353 q^{46} +0.0266212 q^{47} +19.4001 q^{48} -5.74831 q^{49} +13.0564 q^{50} +8.50833 q^{51} +20.8626 q^{52} -5.81453 q^{53} +13.1396 q^{54} +0.117747 q^{55} +8.78570 q^{56} -1.32654 q^{57} +26.9546 q^{58} +2.97507 q^{59} -2.11317 q^{60} -1.08190 q^{61} +23.6959 q^{62} -0.260180 q^{63} +12.1921 q^{64} -0.991220 q^{65} +2.36577 q^{66} -1.05609 q^{67} +23.5370 q^{68} +12.3471 q^{69} -0.698170 q^{70} +4.92159 q^{71} -1.82622 q^{72} -0.646654 q^{73} +3.74197 q^{74} -8.88925 q^{75} -3.66968 q^{76} +0.557463 q^{77} -19.9156 q^{78} -4.56776 q^{79} -2.54984 q^{80} -9.64358 q^{81} -14.5366 q^{82} +0.182483 q^{83} -10.0046 q^{84} -1.11829 q^{85} +33.2327 q^{86} -18.3516 q^{87} +3.91287 q^{88} +15.1571 q^{89} +0.145124 q^{90} -4.69286 q^{91} +34.1563 q^{92} -16.1330 q^{93} -0.0703005 q^{94} +0.174353 q^{95} -22.9936 q^{96} -0.149862 q^{97} +15.1800 q^{98} -0.115876 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 139 q - 13 q^{2} - 22 q^{3} + 113 q^{4} - 16 q^{5} - 15 q^{6} - 44 q^{7} - 36 q^{8} + 87 q^{9}+O(q^{10}) $ 139 * q - 13 * q^2 - 22 * q^3 + 113 * q^4 - 16 * q^5 - 15 * q^6 - 44 * q^7 - 36 * q^8 + 87 * q^9	\( 139 q - 13 q^{2} - 22 q^{3} + 113 q^{4} - 16 q^{5} - 15 q^{6} - 44 q^{7} - 36 q^{8} + 87 q^{9} - 40 q^{10} - 17 q^{11} - 59 q^{12} - 89 q^{13} - 15 q^{14} - 29 q^{15} + 73 q^{16} - 58 q^{17} - 51 q^{18} - 37 q^{19} - 24 q^{20} - 37 q^{21} - 99 q^{22} - 42 q^{23} - 27 q^{24} - 11 q^{25} + 2 q^{26} - 73 q^{27} - 113 q^{28} - 57 q^{29} - 29 q^{30} - 51 q^{31} - 80 q^{32} - 78 q^{33} - 28 q^{34} - 34 q^{35} + 28 q^{36} - 117 q^{37} - 31 q^{38} - 36 q^{39} - 107 q^{40} - 60 q^{41} - 41 q^{42} - 109 q^{43} - 21 q^{44} - 62 q^{45} - 92 q^{46} - 26 q^{47} - 90 q^{48} - 7 q^{49} - 22 q^{50} - 47 q^{51} - 182 q^{52} - 83 q^{53} - 19 q^{54} - 53 q^{55} - 23 q^{56} - 201 q^{57} - 112 q^{58} + 14 q^{59} - 64 q^{60} - 73 q^{61} - 21 q^{62} - 94 q^{63} + 14 q^{64} - 123 q^{65} - 10 q^{66} - 135 q^{67} - 84 q^{68} - 50 q^{69} - 35 q^{70} - 29 q^{71} - 143 q^{72} - 266 q^{73} - 53 q^{74} - 32 q^{75} - 66 q^{76} - 69 q^{77} - 59 q^{78} - 124 q^{79} - 20 q^{80} - 33 q^{81} - 93 q^{82} - 28 q^{83} - 4 q^{84} - 179 q^{85} + 6 q^{86} - 40 q^{87} - 259 q^{88} - 41 q^{89} + 2 q^{90} - 50 q^{91} - 77 q^{92} - 60 q^{93} - 48 q^{94} - 37 q^{95} + 3 q^{96} - 220 q^{97} - 9 q^{98} - 35 q^{99}+O(q^{100}) \) 139 * q - 13 * q^2 - 22 * q^3 + 113 * q^4 - 16 * q^5 - 15 * q^6 - 44 * q^7 - 36 * q^8 + 87 * q^9 - 40 * q^10 - 17 * q^11 - 59 * q^12 - 89 * q^13 - 15 * q^14 - 29 * q^15 + 73 * q^16 - 58 * q^17 - 51 * q^18 - 37 * q^19 - 24 * q^20 - 37 * q^21 - 99 * q^22 - 42 * q^23 - 27 * q^24 - 11 * q^25 + 2 * q^26 - 73 * q^27 - 113 * q^28 - 57 * q^29 - 29 * q^30 - 51 * q^31 - 80 * q^32 - 78 * q^33 - 28 * q^34 - 34 * q^35 + 28 * q^36 - 117 * q^37 - 31 * q^38 - 36 * q^39 - 107 * q^40 - 60 * q^41 - 41 * q^42 - 109 * q^43 - 21 * q^44 - 62 * q^45 - 92 * q^46 - 26 * q^47 - 90 * q^48 - 7 * q^49 - 22 * q^50 - 47 * q^51 - 182 * q^52 - 83 * q^53 - 19 * q^54 - 53 * q^55 - 23 * q^56 - 201 * q^57 - 112 * q^58 + 14 * q^59 - 64 * q^60 - 73 * q^61 - 21 * q^62 - 94 * q^63 + 14 * q^64 - 123 * q^65 - 10 * q^66 - 135 * q^67 - 84 * q^68 - 50 * q^69 - 35 * q^70 - 29 * q^71 - 143 * q^72 - 266 * q^73 - 53 * q^74 - 32 * q^75 - 66 * q^76 - 69 * q^77 - 59 * q^78 - 124 * q^79 - 20 * q^80 - 33 * q^81 - 93 * q^82 - 28 * q^83 - 4 * q^84 - 179 * q^85 + 6 * q^86 - 40 * q^87 - 259 * q^88 - 41 * q^89 + 2 * q^90 - 50 * q^91 - 77 * q^92 - 60 * q^93 - 48 * q^94 - 37 * q^95 + 3 * q^96 - 220 * q^97 - 9 * q^98 - 35 * 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 $

$2$	−2.64078	−1.86731	−0.933655	−	0.358173i	$-0.883400\pi$
$2$	−2.64078	−1.86731	−0.933655	+	0.358173i	$0.883400\pi$
$3$	1.79793	1.03804	0.519018	−	0.854763i	$-0.326298\pi$
$3$	1.79793	1.03804	0.519018	+	0.854763i	$0.326298\pi$
$4$	4.97369	2.48685
$5$	−0.236310	−0.105681	−0.0528404	−	0.998603i	$-0.516827\pi$
$5$	−0.236310	−0.105681	−0.0528404	+	0.998603i	$0.516827\pi$
$6$	−4.74793	−1.93833
$7$	−1.11879	−0.422863	−0.211431	−	0.977393i	$-0.567812\pi$
$7$	−1.11879	−0.422863	−0.211431	+	0.977393i	$0.567812\pi$
$8$	−7.85286	−2.77640
$9$	0.232555	0.0775182
$10$	0.624041	0.197339
$11$	−0.498274	−0.150235	−0.0751176	−	0.997175i	$-0.523933\pi$
$11$	−0.498274	−0.150235	−0.0751176	+	0.997175i	$0.523933\pi$
$12$	8.94236	2.58144
$13$	4.19458	1.16337	0.581684	−	0.813415i	$-0.302394\pi$
$13$	4.19458	1.16337	0.581684	+	0.813415i	$0.302394\pi$
$14$	2.95447	0.789616
$15$	−0.424868	−0.109701
$16$	10.7902	2.69756
$17$	4.73229	1.14775	0.573874	−	0.818943i	$-0.305440\pi$
$17$	4.73229	1.14775	0.573874	+	0.818943i	$0.305440\pi$
$18$	−0.614124	−0.144751
$19$	−0.737817	−0.169267	−0.0846334	−	0.996412i	$-0.526972\pi$
$19$	−0.737817	−0.169267	−0.0846334	+	0.996412i	$0.526972\pi$
$20$	−1.17533	−0.262812
$21$	−2.01151	−0.438947
$22$	1.31583	0.280536
$23$	6.86740	1.43195	0.715976	−	0.698125i	$-0.245982\pi$
$23$	6.86740	1.43195	0.715976	+	0.698125i	$0.245982\pi$
$24$	−14.1189	−2.88201
$25$	−4.94416	−0.988832
$26$	−11.0770	−2.17237
$27$	−4.97567	−0.957569
$28$	−5.56452	−1.05159
$29$	−10.2071	−1.89541	−0.947704	−	0.319151i	$-0.896602\pi$
$29$	−10.2071	−1.89541	−0.947704	+	0.319151i	$0.896602\pi$
$30$	1.12198	0.204845
$31$	−8.97308	−1.61161	−0.805806	−	0.592180i	$-0.798268\pi$
$31$	−8.97308	−1.61161	−0.805806	+	0.592180i	$0.798268\pi$
$32$	−12.7889	−2.26078
$33$	−0.895862	−0.155949
$34$	−12.4969	−2.14320
$35$	0.264381	0.0446885
$36$	1.15666	0.192776
$37$	−1.41700	−0.232953	−0.116477	−	0.993193i	$-0.537160\pi$
$37$	−1.41700	−0.232953	−0.116477	+	0.993193i	$0.537160\pi$
$38$	1.94841	0.316074
$39$	7.54157	1.20762
$40$	1.85571	0.293413
$41$	5.50467	0.859686	0.429843	−	0.902904i	$-0.358569\pi$
$41$	5.50467	0.859686	0.429843	+	0.902904i	$0.358569\pi$
$42$	5.31194	0.819649
$43$	−12.5845	−1.91911	−0.959555	−	0.281520i	$-0.909161\pi$
$43$	−12.5845	−1.91911	−0.959555	+	0.281520i	$0.909161\pi$
$44$	−2.47826	−0.373612
$45$	−0.0549549	−0.00819219
$46$	−18.1353	−2.67390
$47$	0.0266212	0.00388310	0.00194155	−	0.999998i	$-0.499382\pi$
$47$	0.0266212	0.00388310	0.00194155	+	0.999998i	$0.499382\pi$
$48$	19.4001	2.80017
$49$	−5.74831	−0.821187
$50$	13.0564	1.84646
$51$	8.50833	1.19140
$52$	20.8626	2.89312
$53$	−5.81453	−0.798687	−0.399344	−	0.916801i	$-0.630762\pi$
$53$	−5.81453	−0.798687	−0.399344	+	0.916801i	$0.630762\pi$
$54$	13.1396	1.78808
$55$	0.117747	0.0158770
$56$	8.78570	1.17404
$57$	−1.32654	−0.175705
$58$	26.9546	3.53931
$59$	2.97507	0.387321	0.193660	−	0.981069i	$-0.437964\pi$
$59$	2.97507	0.387321	0.193660	+	0.981069i	$0.437964\pi$
$60$	−2.11317	−0.272808
$61$	−1.08190	−0.138523	−0.0692617	−	0.997599i	$-0.522064\pi$
$61$	−1.08190	−0.138523	−0.0692617	+	0.997599i	$0.522064\pi$
$62$	23.6959	3.00938
$63$	−0.260180	−0.0327796
$64$	12.1921	1.52402
$65$	−0.991220	−0.122946
$66$	2.36577	0.291206
$67$	−1.05609	−0.129022	−0.0645108	−	0.997917i	$-0.520549\pi$
$67$	−1.05609	−0.129022	−0.0645108	+	0.997917i	$0.520549\pi$
$68$	23.5370	2.85428
$69$	12.3471	1.48642
$70$	−0.698170	−0.0834473
$71$	4.92159	0.584085	0.292043	−	0.956405i	$-0.405665\pi$
$71$	4.92159	0.584085	0.292043	+	0.956405i	$0.405665\pi$
$72$	−1.82622	−0.215222
$73$	−0.646654	−0.0756851	−0.0378426	−	0.999284i	$-0.512049\pi$
$73$	−0.646654	−0.0756851	−0.0378426	+	0.999284i	$0.512049\pi$
$74$	3.74197	0.434996
$75$	−8.88925	−1.02644
$76$	−3.66968	−0.420941
$77$	0.557463	0.0635288
$78$	−19.9156	−2.25500
$79$	−4.56776	−0.513914	−0.256957	−	0.966423i	$-0.582720\pi$
$79$	−4.56776	−0.513914	−0.256957	+	0.966423i	$0.582720\pi$
$80$	−2.54984	−0.285081
$81$	−9.64358	−1.07151
$82$	−14.5366	−1.60530
$83$	0.182483	0.0200301	0.0100150	−	0.999950i	$-0.496812\pi$
$83$	0.182483	0.0200301	0.0100150	+	0.999950i	$0.496812\pi$
$84$	−10.0046	−1.09159
$85$	−1.11829	−0.121295
$86$	33.2327	3.58358
$87$	−18.3516	−1.96750
$88$	3.91287	0.417114
$89$	15.1571	1.60665	0.803326	−	0.595540i	$-0.203062\pi$
$89$	15.1571	1.60665	0.803326	+	0.595540i	$0.203062\pi$
$90$	0.145124	0.0152974
$91$	−4.69286	−0.491945
$92$	34.1563	3.56104
$93$	−16.1330	−1.67291
$94$	−0.0703005	−0.00725095
$95$	0.174353	0.0178883
$96$	−22.9936	−2.34677
$97$	−0.149862	−0.0152162	−0.00760811	−	0.999971i	$-0.502422\pi$
$97$	−0.149862	−0.0152162	−0.00760811	+	0.999971i	$0.502422\pi$
$98$	15.1800	1.53341
$99$	−0.115876	−0.0116460
$100$	−24.5907	−2.45907
$101$	11.4681	1.14112	0.570559	−	0.821257i	$-0.306727\pi$
$101$	11.4681	1.14112	0.570559	+	0.821257i	$0.306727\pi$
$102$	−22.4686	−2.22472
$103$	9.85148	0.970695	0.485348	−	0.874321i	$-0.338693\pi$
$103$	9.85148	0.970695	0.485348	+	0.874321i	$0.338693\pi$
$104$	−32.9395	−3.22998
$105$	0.475338	0.0463883
$106$	15.3549	1.49140
$107$	1.57324	0.152091	0.0760455	−	0.997104i	$-0.475771\pi$
$107$	1.57324	0.152091	0.0760455	+	0.997104i	$0.475771\pi$
$108$	−24.7475	−2.38133
$109$	−8.74754	−0.837863	−0.418931	−	0.908018i	$-0.637595\pi$
$109$	−8.74754	−0.837863	−0.418931	+	0.908018i	$0.637595\pi$
$110$	−0.310943	−0.0296473
$111$	−2.54766	−0.241814
$112$	−12.0720	−1.14070
$113$	−9.90707	−0.931979	−0.465989	−	0.884790i	$-0.654302\pi$
$113$	−9.90707	−0.931979	−0.465989	+	0.884790i	$0.654302\pi$
$114$	3.50311	0.328096
$115$	−1.62283	−0.151330
$116$	−50.7669	−4.71359
$117$	0.975470	0.0901822
$118$	−7.85648	−0.723248
$119$	−5.29444	−0.485340
$120$	3.33643	0.304573
$121$	−10.7517	−0.977429
$122$	2.85706	0.258666
$123$	9.89702	0.892384
$124$	−44.6293	−4.00783
$125$	2.34990	0.210181
$126$	0.687076	0.0612096
$127$	−6.48955	−0.575854	−0.287927	−	0.957652i	$-0.592966\pi$
$127$	−6.48955	−0.575854	−0.287927	+	0.957652i	$0.592966\pi$
$128$	−6.61885	−0.585029
$129$	−22.6260	−1.99211
$130$	2.61759	0.229578
$131$	−6.67244	−0.582974	−0.291487	−	0.956575i	$-0.594150\pi$
$131$	−6.67244	−0.582974	−0.291487	+	0.956575i	$0.594150\pi$
$132$	−4.45574	−0.387823
$133$	0.825462	0.0715766
$134$	2.78889	0.240923
$135$	1.17580	0.101197
$136$	−37.1620	−3.18662
$137$	8.15363	0.696612	0.348306	−	0.937381i	$-0.386757\pi$
$137$	8.15363	0.696612	0.348306	+	0.937381i	$0.386757\pi$
$138$	−32.6059	−2.77560
$139$	6.95657	0.590048	0.295024	−	0.955490i	$-0.404672\pi$
$139$	6.95657	0.590048	0.295024	+	0.955490i	$0.404672\pi$
$140$	1.31495	0.111133
$141$	0.0478630	0.00403079
$142$	−12.9968	−1.09067
$143$	−2.09005	−0.174779
$144$	2.50932	0.209110
$145$	2.41203	0.200308
$146$	1.70767	0.141328
$147$	−10.3351	−0.852422
$148$	−7.04772	−0.579319
$149$	−3.52623	−0.288880	−0.144440	−	0.989514i	$-0.546138\pi$
$149$	−3.52623	−0.288880	−0.144440	+	0.989514i	$0.546138\pi$
$150$	23.4745	1.91669
$151$	1.12599	0.0916320	0.0458160	−	0.998950i	$-0.485411\pi$
$151$	1.12599	0.0916320	0.0458160	+	0.998950i	$0.485411\pi$
$152$	5.79397	0.469953
$153$	1.10052	0.0889714
$154$	−1.47214	−0.118628
$155$	2.12042	0.170317
$156$	37.5095	3.00316
$157$	16.4668	1.31419	0.657096	−	0.753807i	$-0.271785\pi$
$157$	16.4668	1.31419	0.657096	+	0.753807i	$0.271785\pi$
$158$	12.0624	0.959636
$159$	−10.4541	−0.829066
$160$	3.02214	0.238921
$161$	−7.68317	−0.605519
$162$	25.4665	2.00084
$163$	17.4456	1.36644	0.683222	−	0.730211i	$-0.260578\pi$
$163$	17.4456	1.36644	0.683222	+	0.730211i	$0.260578\pi$
$164$	27.3786	2.13791
$165$	0.211701	0.0164809
$166$	−0.481896	−0.0374024
$167$	1.15427	0.0893203	0.0446602	−	0.999002i	$-0.485779\pi$
$167$	1.15427	0.0893203	0.0446602	+	0.999002i	$0.485779\pi$
$168$	15.7961	1.21869
$169$	4.59453	0.353425
$170$	2.95314	0.226496
$171$	−0.171583	−0.0131213
$172$	−62.5912	−4.77254
$173$	−22.8150	−1.73460	−0.867298	−	0.497789i	$-0.834145\pi$
$173$	−22.8150	−1.73460	−0.867298	+	0.497789i	$0.834145\pi$
$174$	48.4625	3.67393
$175$	5.53147	0.418140
$176$	−5.37650	−0.405269
$177$	5.34896	0.402053
$178$	−40.0265	−3.00012
$179$	−11.6752	−0.872643	−0.436321	−	0.899791i	$-0.643719\pi$
$179$	−11.6752	−0.872643	−0.436321	+	0.899791i	$0.643719\pi$
$180$	−0.273329	−0.0203727
$181$	−2.38847	−0.177533	−0.0887666	−	0.996052i	$-0.528293\pi$
$181$	−2.38847	−0.177533	−0.0887666	+	0.996052i	$0.528293\pi$
$182$	12.3928	0.918614
$183$	−1.94519	−0.143792
$184$	−53.9287	−3.97568
$185$	0.334850	0.0246187
$186$	42.6035	3.12384
$187$	−2.35798	−0.172432
$188$	0.132406	0.00965667
$189$	5.56673	0.404920
$190$	−0.460428	−0.0334030
$191$	−14.5569	−1.05330	−0.526649	−	0.850083i	$-0.676552\pi$
$191$	−14.5569	−1.05330	−0.526649	+	0.850083i	$0.676552\pi$
$192$	21.9206	1.58198
$193$	−24.3260	−1.75102	−0.875511	−	0.483198i	$-0.839475\pi$
$193$	−24.3260	−1.75102	−0.875511	+	0.483198i	$0.839475\pi$
$194$	0.395753	0.0284134
$195$	−1.78215	−0.127622
$196$	−28.5903	−2.04217
$197$	−2.59575	−0.184940	−0.0924699	−	0.995715i	$-0.529476\pi$
$197$	−2.59575	−0.184940	−0.0924699	+	0.995715i	$0.529476\pi$
$198$	0.306002	0.0217466
$199$	−7.69014	−0.545139	−0.272570	−	0.962136i	$-0.587874\pi$
$199$	−7.69014	−0.545139	−0.272570	+	0.962136i	$0.587874\pi$
$200$	38.8258	2.74540
$201$	−1.89877	−0.133929
$202$	−30.2847	−2.13082
$203$	11.4196	0.801497
$204$	42.3178	2.96284
$205$	−1.30081	−0.0908523
$206$	−26.0156	−1.81259
$207$	1.59704	0.111002
$208$	45.2606	3.13826
$209$	0.367635	0.0254298
$210$	−1.25526	−0.0866213
$211$	8.49624	0.584905	0.292452	−	0.956280i	$-0.405529\pi$
$211$	8.49624	0.584905	0.292452	+	0.956280i	$0.405529\pi$
$212$	−28.9197	−1.98621
$213$	8.84868	0.606302
$214$	−4.15458	−0.284001
$215$	2.97383	0.202813
$216$	39.0733	2.65860
$217$	10.0390	0.681490
$218$	23.1003	1.56455
$219$	−1.16264	−0.0785639
$220$	0.585637	0.0394836
$221$	19.8500	1.33525
$222$	6.72781	0.451541
$223$	10.4336	0.698686	0.349343	−	0.936995i	$-0.386405\pi$
$223$	10.4336	0.698686	0.349343	+	0.936995i	$0.386405\pi$
$224$	14.3081	0.955999
$225$	−1.14979	−0.0766524
$226$	26.1623	1.74029
$227$	−7.41668	−0.492262	−0.246131	−	0.969237i	$-0.579159\pi$
$227$	−7.41668	−0.492262	−0.246131	+	0.969237i	$0.579159\pi$
$228$	−6.59783	−0.436952
$229$	−14.6996	−0.971377	−0.485688	−	0.874132i	$-0.661431\pi$
$229$	−14.6996	−0.971377	−0.485688	+	0.874132i	$0.661431\pi$
$230$	4.28553	0.282580
$231$	1.00228	0.0659452
$232$	80.1548	5.26242
$233$	−1.34508	−0.0881194	−0.0440597	−	0.999029i	$-0.514029\pi$
$233$	−1.34508	−0.0881194	−0.0440597	+	0.999029i	$0.514029\pi$
$234$	−2.57600	−0.168398
$235$	−0.00629084	−0.000410369 0
$236$	14.7971	0.963207
$237$	−8.21252	−0.533461
$238$	13.9814	0.906281
$239$	11.3362	0.733279	0.366639	−	0.930363i	$-0.380508\pi$
$239$	11.3362	0.733279	0.366639	+	0.930363i	$0.380508\pi$
$240$	−4.58443	−0.295924
$241$	9.82931	0.633162	0.316581	−	0.948566i	$-0.397465\pi$
$241$	9.82931	0.633162	0.316581	+	0.948566i	$0.397465\pi$
$242$	28.3929	1.82516
$243$	−2.41147	−0.154696
$244$	−5.38105	−0.344487
$245$	1.35838	0.0867838
$246$	−26.1358	−1.66636
$247$	−3.09484	−0.196920
$248$	70.4643	4.47449
$249$	0.328091	0.0207919
$250$	−6.20556	−0.392474
$251$	−23.5838	−1.48860	−0.744298	−	0.667848i	$-0.767216\pi$
$251$	−23.5838	−1.48860	−0.744298	+	0.667848i	$0.767216\pi$
$252$	−1.29405	−0.0815177
$253$	−3.42184	−0.215129
$254$	17.1374	1.07530
$255$	−2.01060	−0.125909
$256$	−6.90535	−0.431584
$257$	−10.5897	−0.660570	−0.330285	−	0.943881i	$-0.607145\pi$
$257$	−10.5897	−0.660570	−0.330285	+	0.943881i	$0.607145\pi$
$258$	59.7501	3.71988
$259$	1.58532	0.0985072
$260$	−4.93003	−0.305747
$261$	−2.37370	−0.146929
$262$	17.6204	1.08859
$263$	−10.3455	−0.637932	−0.318966	−	0.947766i	$-0.603336\pi$
$263$	−10.3455	−0.637932	−0.318966	+	0.947766i	$0.603336\pi$
$264$	7.03507	0.432979
$265$	1.37403	0.0844060
$266$	−2.17986	−0.133656
$267$	27.2514	1.66776
$268$	−5.25265	−0.320857
$269$	−20.6549	−1.25935	−0.629676	−	0.776858i	$-0.716812\pi$
$269$	−20.6549	−1.25935	−0.629676	+	0.776858i	$0.716812\pi$
$270$	−3.10502	−0.188966
$271$	−30.3043	−1.84085	−0.920427	−	0.390915i	$-0.872159\pi$
$271$	−30.3043	−1.84085	−0.920427	+	0.390915i	$0.872159\pi$
$272$	51.0626	3.09612
$273$	−8.43743	−0.510656
$274$	−21.5319	−1.30079
$275$	2.46354	0.148557
$276$	61.4107	3.69649
$277$	10.1933	0.612454	0.306227	−	0.951959i	$-0.400933\pi$
$277$	10.1933	0.612454	0.306227	+	0.951959i	$0.400933\pi$
$278$	−18.3707	−1.10180
$279$	−2.08673	−0.124929
$280$	−2.07614	−0.124073
$281$	4.67549	0.278916	0.139458	−	0.990228i	$-0.455464\pi$
$281$	4.67549	0.278916	0.139458	+	0.990228i	$0.455464\pi$
$282$	−0.126395	−0.00752674
$283$	−7.48797	−0.445113	−0.222557	−	0.974920i	$-0.571440\pi$
$283$	−7.48797	−0.445113	−0.222557	+	0.974920i	$0.571440\pi$
$284$	24.4785	1.45253
$285$	0.313475	0.0185687
$286$	5.51935	0.326366
$287$	−6.15857	−0.363529
$288$	−2.97412	−0.175252
$289$	5.39457	0.317327
$290$	−6.36964	−0.374038
$291$	−0.269442	−0.0157950
$292$	−3.21626	−0.188217
$293$	−20.4180	−1.19283	−0.596417	−	0.802675i	$-0.703410\pi$
$293$	−20.4180	−1.19283	−0.596417	+	0.802675i	$0.703410\pi$
$294$	27.2926	1.59174
$295$	−0.703037	−0.0409324
$296$	11.1275	0.646772
$297$	2.47925	0.143861
$298$	9.31199	0.539429
$299$	28.8059	1.66589
$300$	−44.2124	−2.55261
$301$	14.0794	0.811520
$302$	−2.97350	−0.171105
$303$	20.6188	1.18452
$304$	−7.96123	−0.456608
$305$	0.255664	0.0146393
$306$	−2.90621	−0.166137
$307$	0.753628	0.0430118	0.0215059	−	0.999769i	$-0.493154\pi$
$307$	0.753628	0.0430118	0.0215059	+	0.999769i	$0.493154\pi$
$308$	2.77265	0.157987
$309$	17.7123	1.00762
$310$	−5.59956	−0.318034
$311$	14.3891	0.815931	0.407966	−	0.912997i	$-0.366238\pi$
$311$	14.3891	0.815931	0.407966	+	0.912997i	$0.366238\pi$
$312$	−59.2229	−3.35284
$313$	−3.52882	−0.199461	−0.0997305	−	0.995014i	$-0.531798\pi$
$313$	−3.52882	−0.199461	−0.0997305	+	0.995014i	$0.531798\pi$
$314$	−43.4850	−2.45400
$315$	0.0614830	0.00346417
$316$	−22.7187	−1.27802
$317$	16.5336	0.928619	0.464310	−	0.885673i	$-0.346303\pi$
$317$	16.5336	0.928619	0.464310	+	0.885673i	$0.346303\pi$
$318$	27.6070	1.54812
$319$	5.08592	0.284757
$320$	−2.88112	−0.161059
$321$	2.82858	0.157876
$322$	20.2895	1.13069
$323$	−3.49157	−0.194276
$324$	−47.9642	−2.66468
$325$	−20.7387	−1.15037
$326$	−46.0699	−2.55157
$327$	−15.7275	−0.869732
$328$	−43.2274	−2.38684
$329$	−0.0297835	−0.00164202
$330$	−0.559054	−0.0307749
$331$	19.4052	1.06661	0.533304	−	0.845924i	$-0.320950\pi$
$331$	19.4052	1.06661	0.533304	+	0.845924i	$0.320950\pi$
$332$	0.907613	0.0498118
$333$	−0.329529	−0.0180581
$334$	−3.04818	−0.166789
$335$	0.249563	0.0136351
$336$	−21.7046	−1.18409
$337$	−4.14846	−0.225981	−0.112990	−	0.993596i	$-0.536043\pi$
$337$	−4.14846	−0.225981	−0.112990	+	0.993596i	$0.536043\pi$
$338$	−12.1331	−0.659954
$339$	−17.8122	−0.967427
$340$	−5.56201	−0.301642
$341$	4.47105	0.242121
$342$	0.453112	0.0245015
$343$	14.2627	0.770112
$344$	98.8239	5.32823
$345$	−2.91774	−0.157086
$346$	60.2494	3.23903
$347$	−11.0614	−0.593809	−0.296905	−	0.954907i	$-0.595954\pi$
$347$	−11.0614	−0.593809	−0.296905	+	0.954907i	$0.595954\pi$
$348$	−91.2754	−4.89287
$349$	−28.8859	−1.54623	−0.773113	−	0.634268i	$-0.781301\pi$
$349$	−28.8859	−1.54623	−0.773113	+	0.634268i	$0.781301\pi$
$350$	−14.6074	−0.780797
$351$	−20.8709	−1.11401
$352$	6.37237	0.339649
$353$	−29.5017	−1.57022	−0.785109	−	0.619357i	$-0.787393\pi$
$353$	−29.5017	−1.57022	−0.785109	+	0.619357i	$0.787393\pi$
$354$	−14.1254	−0.750757
$355$	−1.16302	−0.0617267
$356$	75.3869	3.99550
$357$	−9.51903	−0.503800
$358$	30.8315	1.62949
$359$	−24.2256	−1.27858	−0.639288	−	0.768967i	$-0.720771\pi$
$359$	−24.2256	−1.27858	−0.639288	+	0.768967i	$0.720771\pi$
$360$	0.431553	0.0227448
$361$	−18.4556	−0.971349
$362$	6.30740	0.331510
$363$	−19.3309	−1.01461
$364$	−23.3408	−1.22339
$365$	0.152811	0.00799847
$366$	5.13680	0.268505
$367$	0.322354	0.0168267	0.00841336	−	0.999965i	$-0.497322\pi$
$367$	0.322354	0.0168267	0.00841336	+	0.999965i	$0.497322\pi$
$368$	74.1009	3.86278
$369$	1.28014	0.0666413
$370$	−0.884265	−0.0459707
$371$	6.50523	0.337735
$372$	−80.2404	−4.16027
$373$	−16.6836	−0.863842	−0.431921	−	0.901911i	$-0.642164\pi$
$373$	−16.6836	−0.863842	−0.431921	+	0.901911i	$0.642164\pi$
$374$	6.22688	0.321984
$375$	4.22496	0.218176
$376$	−0.209052	−0.0107810
$377$	−42.8145	−2.20506
$378$	−14.7005	−0.756112
$379$	15.1762	0.779550	0.389775	−	0.920910i	$-0.372553\pi$
$379$	15.1762	0.779550	0.389775	+	0.920910i	$0.372553\pi$
$380$	0.867180	0.0444854
$381$	−11.6678	−0.597757
$382$	38.4414	1.96683
$383$	−26.5440	−1.35633	−0.678167	−	0.734907i	$-0.737226\pi$
$383$	−26.5440	−1.35633	−0.678167	+	0.734907i	$0.737226\pi$
$384$	−11.9002	−0.607281
$385$	−0.131734	−0.00671378
$386$	64.2394	3.26970
$387$	−2.92657	−0.148766
$388$	−0.745370	−0.0378404
$389$	28.9093	1.46576	0.732879	−	0.680359i	$-0.238176\pi$
$389$	28.9093	1.46576	0.732879	+	0.680359i	$0.238176\pi$
$390$	4.70625	0.238310
$391$	32.4985	1.64352
$392$	45.1407	2.27995
$393$	−11.9966	−0.605148
$394$	6.85480	0.345340
$395$	1.07941	0.0543108
$396$	−0.576331	−0.0289617
$397$	5.61709	0.281914	0.140957	−	0.990016i	$-0.454982\pi$
$397$	5.61709	0.281914	0.140957	+	0.990016i	$0.454982\pi$
$398$	20.3079	1.01794
$399$	1.48412	0.0742991
$400$	−53.3487	−2.66743
$401$	37.1947	1.85741	0.928706	−	0.370816i	$-0.120922\pi$
$401$	37.1947	1.85741	0.928706	+	0.370816i	$0.120922\pi$
$402$	5.01423	0.250087
$403$	−37.6383	−1.87490
$404$	57.0388	2.83779
$405$	2.27887	0.113238
$406$	−30.1565	−1.49664
$407$	0.706053	0.0349977
$408$	−66.8147	−3.30782
$409$	−38.8567	−1.92134	−0.960671	−	0.277690i	$-0.910431\pi$
$409$	−38.8567	−1.92134	−0.960671	+	0.277690i	$0.910431\pi$
$410$	3.43514	0.169650
$411$	14.6597	0.723108
$412$	48.9983	2.41397
$413$	−3.32847	−0.163783
$414$	−4.21744	−0.207276
$415$	−0.0431224	−0.00211680
$416$	−53.6441	−2.63012
$417$	12.5074	0.612491
$418$	−0.970841	−0.0474854
$419$	39.1558	1.91289	0.956443	−	0.291921i	$-0.0942944\pi$
$419$	39.1558	1.91289	0.956443	+	0.291921i	$0.0942944\pi$
$420$	2.36419	0.115361
$421$	17.2506	0.840743	0.420371	−	0.907352i	$-0.361900\pi$
$421$	17.2506	0.840743	0.420371	+	0.907352i	$0.361900\pi$
$422$	−22.4367	−1.09220
$423$	0.00619088	0.000301011 0
$424$	45.6607	2.21748
$425$	−23.3972	−1.13493
$426$	−23.3674	−1.13215
$427$	1.21042	0.0585764
$428$	7.82482	0.378227
$429$	−3.75777	−0.181427
$430$	−7.85321	−0.378715
$431$	14.4320	0.695163	0.347582	−	0.937650i	$-0.387003\pi$
$431$	14.4320	0.695163	0.347582	+	0.937650i	$0.387003\pi$
$432$	−53.6888	−2.58310
$433$	−19.1190	−0.918802	−0.459401	−	0.888229i	$-0.651936\pi$
$433$	−19.1190	−0.918802	−0.459401	+	0.888229i	$0.651936\pi$
$434$	−26.5107	−1.27255
$435$	4.33667	0.207927
$436$	−43.5076	−2.08364
$437$	−5.06688	−0.242382
$438$	3.07027	0.146703
$439$	−0.703363	−0.0335697	−0.0167848	−	0.999859i	$-0.505343\pi$
$439$	−0.703363	−0.0335697	−0.0167848	+	0.999859i	$0.505343\pi$
$440$	−0.924650	−0.0440809
$441$	−1.33680	−0.0636570
$442$	−52.4193	−2.49333
$443$	1.82342	0.0866332	0.0433166	−	0.999061i	$-0.486208\pi$
$443$	1.82342	0.0866332	0.0433166	+	0.999061i	$0.486208\pi$
$444$	−12.6713	−0.601354
$445$	−3.58177	−0.169792
$446$	−27.5528	−1.30466
$447$	−6.33992	−0.299868
$448$	−13.6404	−0.644449
$449$	3.34140	0.157691	0.0788453	−	0.996887i	$-0.474877\pi$
$449$	3.34140	0.157691	0.0788453	+	0.996887i	$0.474877\pi$
$450$	3.03633	0.143134
$451$	−2.74283	−0.129155
$452$	−49.2747	−2.31769
$453$	2.02446	0.0951173
$454$	19.5858	0.919206
$455$	1.10897	0.0519892
$456$	10.4172	0.487828
$457$	−34.4553	−1.61175	−0.805875	−	0.592086i	$-0.798304\pi$
$457$	−34.4553	−1.61175	−0.805875	+	0.592086i	$0.798304\pi$
$458$	38.8183	1.81386
$459$	−23.5463	−1.09905
$460$	−8.07147	−0.376334
$461$	37.0941	1.72764	0.863822	−	0.503798i	$-0.168064\pi$
$461$	37.0941	1.72764	0.863822	+	0.503798i	$0.168064\pi$
$462$	−2.64680	−0.123140
$463$	−35.4364	−1.64687	−0.823434	−	0.567413i	$-0.807944\pi$
$463$	−35.4364	−1.64687	−0.823434	+	0.567413i	$0.807944\pi$
$464$	−110.137	−5.11298
$465$	3.81238	0.176795
$466$	3.55207	0.164546
$467$	−33.1535	−1.53416	−0.767080	−	0.641552i	$-0.778291\pi$
$467$	−33.1535	−1.53416	−0.767080	+	0.641552i	$0.778291\pi$
$468$	4.85169	0.224269
$469$	1.18154	0.0545584
$470$	0.0166127	0.000766286 0
$471$	29.6061	1.36418
$472$	−23.3628	−1.07536
$473$	6.27050	0.288318
$474$	21.6874	0.996136
$475$	3.64788	0.167376
$476$	−26.3329	−1.20697
$477$	−1.35220	−0.0619128
$478$	−29.9364	−1.36926
$479$	−17.2258	−0.787068	−0.393534	−	0.919310i	$-0.628748\pi$
$479$	−17.2258	−0.787068	−0.393534	+	0.919310i	$0.628748\pi$
$480$	5.43360	0.248009
$481$	−5.94372	−0.271010
$482$	−25.9570	−1.18231
$483$	−13.8138	−0.628550
$484$	−53.4758	−2.43072
$485$	0.0354139	0.00160806
$486$	6.36814	0.288865
$487$	32.0414	1.45194	0.725968	−	0.687728i	$-0.241392\pi$
$487$	32.0414	1.45194	0.725968	+	0.687728i	$0.241392\pi$
$488$	8.49603	0.384597
$489$	31.3659	1.41842
$490$	−3.58718	−0.162052
$491$	26.9888	1.21799	0.608994	−	0.793175i	$-0.291573\pi$
$491$	26.9888	1.21799	0.608994	+	0.793175i	$0.291573\pi$
$492$	49.2248	2.21922
$493$	−48.3029	−2.17545
$494$	8.17277	0.367710
$495$	0.0273826	0.00123076
$496$	−96.8217	−4.34742
$497$	−5.50622	−0.246988
$498$	−0.866416	−0.0388250
$499$	37.3524	1.67212	0.836062	−	0.548635i	$-0.184852\pi$
$499$	37.3524	1.67212	0.836062	+	0.548635i	$0.184852\pi$
$500$	11.6877	0.522689
$501$	2.07530	0.0927177
$502$	62.2795	2.77967
$503$	7.20047	0.321053	0.160527	−	0.987032i	$-0.448681\pi$
$503$	7.20047	0.321053	0.160527	+	0.987032i	$0.448681\pi$
$504$	2.04315	0.0910093
$505$	−2.71002	−0.120594
$506$	9.03632	0.401713
$507$	8.26064	0.366868
$508$	−32.2770	−1.43206
$509$	−13.7075	−0.607573	−0.303787	−	0.952740i	$-0.598251\pi$
$509$	−13.7075	−0.607573	−0.303787	+	0.952740i	$0.598251\pi$
$510$	5.30954	0.235111
$511$	0.723470	0.0320044
$512$	31.4732	1.39093
$513$	3.67114	0.162085
$514$	27.9651	1.23349
$515$	−2.32800	−0.102584
$516$	−112.535	−4.95406
$517$	−0.0132646	−0.000583378 0
$518$	−4.18648	−0.183943
$519$	−41.0199	−1.80057
$520$	7.78391	0.341347
$521$	−36.9297	−1.61792	−0.808959	−	0.587865i	$-0.799969\pi$
$521$	−36.9297	−1.61792	−0.808959	+	0.587865i	$0.799969\pi$
$522$	6.26842	0.274361
$523$	−21.6467	−0.946545	−0.473273	−	0.880916i	$-0.656927\pi$
$523$	−21.6467	−0.946545	−0.473273	+	0.880916i	$0.656927\pi$
$524$	−33.1867	−1.44977
$525$	9.94520	0.434044
$526$	27.3202	1.19122
$527$	−42.4632	−1.84973
$528$	−9.66657	−0.420683
$529$	24.1611	1.05048
$530$	−3.62850	−0.157612
$531$	0.691865	0.0300244
$532$	4.10560	0.178000
$533$	23.0898	1.00013
$534$	−71.9649	−3.11423
$535$	−0.371772	−0.0160731
$536$	8.29330	0.358216
$537$	−20.9911	−0.905834
$538$	54.5450	2.35160
$539$	2.86423	0.123371
$540$	5.84807	0.251661
$541$	−12.5735	−0.540576	−0.270288	−	0.962780i	$-0.587119\pi$
$541$	−12.5735	−0.540576	−0.270288	+	0.962780i	$0.587119\pi$
$542$	80.0268	3.43744
$543$	−4.29430	−0.184286
$544$	−60.5208	−2.59481
$545$	2.06713	0.0885461
$546$	22.2814	0.953554
$547$	−27.0538	−1.15674	−0.578368	−	0.815776i	$-0.696310\pi$
$547$	−27.0538	−1.15674	−0.578368	+	0.815776i	$0.696310\pi$
$548$	40.5537	1.73237
$549$	−0.251601	−0.0107381
$550$	−6.50567	−0.277403
$551$	7.53096	0.320830
$552$	−96.9601	−4.12689
$553$	5.11037	0.217315
$554$	−26.9181	−1.14364
$555$	0.602038	0.0255551
$556$	34.5998	1.46736
$557$	−28.4133	−1.20391	−0.601954	−	0.798530i	$-0.705611\pi$
$557$	−28.4133	−1.20391	−0.601954	+	0.798530i	$0.705611\pi$
$558$	5.51058	0.233282
$559$	−52.7865	−2.23263
$560$	2.85273	0.120550
$561$	−4.23948	−0.178991
$562$	−12.3469	−0.520823
$563$	−0.884248	−0.0372666	−0.0186333	−	0.999826i	$-0.505932\pi$
$563$	−0.884248	−0.0372666	−0.0186333	+	0.999826i	$0.505932\pi$
$564$	0.238056	0.0100240
$565$	2.34114	0.0984924
$566$	19.7740	0.831165
$567$	10.7891	0.453101
$568$	−38.6486	−1.62166
$569$	−28.3416	−1.18814	−0.594071	−	0.804413i	$-0.702480\pi$
$569$	−28.3416	−1.18814	−0.594071	+	0.804413i	$0.702480\pi$
$570$	−0.827818	−0.0346735
$571$	32.1313	1.34465	0.672326	−	0.740255i	$-0.265295\pi$
$571$	32.1313	1.34465	0.672326	+	0.740255i	$0.265295\pi$
$572$	−10.3953	−0.434648
$573$	−26.1722	−1.09336
$574$	16.2634	0.678821
$575$	−33.9535	−1.41596
$576$	2.83533	0.118139
$577$	13.0401	0.542865	0.271432	−	0.962457i	$-0.412503\pi$
$577$	13.0401	0.542865	0.271432	+	0.962457i	$0.412503\pi$
$578$	−14.2458	−0.592549
$579$	−43.7364	−1.81762
$580$	11.9967	0.498136
$581$	−0.204160	−0.00846997
$582$	0.711536	0.0294941
$583$	2.89723	0.119991
$584$	5.07808	0.210133
$585$	−0.230513	−0.00953053
$586$	53.9194	2.22739
$587$	5.45141	0.225004	0.112502	−	0.993652i	$-0.464114\pi$
$587$	5.45141	0.225004	0.112502	+	0.993652i	$0.464114\pi$
$588$	−51.4034	−2.11984
$589$	6.62049	0.272793
$590$	1.85656	0.0764334
$591$	−4.66698	−0.191974
$592$	−15.2898	−0.628405
$593$	−1.02140	−0.0419439	−0.0209720	−	0.999780i	$-0.506676\pi$
$593$	−1.02140	−0.0419439	−0.0209720	+	0.999780i	$0.506676\pi$
$594$	−6.54714	−0.268632
$595$	1.25113	0.0512912
$596$	−17.5384	−0.718401
$597$	−13.8263	−0.565874
$598$	−76.0698	−3.11073
$599$	15.6744	0.640440	0.320220	−	0.947343i	$-0.396243\pi$
$599$	15.6744	0.640440	0.320220	+	0.947343i	$0.396243\pi$
$600$	69.8060	2.84982
$601$	−2.42899	−0.0990805	−0.0495402	−	0.998772i	$-0.515776\pi$
$601$	−2.42899	−0.0990805	−0.0495402	+	0.998772i	$0.515776\pi$
$602$	−37.1804	−1.51536
$603$	−0.245598	−0.0100015
$604$	5.60035	0.227875
$605$	2.54074	0.103296
$606$	−54.4497	−2.21187
$607$	26.0796	1.05854	0.529270	−	0.848453i	$-0.322466\pi$
$607$	26.0796	1.05854	0.529270	+	0.848453i	$0.322466\pi$
$608$	9.43587	0.382675
$609$	20.5316	0.831983
$610$	−0.675151	−0.0273361
$611$	0.111665	0.00451747
$612$	5.47363	0.221258
$613$	−7.49372	−0.302669	−0.151334	−	0.988483i	$-0.548357\pi$
$613$	−7.49372	−0.302669	−0.151334	+	0.988483i	$0.548357\pi$
$614$	−1.99016	−0.0803164
$615$	−2.33876	−0.0943080
$616$	−4.37768	−0.176382
$617$	3.32895	0.134019	0.0670093	−	0.997752i	$-0.478654\pi$
$617$	3.32895	0.134019	0.0670093	+	0.997752i	$0.478654\pi$
$618$	−46.7742	−1.88153
$619$	21.0954	0.847896	0.423948	−	0.905687i	$-0.360644\pi$
$619$	21.0954	0.847896	0.423948	+	0.905687i	$0.360644\pi$
$620$	10.5463	0.423551
$621$	−34.1699	−1.37119
$622$	−37.9984	−1.52360
$623$	−16.9576	−0.679393
$624$	81.3754	3.25762
$625$	24.1655	0.966619
$626$	9.31883	0.372456
$627$	0.660982	0.0263971
$628$	81.9007	3.26819
$629$	−6.70565	−0.267372
$630$	−0.162363	−0.00646868
$631$	13.6181	0.542126	0.271063	−	0.962562i	$-0.412625\pi$
$631$	13.6181	0.542126	0.271063	+	0.962562i	$0.412625\pi$
$632$	35.8700	1.42683
$633$	15.2756	0.607152
$634$	−43.6615	−1.73402
$635$	1.53354	0.0608568
$636$	−51.9956	−2.06176
$637$	−24.1118	−0.955343
$638$	−13.4308	−0.531730
$639$	1.14454	0.0452773
$640$	1.56410	0.0618264
$641$	−17.8722	−0.705911	−0.352955	−	0.935640i	$-0.614823\pi$
$641$	−17.8722	−0.705911	−0.352955	+	0.935640i	$0.614823\pi$
$642$	−7.46964	−0.294803
$643$	31.4386	1.23982	0.619909	−	0.784674i	$-0.287170\pi$
$643$	31.4386	1.23982	0.619909	+	0.784674i	$0.287170\pi$
$644$	−38.2137	−1.50583
$645$	5.34674	0.210527
$646$	9.22044	0.362773
$647$	−25.9660	−1.02083	−0.510414	−	0.859929i	$-0.670508\pi$
$647$	−25.9660	−1.02083	−0.510414	+	0.859929i	$0.670508\pi$
$648$	75.7297	2.97494
$649$	−1.48240	−0.0581892
$650$	54.7662	2.14811
$651$	18.0494	0.707411
$652$	86.7690	3.39814
$653$	8.07678	0.316069	0.158034	−	0.987434i	$-0.449484\pi$
$653$	8.07678	0.316069	0.158034	+	0.987434i	$0.449484\pi$
$654$	41.5327	1.62406
$655$	1.57676	0.0616092
$656$	59.3968	2.31906
$657$	−0.150382	−0.00586697
$658$	0.0786515	0.00306615
$659$	−33.4525	−1.30312	−0.651562	−	0.758595i	$-0.725886\pi$
$659$	−33.4525	−1.30312	−0.651562	+	0.758595i	$0.725886\pi$
$660$	1.05293	0.0409854
$661$	44.5532	1.73292	0.866459	−	0.499249i	$-0.166391\pi$
$661$	44.5532	1.73292	0.866459	+	0.499249i	$0.166391\pi$
$662$	−51.2449	−1.99169
$663$	35.6889	1.38604
$664$	−1.43301	−0.0556116
$665$	−0.195065	−0.00756428
$666$	0.870213	0.0337201
$667$	−70.0961	−2.71413
$668$	5.74100	0.222126
$669$	18.7589	0.725261
$670$	−0.659041	−0.0254610
$671$	0.539084	0.0208111
$672$	25.7250	0.992362
$673$	−13.7385	−0.529580	−0.264790	−	0.964306i	$-0.585303\pi$
$673$	−13.7385	−0.529580	−0.264790	+	0.964306i	$0.585303\pi$
$674$	10.9551	0.421976
$675$	24.6005	0.946875
$676$	22.8518	0.878914
$677$	19.1556	0.736210	0.368105	−	0.929784i	$-0.380007\pi$
$677$	19.1556	0.736210	0.368105	+	0.929784i	$0.380007\pi$
$678$	47.0381	1.80649
$679$	0.167664	0.00643437
$680$	8.78174	0.336764
$681$	−13.3347	−0.510986
$682$	−11.8070	−0.452115
$683$	−22.1248	−0.846583	−0.423291	−	0.905994i	$-0.639125\pi$
$683$	−22.1248	−0.846583	−0.423291	+	0.905994i	$0.639125\pi$
$684$	−0.853400	−0.0326306
$685$	−1.92678	−0.0736186
$686$	−37.6645	−1.43804
$687$	−26.4289	−1.00832
$688$	−135.789	−5.17692
$689$	−24.3895	−0.929167
$690$	7.70509	0.293328
$691$	18.1494	0.690435	0.345217	−	0.938523i	$-0.387805\pi$
$691$	18.1494	0.690435	0.345217	+	0.938523i	$0.387805\pi$
$692$	−113.475	−4.31368
$693$	0.129641	0.00492464
$694$	29.2108	1.10883
$695$	−1.64390	−0.0623568
$696$	144.113	5.46258
$697$	26.0497	0.986703
$698$	76.2811	2.88728
$699$	−2.41837	−0.0914711
$700$	27.5118	1.03985
$701$	41.3816	1.56296	0.781481	−	0.623929i	$-0.214465\pi$
$701$	41.3816	1.56296	0.781481	+	0.623929i	$0.214465\pi$
$702$	55.1153	2.08019
$703$	1.04549	0.0394312
$704$	−6.07501	−0.228961
$705$	−0.0113105	−0.000425978 0
$706$	77.9074	2.93208
$707$	−12.8304	−0.482536
$708$	26.6041	0.999843
$709$	−13.4468	−0.505007	−0.252503	−	0.967596i	$-0.581254\pi$
$709$	−13.4468	−0.505007	−0.252503	+	0.967596i	$0.581254\pi$
$710$	3.07127	0.115263
$711$	−1.06225	−0.0398377
$712$	−119.027	−4.46071
$713$	−61.6217	−2.30775
$714$	25.1376	0.940752
$715$	0.493899	0.0184708
$716$	−58.0687	−2.17013
$717$	20.3817	0.761170
$718$	63.9743	2.38750
$719$	46.3677	1.72922	0.864612	−	0.502441i	$-0.167564\pi$
$719$	46.3677	1.72922	0.864612	+	0.502441i	$0.167564\pi$
$720$	−0.592977	−0.0220989
$721$	−11.0217	−0.410471
$722$	48.7372	1.81381
$723$	17.6724	0.657244
$724$	−11.8795	−0.441498
$725$	50.4654	1.87424
$726$	51.0484	1.89459
$727$	−24.8480	−0.921561	−0.460780	−	0.887514i	$-0.652430\pi$
$727$	−24.8480	−0.921561	−0.460780	+	0.887514i	$0.652430\pi$
$728$	36.8523	1.36584
$729$	24.5951	0.910929
$730$	−0.403538	−0.0149356
$731$	−59.5533	−2.20266
$732$	−9.67476	−0.357589
$733$	30.7208	1.13470	0.567349	−	0.823478i	$-0.307969\pi$
$733$	30.7208	1.13470	0.567349	+	0.823478i	$0.307969\pi$
$734$	−0.851264	−0.0314207
$735$	2.44228	0.0900847
$736$	−87.8265	−3.23733
$737$	0.526220	0.0193836
$738$	−3.38056	−0.124440
$739$	−7.86606	−0.289358	−0.144679	−	0.989479i	$-0.546215\pi$
$739$	−7.86606	−0.289358	−0.144679	+	0.989479i	$0.546215\pi$
$740$	1.66544	0.0612229
$741$	−5.56430	−0.204410
$742$	−17.1789	−0.630656
$743$	33.5072	1.22926	0.614631	−	0.788815i	$-0.289305\pi$
$743$	33.5072	1.22926	0.614631	+	0.788815i	$0.289305\pi$
$744$	126.690	4.64468
$745$	0.833283	0.0305291
$746$	44.0575	1.61306
$747$	0.0424372	0.00155270
$748$	−11.7278	−0.428813
$749$	−1.76013	−0.0643136
$750$	−11.1572	−0.407402
$751$	−10.9533	−0.399692	−0.199846	−	0.979827i	$-0.564044\pi$
$751$	−10.9533	−0.399692	−0.199846	+	0.979827i	$0.564044\pi$
$752$	0.287249	0.0104749
$753$	−42.4020	−1.54522
$754$	113.063	4.11753
$755$	−0.266083	−0.00968375
$756$	27.6872	1.00697
$757$	18.2818	0.664465	0.332232	−	0.943198i	$-0.392198\pi$
$757$	18.2818	0.664465	0.332232	+	0.943198i	$0.392198\pi$
$758$	−40.0770	−1.45566
$759$	−6.15224	−0.223312
$760$	−1.36917	−0.0496651
$761$	36.2894	1.31549	0.657746	−	0.753240i	$-0.271510\pi$
$761$	36.2894	1.31549	0.657746	+	0.753240i	$0.271510\pi$
$762$	30.8119	1.11620
$763$	9.78666	0.354301
$764$	−72.4014	−2.61939
$765$	−0.260062	−0.00940258
$766$	70.0967	2.53270
$767$	12.4792	0.450596
$768$	−12.4153	−0.448000
$769$	−40.3527	−1.45516	−0.727578	−	0.686025i	$-0.759354\pi$
$769$	−40.3527	−1.45516	−0.727578	+	0.686025i	$0.759354\pi$
$770$	0.347880	0.0125367
$771$	−19.0396	−0.685695
$772$	−120.990	−4.35452
$773$	44.2705	1.59230	0.796149	−	0.605100i	$-0.206867\pi$
$773$	44.2705	1.59230	0.796149	+	0.605100i	$0.206867\pi$
$774$	7.72842	0.277792
$775$	44.3643	1.59361
$776$	1.17685	0.0422464
$777$	2.85030	0.102254
$778$	−76.3429	−2.73703
$779$	−4.06144	−0.145516
$780$	−8.86385	−0.317377
$781$	−2.45230	−0.0877502
$782$	−85.8213	−3.06896
$783$	50.7871	1.81498
$784$	−62.0257	−2.21520
$785$	−3.89126	−0.138885
$786$	31.6803	1.13000
$787$	−20.1904	−0.719708	−0.359854	−	0.933009i	$-0.617174\pi$
$787$	−20.1904	−0.719708	−0.359854	+	0.933009i	$0.617174\pi$
$788$	−12.9105	−0.459917
$789$	−18.6005	−0.662197
$790$	−2.85047	−0.101415
$791$	11.0839	0.394099
$792$	0.909957	0.0323339
$793$	−4.53813	−0.161154
$794$	−14.8335	−0.526420
$795$	2.47041	0.0876164
$796$	−38.2484	−1.35568
$797$	31.5060	1.11600	0.557999	−	0.829841i	$-0.311569\pi$
$797$	31.5060	1.11600	0.557999	+	0.829841i	$0.311569\pi$
$798$	−3.91924	−0.138739
$799$	0.125979	0.00445682
$800$	63.2303	2.23553
$801$	3.52486	0.124545
$802$	−98.2227	−3.46837
$803$	0.322211	0.0113706
$804$	−9.44391	−0.333061
$805$	1.81561	0.0639918
$806$	99.3943	3.50102
$807$	−37.1361	−1.30725
$808$	−90.0573	−3.16820
$809$	45.0648	1.58439	0.792196	−	0.610266i	$-0.208937\pi$
$809$	45.0648	1.58439	0.792196	+	0.610266i	$0.208937\pi$
$810$	−6.01799	−0.211451
$811$	24.6511	0.865616	0.432808	−	0.901486i	$-0.357523\pi$
$811$	24.6511	0.865616	0.432808	+	0.901486i	$0.357523\pi$
$812$	56.7975	1.99320
$813$	−54.4850	−1.91087
$814$	−1.86453	−0.0653517
$815$	−4.12256	−0.144407
$816$	91.8070	3.21389
$817$	9.28503	0.324842
$818$	102.612	3.58774
$819$	−1.09135	−0.0381347
$820$	−6.46982	−0.225936
$821$	44.3741	1.54867	0.774334	−	0.632777i	$-0.218085\pi$
$821$	44.3741	1.54867	0.774334	+	0.632777i	$0.218085\pi$
$822$	−38.7129	−1.35027
$823$	7.27716	0.253666	0.126833	−	0.991924i	$-0.459519\pi$
$823$	7.27716	0.253666	0.126833	+	0.991924i	$0.459519\pi$
$824$	−77.3623	−2.69504
$825$	4.42928	0.154208
$826$	8.78975	0.305834
$827$	−35.3635	−1.22971	−0.614855	−	0.788640i	$-0.710785\pi$
$827$	−35.3635	−1.22971	−0.614855	+	0.788640i	$0.710785\pi$
$828$	7.94321	0.276046
$829$	−36.8563	−1.28007	−0.640037	−	0.768344i	$-0.721081\pi$
$829$	−36.8563	−1.28007	−0.640037	+	0.768344i	$0.721081\pi$
$830$	0.113877	0.00395272
$831$	18.3268	0.635749
$832$	51.1409	1.77299
$833$	−27.2027	−0.942517
$834$	−33.0293	−1.14371
$835$	−0.272766	−0.00943945
$836$	1.82850	0.0632401
$837$	44.6471	1.54323
$838$	−103.402	−3.57195
$839$	−9.19450	−0.317430	−0.158715	−	0.987324i	$-0.550735\pi$
$839$	−9.19450	−0.317430	−0.158715	+	0.987324i	$0.550735\pi$
$840$	−3.73276	−0.128793
$841$	75.1846	2.59257
$842$	−45.5550	−1.56993
$843$	8.40621	0.289525
$844$	42.2577	1.45457
$845$	−1.08573	−0.0373503
$846$	−0.0163487	−0.000562080 0
$847$	12.0289	0.413318
$848$	−62.7402	−2.15451
$849$	−13.4628	−0.462044
$850$	61.7867	2.11927
$851$	−9.73109	−0.333577
$852$	44.0106	1.50778
$853$	28.4161	0.972948	0.486474	−	0.873695i	$-0.338283\pi$
$853$	28.4161	0.972948	0.486474	+	0.873695i	$0.338283\pi$
$854$	−3.19645	−0.109380
$855$	0.0405467	0.00138667
$856$	−12.3544	−0.422266
$857$	22.9632	0.784409	0.392204	−	0.919878i	$-0.371713\pi$
$857$	22.9632	0.784409	0.392204	+	0.919878i	$0.371713\pi$
$858$	9.92341	0.338780
$859$	1.96021	0.0668816	0.0334408	−	0.999441i	$-0.489353\pi$
$859$	1.96021	0.0668816	0.0334408	+	0.999441i	$0.489353\pi$
$860$	14.7909	0.504366
$861$	−11.0727	−0.377356
$862$	−38.1116	−1.29809
$863$	−41.7347	−1.42067	−0.710333	−	0.703866i	$-0.751455\pi$
$863$	−41.7347	−1.42067	−0.710333	+	0.703866i	$0.751455\pi$
$864$	63.6334	2.16485
$865$	5.39142	0.183314
$866$	50.4891	1.71569
$867$	9.69906	0.329397
$868$	49.9308	1.69476
$869$	2.27600	0.0772079
$870$	−11.4522	−0.388265
$871$	−4.42984	−0.150100
$872$	68.6932	2.32625
$873$	−0.0348512	−0.00117953
$874$	13.3805	0.452602
$875$	−2.62904	−0.0888779
$876$	−5.78261	−0.195376
$877$	−9.81896	−0.331563	−0.165781	−	0.986163i	$-0.553015\pi$
$877$	−9.81896	−0.331563	−0.165781	+	0.986163i	$0.553015\pi$
$878$	1.85742	0.0626850
$879$	−36.7102	−1.23820
$880$	1.27052	0.0428292
$881$	35.8079	1.20640	0.603199	−	0.797591i	$-0.293892\pi$
$881$	35.8079	1.20640	0.603199	+	0.797591i	$0.293892\pi$
$882$	3.53018	0.118867
$883$	17.7442	0.597141	0.298571	−	0.954388i	$-0.403490\pi$
$883$	17.7442	0.597141	0.298571	+	0.954388i	$0.403490\pi$
$884$	98.7277	3.32057
$885$	−1.26401	−0.0424893
$886$	−4.81524	−0.161771
$887$	21.9307	0.736362	0.368181	−	0.929754i	$-0.379981\pi$
$887$	21.9307	0.736362	0.368181	+	0.929754i	$0.379981\pi$
$888$	20.0065	0.671373
$889$	7.26044	0.243507
$890$	9.45866	0.317055
$891$	4.80514	0.160978
$892$	51.8936	1.73753
$893$	−0.0196416	−0.000657280 0
$894$	16.7423	0.559947
$895$	2.75895	0.0922216
$896$	7.40510	0.247387
$897$	51.7909	1.72925
$898$	−8.82389	−0.294457
$899$	91.5889	3.05466
$900$	−5.71869	−0.190623
$901$	−27.5160	−0.916692
$902$	7.24321	0.241172
$903$	25.3137	0.842387
$904$	77.7988	2.58755
$905$	0.564418	0.0187619
$906$	−5.34614	−0.177614
$907$	−10.4709	−0.347679	−0.173839	−	0.984774i	$-0.555617\pi$
$907$	−10.4709	−0.347679	−0.173839	+	0.984774i	$0.555617\pi$
$908$	−36.8883	−1.22418
$909$	2.66696	0.0884574
$910$	−2.92853	−0.0970799
$911$	17.0800	0.565884	0.282942	−	0.959137i	$-0.408690\pi$
$911$	17.0800	0.565884	0.282942	+	0.959137i	$0.408690\pi$
$912$	−14.3137	−0.473975
$913$	−0.0909264	−0.00300922
$914$	90.9886	3.00964
$915$	0.459666	0.0151961
$916$	−73.1113	−2.41566
$917$	7.46506	0.246518
$918$	62.1806	2.05227
$919$	−16.9267	−0.558359	−0.279179	−	0.960239i	$-0.590062\pi$
$919$	−16.9267	−0.558359	−0.279179	+	0.960239i	$0.590062\pi$
$920$	12.7439	0.420153
$921$	1.35497	0.0446478
$922$	−97.9571	−3.22605
$923$	20.6440	0.679506
$924$	4.98504	0.163996
$925$	7.00586	0.230351
$926$	93.5795	3.07521
$927$	2.29101	0.0752466
$928$	130.537	4.28510
$929$	−15.4377	−0.506495	−0.253248	−	0.967401i	$-0.581499\pi$
$929$	−15.4377	−0.506495	−0.253248	+	0.967401i	$0.581499\pi$
$930$	−10.0676	−0.330131
$931$	4.24120	0.139000
$932$	−6.69004	−0.219139
$933$	25.8706	0.846966
$934$	87.5509	2.86475
$935$	0.557212	0.0182228
$936$	−7.66022	−0.250382
$937$	21.4155	0.699615	0.349808	−	0.936822i	$-0.386247\pi$
$937$	21.4155	0.699615	0.349808	+	0.936822i	$0.386247\pi$
$938$	−3.12018	−0.101877
$939$	−6.34458	−0.207048
$940$	−0.0312887	−0.00102053
$941$	−33.0556	−1.07758	−0.538790	−	0.842440i	$-0.681119\pi$
$941$	−33.0556	−1.07758	−0.538790	+	0.842440i	$0.681119\pi$
$942$	−78.1831	−2.54734
$943$	37.8028	1.23103
$944$	32.1017	1.04482
$945$	−1.31547	−0.0427923
$946$	−16.5590	−0.538379
$947$	−24.4759	−0.795360	−0.397680	−	0.917524i	$-0.630185\pi$
$947$	−24.4759	−0.795360	−0.397680	+	0.917524i	$0.630185\pi$
$948$	−40.8466	−1.32664
$949$	−2.71244	−0.0880496
$950$	−9.63324	−0.312544
$951$	29.7263	0.963940
$952$	41.5765	1.34750
$953$	3.93975	0.127621	0.0638105	−	0.997962i	$-0.479675\pi$
$953$	3.93975	0.127621	0.0638105	+	0.997962i	$0.479675\pi$
$954$	3.57085	0.115610
$955$	3.43993	0.111314
$956$	56.3829	1.82355
$957$	9.14413	0.295588
$958$	45.4895	1.46970
$959$	−9.12220	−0.294571
$960$	−5.18005	−0.167185
$961$	49.5161	1.59729
$962$	15.6960	0.506060
$963$	0.365864	0.0117898
$964$	48.8880	1.57458
$965$	5.74846	0.185050
$966$	36.4792	1.17370
$967$	−9.80018	−0.315153	−0.157576	−	0.987507i	$-0.550368\pi$
$967$	−9.80018	−0.315153	−0.157576	+	0.987507i	$0.550368\pi$
$968$	84.4318	2.71374
$969$	−6.27759	−0.201665
$970$	−0.0935202	−0.00300275
$971$	−0.0847918	−0.00272110	−0.00136055	−	0.999999i	$-0.500433\pi$
$971$	−0.0847918	−0.00272110	−0.00136055	+	0.999999i	$0.500433\pi$
$972$	−11.9939	−0.384705
$973$	−7.78293	−0.249509
$974$	−84.6142	−2.71121
$975$	−37.2867	−1.19413
$976$	−11.6740	−0.373676
$977$	43.9367	1.40566	0.702829	−	0.711359i	$-0.251920\pi$
$977$	43.9367	1.40566	0.702829	+	0.711359i	$0.251920\pi$
$978$	−82.8304	−2.64862
$979$	−7.55239	−0.241376
$980$	6.75617	0.215818
$981$	−2.03428	−0.0649496
$982$	−71.2714	−2.27436
$983$	−25.2154	−0.804247	−0.402123	−	0.915586i	$-0.631728\pi$
$983$	−25.2154	−0.804247	−0.402123	+	0.915586i	$0.631728\pi$
$984$	−77.7199	−2.47762
$985$	0.613401	0.0195446
$986$	127.557	4.06224
$987$	−0.0535486	−0.00170447
$988$	−15.3928	−0.489709
$989$	−86.4224	−2.74807
$990$	−0.0723112	−0.00229820
$991$	58.0596	1.84432	0.922162	−	0.386804i	$-0.126421\pi$
$991$	58.0596	1.84432	0.922162	+	0.386804i	$0.126421\pi$
$992$	114.756	3.64350
$993$	34.8893	1.10718
$994$	14.5407	0.461203
$995$	1.81725	0.0576108
$996$	1.63183	0.0517064
$997$	−45.9160	−1.45418	−0.727088	−	0.686545i	$-0.759127\pi$
$997$	−45.9160	−1.45418	−0.727088	+	0.686545i	$0.759127\pi$
$998$	−98.6394	−3.12237
$999$	7.05052	0.223069

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4007.2.a.a.1.5	✓	139

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4007.2.a.a.1.5	✓	139	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:	\( N \)	\(=\)	\( 4007 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4007.a (trivial)

\(f(q)\)	\(=\)	\(q-2.64078 q^{2} +1.79793 q^{3} +4.97369 q^{4} -0.236310 q^{5} -4.74793 q^{6} -1.11879 q^{7} -7.85286 q^{8} +0.232555 q^{9} +O(q^{10})\)	\(q-2.64078 q^{2} +1.79793 q^{3} +4.97369 q^{4} -0.236310 q^{5} -4.74793 q^{6} -1.11879 q^{7} -7.85286 q^{8} +0.232555 q^{9} +0.624041 q^{10} -0.498274 q^{11} +8.94236 q^{12} +4.19458 q^{13} +2.95447 q^{14} -0.424868 q^{15} +10.7902 q^{16} +4.73229 q^{17} -0.614124 q^{18} -0.737817 q^{19} -1.17533 q^{20} -2.01151 q^{21} +1.31583 q^{22} +6.86740 q^{23} -14.1189 q^{24} -4.94416 q^{25} -11.0770 q^{26} -4.97567 q^{27} -5.56452 q^{28} -10.2071 q^{29} +1.12198 q^{30} -8.97308 q^{31} -12.7889 q^{32} -0.895862 q^{33} -12.4969 q^{34} +0.264381 q^{35} +1.15666 q^{36} -1.41700 q^{37} +1.94841 q^{38} +7.54157 q^{39} +1.85571 q^{40} +5.50467 q^{41} +5.31194 q^{42} -12.5845 q^{43} -2.47826 q^{44} -0.0549549 q^{45} -18.1353 q^{46} +0.0266212 q^{47} +19.4001 q^{48} -5.74831 q^{49} +13.0564 q^{50} +8.50833 q^{51} +20.8626 q^{52} -5.81453 q^{53} +13.1396 q^{54} +0.117747 q^{55} +8.78570 q^{56} -1.32654 q^{57} +26.9546 q^{58} +2.97507 q^{59} -2.11317 q^{60} -1.08190 q^{61} +23.6959 q^{62} -0.260180 q^{63} +12.1921 q^{64} -0.991220 q^{65} +2.36577 q^{66} -1.05609 q^{67} +23.5370 q^{68} +12.3471 q^{69} -0.698170 q^{70} +4.92159 q^{71} -1.82622 q^{72} -0.646654 q^{73} +3.74197 q^{74} -8.88925 q^{75} -3.66968 q^{76} +0.557463 q^{77} -19.9156 q^{78} -4.56776 q^{79} -2.54984 q^{80} -9.64358 q^{81} -14.5366 q^{82} +0.182483 q^{83} -10.0046 q^{84} -1.11829 q^{85} +33.2327 q^{86} -18.3516 q^{87} +3.91287 q^{88} +15.1571 q^{89} +0.145124 q^{90} -4.69286 q^{91} +34.1563 q^{92} -16.1330 q^{93} -0.0703005 q^{94} +0.174353 q^{95} -22.9936 q^{96} -0.149862 q^{97} +15.1800 q^{98} -0.115876 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 139 q - 13 q^{2} - 22 q^{3} + 113 q^{4} - 16 q^{5} - 15 q^{6} - 44 q^{7} - 36 q^{8} + 87 q^{9}+O(q^{10}) \) 139 * q - 13 * q^2 - 22 * q^3 + 113 * q^4 - 16 * q^5 - 15 * q^6 - 44 * q^7 - 36 * q^8 + 87 * q^9	\( 139 q - 13 q^{2} - 22 q^{3} + 113 q^{4} - 16 q^{5} - 15 q^{6} - 44 q^{7} - 36 q^{8} + 87 q^{9} - 40 q^{10} - 17 q^{11} - 59 q^{12} - 89 q^{13} - 15 q^{14} - 29 q^{15} + 73 q^{16} - 58 q^{17} - 51 q^{18} - 37 q^{19} - 24 q^{20} - 37 q^{21} - 99 q^{22} - 42 q^{23} - 27 q^{24} - 11 q^{25} + 2 q^{26} - 73 q^{27} - 113 q^{28} - 57 q^{29} - 29 q^{30} - 51 q^{31} - 80 q^{32} - 78 q^{33} - 28 q^{34} - 34 q^{35} + 28 q^{36} - 117 q^{37} - 31 q^{38} - 36 q^{39} - 107 q^{40} - 60 q^{41} - 41 q^{42} - 109 q^{43} - 21 q^{44} - 62 q^{45} - 92 q^{46} - 26 q^{47} - 90 q^{48} - 7 q^{49} - 22 q^{50} - 47 q^{51} - 182 q^{52} - 83 q^{53} - 19 q^{54} - 53 q^{55} - 23 q^{56} - 201 q^{57} - 112 q^{58} + 14 q^{59} - 64 q^{60} - 73 q^{61} - 21 q^{62} - 94 q^{63} + 14 q^{64} - 123 q^{65} - 10 q^{66} - 135 q^{67} - 84 q^{68} - 50 q^{69} - 35 q^{70} - 29 q^{71} - 143 q^{72} - 266 q^{73} - 53 q^{74} - 32 q^{75} - 66 q^{76} - 69 q^{77} - 59 q^{78} - 124 q^{79} - 20 q^{80} - 33 q^{81} - 93 q^{82} - 28 q^{83} - 4 q^{84} - 179 q^{85} + 6 q^{86} - 40 q^{87} - 259 q^{88} - 41 q^{89} + 2 q^{90} - 50 q^{91} - 77 q^{92} - 60 q^{93} - 48 q^{94} - 37 q^{95} + 3 q^{96} - 220 q^{97} - 9 q^{98} - 35 q^{99}+O(q^{100}) \) 139 * q - 13 * q^2 - 22 * q^3 + 113 * q^4 - 16 * q^5 - 15 * q^6 - 44 * q^7 - 36 * q^8 + 87 * q^9 - 40 * q^10 - 17 * q^11 - 59 * q^12 - 89 * q^13 - 15 * q^14 - 29 * q^15 + 73 * q^16 - 58 * q^17 - 51 * q^18 - 37 * q^19 - 24 * q^20 - 37 * q^21 - 99 * q^22 - 42 * q^23 - 27 * q^24 - 11 * q^25 + 2 * q^26 - 73 * q^27 - 113 * q^28 - 57 * q^29 - 29 * q^30 - 51 * q^31 - 80 * q^32 - 78 * q^33 - 28 * q^34 - 34 * q^35 + 28 * q^36 - 117 * q^37 - 31 * q^38 - 36 * q^39 - 107 * q^40 - 60 * q^41 - 41 * q^42 - 109 * q^43 - 21 * q^44 - 62 * q^45 - 92 * q^46 - 26 * q^47 - 90 * q^48 - 7 * q^49 - 22 * q^50 - 47 * q^51 - 182 * q^52 - 83 * q^53 - 19 * q^54 - 53 * q^55 - 23 * q^56 - 201 * q^57 - 112 * q^58 + 14 * q^59 - 64 * q^60 - 73 * q^61 - 21 * q^62 - 94 * q^63 + 14 * q^64 - 123 * q^65 - 10 * q^66 - 135 * q^67 - 84 * q^68 - 50 * q^69 - 35 * q^70 - 29 * q^71 - 143 * q^72 - 266 * q^73 - 53 * q^74 - 32 * q^75 - 66 * q^76 - 69 * q^77 - 59 * q^78 - 124 * q^79 - 20 * q^80 - 33 * q^81 - 93 * q^82 - 28 * q^83 - 4 * q^84 - 179 * q^85 + 6 * q^86 - 40 * q^87 - 259 * q^88 - 41 * q^89 + 2 * q^90 - 50 * q^91 - 77 * q^92 - 60 * q^93 - 48 * q^94 - 37 * q^95 + 3 * q^96 - 220 * q^97 - 9 * q^98 - 35 * q^99

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