LMFDB - Embedded newform 1334.2.a.j.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1334,2,Mod(1,1334)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1334, 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("1334.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1334 = 2 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1334.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:	$10.6520436296$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 3x^{8} - 16x^{7} + 44x^{6} + 87x^{5} - 209x^{4} - 160x^{3} + 348x^{2} + 12x - 100 $ x^9 - 3x^8 - 16x^7 + 44x^6 + 87x^5 - 209x^4 - 160x^3 + 348x^2 + 12x - 100
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.07506$ of defining polynomial
Character	$\chi$	$=$	1334.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.07506 q^{3} +1.00000 q^{4} -2.43466 q^{5} +1.07506 q^{6} +1.40780 q^{7} -1.00000 q^{8} -1.84425 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.07506 q^{3} +1.00000 q^{4} -2.43466 q^{5} +1.07506 q^{6} +1.40780 q^{7} -1.00000 q^{8} -1.84425 q^{9} +2.43466 q^{10} -6.33258 q^{11} -1.07506 q^{12} +2.92714 q^{13} -1.40780 q^{14} +2.61740 q^{15} +1.00000 q^{16} -0.283266 q^{17} +1.84425 q^{18} -2.36360 q^{19} -2.43466 q^{20} -1.51347 q^{21} +6.33258 q^{22} -1.00000 q^{23} +1.07506 q^{24} +0.927554 q^{25} -2.92714 q^{26} +5.20785 q^{27} +1.40780 q^{28} +1.00000 q^{29} -2.61740 q^{30} -4.33258 q^{31} -1.00000 q^{32} +6.80788 q^{33} +0.283266 q^{34} -3.42751 q^{35} -1.84425 q^{36} +6.76879 q^{37} +2.36360 q^{38} -3.14685 q^{39} +2.43466 q^{40} +4.76879 q^{41} +1.51347 q^{42} +4.18246 q^{43} -6.33258 q^{44} +4.49012 q^{45} +1.00000 q^{46} -9.78996 q^{47} -1.07506 q^{48} -5.01810 q^{49} -0.927554 q^{50} +0.304528 q^{51} +2.92714 q^{52} -3.25385 q^{53} -5.20785 q^{54} +15.4177 q^{55} -1.40780 q^{56} +2.54100 q^{57} -1.00000 q^{58} +7.80462 q^{59} +2.61740 q^{60} -1.83744 q^{61} +4.33258 q^{62} -2.59634 q^{63} +1.00000 q^{64} -7.12659 q^{65} -6.80788 q^{66} +10.8071 q^{67} -0.283266 q^{68} +1.07506 q^{69} +3.42751 q^{70} +6.14324 q^{71} +1.84425 q^{72} +4.63583 q^{73} -6.76879 q^{74} -0.997173 q^{75} -2.36360 q^{76} -8.91501 q^{77} +3.14685 q^{78} +0.484928 q^{79} -2.43466 q^{80} -0.0659779 q^{81} -4.76879 q^{82} +13.1695 q^{83} -1.51347 q^{84} +0.689656 q^{85} -4.18246 q^{86} -1.07506 q^{87} +6.33258 q^{88} -8.83682 q^{89} -4.49012 q^{90} +4.12083 q^{91} -1.00000 q^{92} +4.65777 q^{93} +9.78996 q^{94} +5.75454 q^{95} +1.07506 q^{96} -5.92654 q^{97} +5.01810 q^{98} +11.6789 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 9 q^{2} - 3 q^{3} + 9 q^{4} + 5 q^{5} + 3 q^{6} + 6 q^{7} - 9 q^{8} + 14 q^{9}+O(q^{10}) $ 9 * q - 9 * q^2 - 3 * q^3 + 9 * q^4 + 5 * q^5 + 3 * q^6 + 6 * q^7 - 9 * q^8 + 14 * q^9	\( 9 q - 9 q^{2} - 3 q^{3} + 9 q^{4} + 5 q^{5} + 3 q^{6} + 6 q^{7} - 9 q^{8} + 14 q^{9} - 5 q^{10} - 3 q^{11} - 3 q^{12} + 13 q^{13} - 6 q^{14} - 3 q^{15} + 9 q^{16} + 2 q^{17} - 14 q^{18} + 16 q^{19} + 5 q^{20} + 8 q^{21} + 3 q^{22} - 9 q^{23} + 3 q^{24} + 20 q^{25} - 13 q^{26} - 21 q^{27} + 6 q^{28} + 9 q^{29} + 3 q^{30} + 15 q^{31} - 9 q^{32} + 13 q^{33} - 2 q^{34} + 14 q^{36} + 12 q^{37} - 16 q^{38} - 5 q^{39} - 5 q^{40} - 6 q^{41} - 8 q^{42} - 3 q^{43} - 3 q^{44} + 20 q^{45} + 9 q^{46} - 19 q^{47} - 3 q^{48} + 37 q^{49} - 20 q^{50} + 6 q^{51} + 13 q^{52} + 5 q^{53} + 21 q^{54} + q^{55} - 6 q^{56} - 20 q^{57} - 9 q^{58} + 12 q^{59} - 3 q^{60} + 12 q^{61} - 15 q^{62} + 6 q^{63} + 9 q^{64} + 19 q^{65} - 13 q^{66} - 6 q^{67} + 2 q^{68} + 3 q^{69} + 12 q^{71} - 14 q^{72} - 12 q^{74} + 16 q^{75} + 16 q^{76} + 34 q^{77} + 5 q^{78} + 29 q^{79} + 5 q^{80} + 5 q^{81} + 6 q^{82} + 24 q^{83} + 8 q^{84} + 12 q^{85} + 3 q^{86} - 3 q^{87} + 3 q^{88} - 2 q^{89} - 20 q^{90} + 58 q^{91} - 9 q^{92} + 7 q^{93} + 19 q^{94} + 6 q^{95} + 3 q^{96} + 12 q^{97} - 37 q^{98} - 20 q^{99}+O(q^{100}) \) 9 * q - 9 * q^2 - 3 * q^3 + 9 * q^4 + 5 * q^5 + 3 * q^6 + 6 * q^7 - 9 * q^8 + 14 * q^9 - 5 * q^10 - 3 * q^11 - 3 * q^12 + 13 * q^13 - 6 * q^14 - 3 * q^15 + 9 * q^16 + 2 * q^17 - 14 * q^18 + 16 * q^19 + 5 * q^20 + 8 * q^21 + 3 * q^22 - 9 * q^23 + 3 * q^24 + 20 * q^25 - 13 * q^26 - 21 * q^27 + 6 * q^28 + 9 * q^29 + 3 * q^30 + 15 * q^31 - 9 * q^32 + 13 * q^33 - 2 * q^34 + 14 * q^36 + 12 * q^37 - 16 * q^38 - 5 * q^39 - 5 * q^40 - 6 * q^41 - 8 * q^42 - 3 * q^43 - 3 * q^44 + 20 * q^45 + 9 * q^46 - 19 * q^47 - 3 * q^48 + 37 * q^49 - 20 * q^50 + 6 * q^51 + 13 * q^52 + 5 * q^53 + 21 * q^54 + q^55 - 6 * q^56 - 20 * q^57 - 9 * q^58 + 12 * q^59 - 3 * q^60 + 12 * q^61 - 15 * q^62 + 6 * q^63 + 9 * q^64 + 19 * q^65 - 13 * q^66 - 6 * q^67 + 2 * q^68 + 3 * q^69 + 12 * q^71 - 14 * q^72 - 12 * q^74 + 16 * q^75 + 16 * q^76 + 34 * q^77 + 5 * q^78 + 29 * q^79 + 5 * q^80 + 5 * q^81 + 6 * q^82 + 24 * q^83 + 8 * q^84 + 12 * q^85 + 3 * q^86 - 3 * q^87 + 3 * q^88 - 2 * q^89 - 20 * q^90 + 58 * q^91 - 9 * q^92 + 7 * q^93 + 19 * q^94 + 6 * q^95 + 3 * q^96 + 12 * q^97 - 37 * q^98 - 20 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.07506	−0.620685	−0.310342	−	0.950625i	$-0.600444\pi$
$3$	−1.07506	−0.620685	−0.310342	+	0.950625i	$0.600444\pi$
$4$	1.00000	0.500000
$5$	−2.43466	−1.08881	−0.544406	−	0.838822i	$-0.683245\pi$
$5$	−2.43466	−1.08881	−0.544406	+	0.838822i	$0.683245\pi$
$6$	1.07506	0.438890
$7$	1.40780	0.532099	0.266049	−	0.963959i	$-0.414282\pi$
$7$	1.40780	0.532099	0.266049	+	0.963959i	$0.414282\pi$
$8$	−1.00000	−0.353553
$9$	−1.84425	−0.614751
$10$	2.43466	0.769906
$11$	−6.33258	−1.90934	−0.954672	−	0.297660i	$-0.903794\pi$
$11$	−6.33258	−1.90934	−0.954672	+	0.297660i	$0.903794\pi$
$12$	−1.07506	−0.310342
$13$	2.92714	0.811844	0.405922	−	0.913908i	$-0.366951\pi$
$13$	2.92714	0.811844	0.405922	+	0.913908i	$0.366951\pi$
$14$	−1.40780	−0.376250
$15$	2.61740	0.675809
$16$	1.00000	0.250000
$17$	−0.283266	−0.0687022	−0.0343511	−	0.999410i	$-0.510936\pi$
$17$	−0.283266	−0.0687022	−0.0343511	+	0.999410i	$0.510936\pi$
$18$	1.84425	0.434694
$19$	−2.36360	−0.542246	−0.271123	−	0.962545i	$-0.587395\pi$
$19$	−2.36360	−0.542246	−0.271123	+	0.962545i	$0.587395\pi$
$20$	−2.43466	−0.544406
$21$	−1.51347	−0.330265
$22$	6.33258	1.35011
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	1.07506	0.219445
$25$	0.927554	0.185511
$26$	−2.92714	−0.574060
$27$	5.20785	1.00225
$28$	1.40780	0.266049
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	−2.61740	−0.477869
$31$	−4.33258	−0.778154	−0.389077	−	0.921205i	$-0.627206\pi$
$31$	−4.33258	−0.778154	−0.389077	+	0.921205i	$0.627206\pi$
$32$	−1.00000	−0.176777
$33$	6.80788	1.18510
$34$	0.283266	0.0485798
$35$	−3.42751	−0.579355
$36$	−1.84425	−0.307375
$37$	6.76879	1.11278	0.556391	−	0.830921i	$-0.312186\pi$
$37$	6.76879	1.11278	0.556391	+	0.830921i	$0.312186\pi$
$38$	2.36360	0.383426
$39$	−3.14685	−0.503899
$40$	2.43466	0.384953
$41$	4.76879	0.744759	0.372380	−	0.928081i	$-0.378542\pi$
$41$	4.76879	0.744759	0.372380	+	0.928081i	$0.378542\pi$
$42$	1.51347	0.233533
$43$	4.18246	0.637820	0.318910	−	0.947785i	$-0.396683\pi$
$43$	4.18246	0.637820	0.318910	+	0.947785i	$0.396683\pi$
$44$	−6.33258	−0.954672
$45$	4.49012	0.669348
$46$	1.00000	0.147442
$47$	−9.78996	−1.42801	−0.714007	−	0.700139i	$-0.753121\pi$
$47$	−9.78996	−1.42801	−0.714007	+	0.700139i	$0.753121\pi$
$48$	−1.07506	−0.155171
$49$	−5.01810	−0.716871
$50$	−0.927554	−0.131176
$51$	0.304528	0.0426424
$52$	2.92714	0.405922
$53$	−3.25385	−0.446950	−0.223475	−	0.974710i	$-0.571740\pi$
$53$	−3.25385	−0.446950	−0.223475	+	0.974710i	$0.571740\pi$
$54$	−5.20785	−0.708698
$55$	15.4177	2.07892
$56$	−1.40780	−0.188125
$57$	2.54100	0.336564
$58$	−1.00000	−0.131306
$59$	7.80462	1.01607	0.508037	−	0.861335i	$-0.330371\pi$
$59$	7.80462	1.01607	0.508037	+	0.861335i	$0.330371\pi$
$60$	2.61740	0.337904
$61$	−1.83744	−0.235260	−0.117630	−	0.993058i	$-0.537530\pi$
$61$	−1.83744	−0.235260	−0.117630	+	0.993058i	$0.537530\pi$
$62$	4.33258	0.550238
$63$	−2.59634	−0.327108
$64$	1.00000	0.125000
$65$	−7.12659	−0.883945
$66$	−6.80788	−0.837993
$67$	10.8071	1.32030	0.660148	−	0.751136i	$-0.270494\pi$
$67$	10.8071	1.32030	0.660148	+	0.751136i	$0.270494\pi$
$68$	−0.283266	−0.0343511
$69$	1.07506	0.129422
$70$	3.42751	0.409666
$71$	6.14324	0.729068	0.364534	−	0.931190i	$-0.381228\pi$
$71$	6.14324	0.729068	0.364534	+	0.931190i	$0.381228\pi$
$72$	1.84425	0.217347
$73$	4.63583	0.542583	0.271291	−	0.962497i	$-0.412549\pi$
$73$	4.63583	0.542583	0.271291	+	0.962497i	$0.412549\pi$
$74$	−6.76879	−0.786855
$75$	−0.997173	−0.115144
$76$	−2.36360	−0.271123
$77$	−8.91501	−1.01596
$78$	3.14685	0.356310
$79$	0.484928	0.0545586	0.0272793	−	0.999628i	$-0.491316\pi$
$79$	0.484928	0.0545586	0.0272793	+	0.999628i	$0.491316\pi$
$80$	−2.43466	−0.272203
$81$	−0.0659779	−0.00733088
$82$	−4.76879	−0.526624
$83$	13.1695	1.44554	0.722770	−	0.691089i	$-0.242869\pi$
$83$	13.1695	1.44554	0.722770	+	0.691089i	$0.242869\pi$
$84$	−1.51347	−0.165133
$85$	0.689656	0.0748037
$86$	−4.18246	−0.451007
$87$	−1.07506	−0.115258
$88$	6.33258	0.675055
$89$	−8.83682	−0.936701	−0.468351	−	0.883543i	$-0.655152\pi$
$89$	−8.83682	−0.936701	−0.468351	+	0.883543i	$0.655152\pi$
$90$	−4.49012	−0.473300
$91$	4.12083	0.431981
$92$	−1.00000	−0.104257
$93$	4.65777	0.482988
$94$	9.78996	1.00976
$95$	5.75454	0.590404
$96$	1.07506	0.109723
$97$	−5.92654	−0.601749	−0.300874	−	0.953664i	$-0.597279\pi$
$97$	−5.92654	−0.601749	−0.300874	+	0.953664i	$0.597279\pi$
$98$	5.01810	0.506904
$99$	11.6789	1.17377
$100$	0.927554	0.0927554
$101$	8.48804	0.844592	0.422296	−	0.906458i	$-0.361224\pi$
$101$	8.48804	0.844592	0.422296	+	0.906458i	$0.361224\pi$
$102$	−0.304528	−0.0301527
$103$	6.47369	0.637872	0.318936	−	0.947776i	$-0.396675\pi$
$103$	6.47369	0.637872	0.318936	+	0.947776i	$0.396675\pi$
$104$	−2.92714	−0.287030
$105$	3.68477	0.359597
$106$	3.25385	0.316042
$107$	−1.65548	−0.160041	−0.0800207	−	0.996793i	$-0.525499\pi$
$107$	−1.65548	−0.160041	−0.0800207	+	0.996793i	$0.525499\pi$
$108$	5.20785	0.501125
$109$	17.8784	1.71244	0.856222	−	0.516607i	$-0.172805\pi$
$109$	17.8784	1.71244	0.856222	+	0.516607i	$0.172805\pi$
$110$	−15.4177	−1.47002
$111$	−7.27683	−0.690686
$112$	1.40780	0.133025
$113$	4.83335	0.454683	0.227341	−	0.973815i	$-0.426997\pi$
$113$	4.83335	0.454683	0.227341	+	0.973815i	$0.426997\pi$
$114$	−2.54100	−0.237987
$115$	2.43466	0.227033
$116$	1.00000	0.0928477
$117$	−5.39839	−0.499081
$118$	−7.80462	−0.718473
$119$	−0.398783	−0.0365563
$120$	−2.61740	−0.238934
$121$	29.1015	2.64560
$122$	1.83744	0.166354
$123$	−5.12672	−0.462261
$124$	−4.33258	−0.389077
$125$	9.91501	0.886825
$126$	2.59634	0.231300
$127$	−11.2901	−1.00184	−0.500919	−	0.865494i	$-0.667004\pi$
$127$	−11.2901	−1.00184	−0.500919	+	0.865494i	$0.667004\pi$
$128$	−1.00000	−0.0883883
$129$	−4.49639	−0.395885
$130$	7.12659	0.625043
$131$	1.86543	0.162984	0.0814918	−	0.996674i	$-0.474032\pi$
$131$	1.86543	0.162984	0.0814918	+	0.996674i	$0.474032\pi$
$132$	6.80788	0.592550
$133$	−3.32747	−0.288528
$134$	−10.8071	−0.933590
$135$	−12.6793	−1.09126
$136$	0.283266	0.0242899
$137$	22.3212	1.90703	0.953513	−	0.301351i	$-0.0974377\pi$
$137$	22.3212	1.90703	0.953513	+	0.301351i	$0.0974377\pi$
$138$	−1.07506	−0.0915149
$139$	−0.343103	−0.0291016	−0.0145508	−	0.999894i	$-0.504632\pi$
$139$	−0.343103	−0.0291016	−0.0145508	+	0.999894i	$0.504632\pi$
$140$	−3.42751	−0.289678
$141$	10.5248	0.886346
$142$	−6.14324	−0.515529
$143$	−18.5364	−1.55009
$144$	−1.84425	−0.153688
$145$	−2.43466	−0.202187
$146$	−4.63583	−0.383664
$147$	5.39474	0.444951
$148$	6.76879	0.556391
$149$	3.54881	0.290730	0.145365	−	0.989378i	$-0.453564\pi$
$149$	3.54881	0.290730	0.145365	+	0.989378i	$0.453564\pi$
$150$	0.997173	0.0814188
$151$	−3.06643	−0.249542	−0.124771	−	0.992186i	$-0.539820\pi$
$151$	−3.06643	−0.249542	−0.124771	+	0.992186i	$0.539820\pi$
$152$	2.36360	0.191713
$153$	0.522415	0.0422347
$154$	8.91501	0.718392
$155$	10.5483	0.847263
$156$	−3.14685	−0.251949
$157$	14.8974	1.18894	0.594470	−	0.804118i	$-0.297362\pi$
$157$	14.8974	1.18894	0.594470	+	0.804118i	$0.297362\pi$
$158$	−0.484928	−0.0385788
$159$	3.49807	0.277415
$160$	2.43466	0.192477
$161$	−1.40780	−0.110950
$162$	0.0659779	0.00518371
$163$	−5.86556	−0.459426	−0.229713	−	0.973258i	$-0.573779\pi$
$163$	−5.86556	−0.459426	−0.229713	+	0.973258i	$0.573779\pi$
$164$	4.76879	0.372380
$165$	−16.5749	−1.29035
$166$	−13.1695	−1.02215
$167$	−9.85834	−0.762861	−0.381431	−	0.924397i	$-0.624569\pi$
$167$	−9.85834	−0.762861	−0.381431	+	0.924397i	$0.624569\pi$
$168$	1.51347	0.116766
$169$	−4.43183	−0.340910
$170$	−0.689656	−0.0528942
$171$	4.35907	0.333346
$172$	4.18246	0.318910
$173$	−8.12015	−0.617364	−0.308682	−	0.951165i	$-0.599888\pi$
$173$	−8.12015	−0.617364	−0.308682	+	0.951165i	$0.599888\pi$
$174$	1.07506	0.0814999
$175$	1.30581	0.0987100
$176$	−6.33258	−0.477336
$177$	−8.39041	−0.630662
$178$	8.83682	0.662348
$179$	11.4247	0.853921	0.426960	−	0.904270i	$-0.359584\pi$
$179$	11.4247	0.853921	0.426960	+	0.904270i	$0.359584\pi$
$180$	4.49012	0.334674
$181$	5.21923	0.387942	0.193971	−	0.981007i	$-0.437863\pi$
$181$	5.21923	0.387942	0.193971	+	0.981007i	$0.437863\pi$
$182$	−4.12083	−0.305457
$183$	1.97535	0.146022
$184$	1.00000	0.0737210
$185$	−16.4797	−1.21161
$186$	−4.65777	−0.341524
$187$	1.79381	0.131176
$188$	−9.78996	−0.714007
$189$	7.33161	0.533296
$190$	−5.75454	−0.417479
$191$	8.25473	0.597291	0.298646	−	0.954364i	$-0.403465\pi$
$191$	8.25473	0.597291	0.298646	+	0.954364i	$0.403465\pi$
$192$	−1.07506	−0.0775856
$193$	−14.5230	−1.04539	−0.522694	−	0.852520i	$-0.675073\pi$
$193$	−14.5230	−1.04539	−0.522694	+	0.852520i	$0.675073\pi$
$194$	5.92654	0.425501
$195$	7.66149	0.548651
$196$	−5.01810	−0.358436
$197$	−0.882862	−0.0629013	−0.0314506	−	0.999505i	$-0.510013\pi$
$197$	−0.882862	−0.0629013	−0.0314506	+	0.999505i	$0.510013\pi$
$198$	−11.6789	−0.829981
$199$	−10.9897	−0.779042	−0.389521	−	0.921018i	$-0.627359\pi$
$199$	−10.9897	−0.779042	−0.389521	+	0.921018i	$0.627359\pi$
$200$	−0.927554	−0.0655879
$201$	−11.6182	−0.819487
$202$	−8.48804	−0.597217
$203$	1.40780	0.0988082
$204$	0.304528	0.0213212
$205$	−11.6104	−0.810902
$206$	−6.47369	−0.451044
$207$	1.84425	0.128184
$208$	2.92714	0.202961
$209$	14.9677	1.03533
$210$	−3.68477	−0.254273
$211$	−23.5726	−1.62280	−0.811402	−	0.584489i	$-0.801295\pi$
$211$	−23.5726	−1.62280	−0.811402	+	0.584489i	$0.801295\pi$
$212$	−3.25385	−0.223475
$213$	−6.60433	−0.452521
$214$	1.65548	0.113166
$215$	−10.1829	−0.694466
$216$	−5.20785	−0.354349
$217$	−6.09941	−0.414055
$218$	−17.8784	−1.21088
$219$	−4.98378	−0.336773
$220$	15.4177	1.03946
$221$	−0.829162	−0.0557754
$222$	7.27683	0.488389
$223$	−20.7377	−1.38870	−0.694351	−	0.719637i	$-0.744308\pi$
$223$	−20.7377	−1.38870	−0.694351	+	0.719637i	$0.744308\pi$
$224$	−1.40780	−0.0940626
$225$	−1.71064	−0.114043
$226$	−4.83335	−0.321509
$227$	−16.4786	−1.09372	−0.546861	−	0.837223i	$-0.684178\pi$
$227$	−16.4786	−1.09372	−0.546861	+	0.837223i	$0.684178\pi$
$228$	2.54100	0.168282
$229$	16.9422	1.11957	0.559786	−	0.828637i	$-0.310883\pi$
$229$	16.9422	1.11957	0.559786	+	0.828637i	$0.310883\pi$
$230$	−2.43466	−0.160537
$231$	9.58414	0.630590
$232$	−1.00000	−0.0656532
$233$	23.6202	1.54741	0.773705	−	0.633546i	$-0.218401\pi$
$233$	23.6202	1.54741	0.773705	+	0.633546i	$0.218401\pi$
$234$	5.39839	0.352904
$235$	23.8352	1.55484
$236$	7.80462	0.508037
$237$	−0.521325	−0.0338637
$238$	0.398783	0.0258492
$239$	11.5234	0.745390	0.372695	−	0.927954i	$-0.378434\pi$
$239$	11.5234	0.745390	0.372695	+	0.927954i	$0.378434\pi$
$240$	2.61740	0.168952
$241$	13.3151	0.857698	0.428849	−	0.903376i	$-0.358919\pi$
$241$	13.3151	0.857698	0.428849	+	0.903376i	$0.358919\pi$
$242$	−29.1015	−1.87072
$243$	−15.5526	−0.997701
$244$	−1.83744	−0.117630
$245$	12.2173	0.780538
$246$	5.12672	0.326868
$247$	−6.91859	−0.440219
$248$	4.33258	0.275119
$249$	−14.1580	−0.897224
$250$	−9.91501	−0.627080
$251$	24.9345	1.57385	0.786925	−	0.617048i	$-0.211672\pi$
$251$	24.9345	1.57385	0.786925	+	0.617048i	$0.211672\pi$
$252$	−2.59634	−0.163554
$253$	6.33258	0.398126
$254$	11.2901	0.708406
$255$	−0.741420	−0.0464295
$256$	1.00000	0.0625000
$257$	10.3767	0.647283	0.323642	−	0.946180i	$-0.395093\pi$
$257$	10.3767	0.647283	0.323642	+	0.946180i	$0.395093\pi$
$258$	4.49639	0.279933
$259$	9.52910	0.592109
$260$	−7.12659	−0.441972
$261$	−1.84425	−0.114156
$262$	−1.86543	−0.115247
$263$	−23.0340	−1.42034	−0.710168	−	0.704032i	$-0.751381\pi$
$263$	−23.0340	−1.42034	−0.710168	+	0.704032i	$0.751381\pi$
$264$	−6.80788	−0.418996
$265$	7.92200	0.486645
$266$	3.32747	0.204020
$267$	9.50009	0.581396
$268$	10.8071	0.660148
$269$	−15.7813	−0.962202	−0.481101	−	0.876665i	$-0.659763\pi$
$269$	−15.7813	−0.962202	−0.481101	+	0.876665i	$0.659763\pi$
$270$	12.6793	0.771639
$271$	−13.5883	−0.825433	−0.412717	−	0.910859i	$-0.635420\pi$
$271$	−13.5883	−0.825433	−0.412717	+	0.910859i	$0.635420\pi$
$272$	−0.283266	−0.0171755
$273$	−4.43013	−0.268124
$274$	−22.3212	−1.34847
$275$	−5.87381	−0.354204
$276$	1.07506	0.0647108
$277$	−18.3380	−1.10183	−0.550913	−	0.834563i	$-0.685720\pi$
$277$	−18.3380	−1.10183	−0.550913	+	0.834563i	$0.685720\pi$
$278$	0.343103	0.0205780
$279$	7.99037	0.478371
$280$	3.42751	0.204833
$281$	1.24607	0.0743341	0.0371671	−	0.999309i	$-0.488167\pi$
$281$	1.24607	0.0743341	0.0371671	+	0.999309i	$0.488167\pi$
$282$	−10.5248	−0.626741
$283$	19.3796	1.15200	0.575998	−	0.817451i	$-0.304614\pi$
$283$	19.3796	1.15200	0.575998	+	0.817451i	$0.304614\pi$
$284$	6.14324	0.364534
$285$	−6.18646	−0.366455
$286$	18.5364	1.09608
$287$	6.71350	0.396285
$288$	1.84425	0.108674
$289$	−16.9198	−0.995280
$290$	2.43466	0.142968
$291$	6.37137	0.373496
$292$	4.63583	0.271291
$293$	12.0036	0.701256	0.350628	−	0.936515i	$-0.385968\pi$
$293$	12.0036	0.701256	0.350628	+	0.936515i	$0.385968\pi$
$294$	−5.39474	−0.314628
$295$	−19.0016	−1.10631
$296$	−6.76879	−0.393428
$297$	−32.9791	−1.91364
$298$	−3.54881	−0.205577
$299$	−2.92714	−0.169281
$300$	−0.997173	−0.0575718
$301$	5.88807	0.339383
$302$	3.06643	0.176453
$303$	−9.12513	−0.524225
$304$	−2.36360	−0.135562
$305$	4.47353	0.256153
$306$	−0.522415	−0.0298645
$307$	10.2661	0.585915	0.292957	−	0.956126i	$-0.405361\pi$
$307$	10.2661	0.585915	0.292957	+	0.956126i	$0.405361\pi$
$308$	−8.91501	−0.507980
$309$	−6.95959	−0.395917
$310$	−10.5483	−0.599105
$311$	−11.9636	−0.678395	−0.339197	−	0.940715i	$-0.610155\pi$
$311$	−11.9636	−0.678395	−0.339197	+	0.940715i	$0.610155\pi$
$312$	3.14685	0.178155
$313$	−1.93119	−0.109158	−0.0545788	−	0.998509i	$-0.517382\pi$
$313$	−1.93119	−0.109158	−0.0545788	+	0.998509i	$0.517382\pi$
$314$	−14.8974	−0.840707
$315$	6.32119	0.356159
$316$	0.484928	0.0272793
$317$	3.24811	0.182432	0.0912160	−	0.995831i	$-0.470925\pi$
$317$	3.24811	0.182432	0.0912160	+	0.995831i	$0.470925\pi$
$318$	−3.49807	−0.196162
$319$	−6.33258	−0.354556
$320$	−2.43466	−0.136101
$321$	1.77974	0.0993352
$322$	1.40780	0.0784537
$323$	0.669527	0.0372535
$324$	−0.0659779	−0.00366544
$325$	2.71508	0.150606
$326$	5.86556	0.324863
$327$	−19.2204	−1.06289
$328$	−4.76879	−0.263312
$329$	−13.7823	−0.759844
$330$	16.5749	0.912416
$331$	−9.15255	−0.503070	−0.251535	−	0.967848i	$-0.580935\pi$
$331$	−9.15255	−0.503070	−0.251535	+	0.967848i	$0.580935\pi$
$332$	13.1695	0.722770
$333$	−12.4833	−0.684083
$334$	9.85834	0.539424
$335$	−26.3115	−1.43755
$336$	−1.51347	−0.0825663
$337$	19.8667	1.08221	0.541103	−	0.840956i	$-0.318007\pi$
$337$	19.8667	1.08221	0.541103	+	0.840956i	$0.318007\pi$
$338$	4.43183	0.241060
$339$	−5.19612	−0.282215
$340$	0.689656	0.0374019
$341$	27.4364	1.48576
$342$	−4.35907	−0.235711
$343$	−16.9191	−0.913545
$344$	−4.18246	−0.225503
$345$	−2.61740	−0.140916
$346$	8.12015	0.436542
$347$	−25.1380	−1.34948	−0.674740	−	0.738055i	$-0.735744\pi$
$347$	−25.1380	−1.34948	−0.674740	+	0.738055i	$0.735744\pi$
$348$	−1.07506	−0.0576291
$349$	31.2708	1.67389	0.836943	−	0.547291i	$-0.184341\pi$
$349$	31.2708	1.67389	0.836943	+	0.547291i	$0.184341\pi$
$350$	−1.30581	−0.0697985
$351$	15.2441	0.813671
$352$	6.33258	0.337528
$353$	−14.1949	−0.755518	−0.377759	−	0.925904i	$-0.623305\pi$
$353$	−14.1949	−0.755518	−0.377759	+	0.925904i	$0.623305\pi$
$354$	8.39041	0.445945
$355$	−14.9567	−0.793818
$356$	−8.83682	−0.468351
$357$	0.428714	0.0226900
$358$	−11.4247	−0.603813
$359$	−20.0879	−1.06020	−0.530099	−	0.847936i	$-0.677845\pi$
$359$	−20.0879	−1.06020	−0.530099	+	0.847936i	$0.677845\pi$
$360$	−4.49012	−0.236650
$361$	−13.4134	−0.705969
$362$	−5.21923	−0.274316
$363$	−31.2858	−1.64208
$364$	4.12083	0.215990
$365$	−11.2867	−0.590770
$366$	−1.97535	−0.103253
$367$	12.5724	0.656272	0.328136	−	0.944630i	$-0.393580\pi$
$367$	12.5724	0.656272	0.328136	+	0.944630i	$0.393580\pi$
$368$	−1.00000	−0.0521286
$369$	−8.79484	−0.457841
$370$	16.4797	0.856737
$371$	−4.58077	−0.237822
$372$	4.65777	0.241494
$373$	−27.5359	−1.42576	−0.712879	−	0.701287i	$-0.752609\pi$
$373$	−27.5359	−1.42576	−0.712879	+	0.701287i	$0.752609\pi$
$374$	−1.79381	−0.0927555
$375$	−10.6592	−0.550439
$376$	9.78996	0.504879
$377$	2.92714	0.150756
$378$	−7.33161	−0.377097
$379$	−7.71967	−0.396533	−0.198266	−	0.980148i	$-0.563531\pi$
$379$	−7.71967	−0.396533	−0.198266	+	0.980148i	$0.563531\pi$
$380$	5.75454	0.295202
$381$	12.1375	0.621825
$382$	−8.25473	−0.422349
$383$	−29.2577	−1.49500	−0.747500	−	0.664262i	$-0.768746\pi$
$383$	−29.2577	−1.49500	−0.747500	+	0.664262i	$0.768746\pi$
$384$	1.07506	0.0548613
$385$	21.7050	1.10619
$386$	14.5230	0.739201
$387$	−7.71352	−0.392100
$388$	−5.92654	−0.300874
$389$	−22.5251	−1.14207	−0.571034	−	0.820926i	$-0.693458\pi$
$389$	−22.5251	−1.14207	−0.571034	+	0.820926i	$0.693458\pi$
$390$	−7.66149	−0.387955
$391$	0.283266	0.0143254
$392$	5.01810	0.253452
$393$	−2.00545	−0.101161
$394$	0.882862	0.0444779
$395$	−1.18063	−0.0594041
$396$	11.6789	0.586885
$397$	2.21829	0.111333	0.0556665	−	0.998449i	$-0.482272\pi$
$397$	2.21829	0.111333	0.0556665	+	0.998449i	$0.482272\pi$
$398$	10.9897	0.550866
$399$	3.57722	0.179085
$400$	0.927554	0.0463777
$401$	22.5036	1.12377	0.561887	−	0.827214i	$-0.310076\pi$
$401$	22.5036	1.12377	0.561887	+	0.827214i	$0.310076\pi$
$402$	11.6182	0.579465
$403$	−12.6821	−0.631739
$404$	8.48804	0.422296
$405$	0.160633	0.00798194
$406$	−1.40780	−0.0698680
$407$	−42.8639	−2.12468
$408$	−0.304528	−0.0150764
$409$	5.54865	0.274363	0.137182	−	0.990546i	$-0.456196\pi$
$409$	5.54865	0.274363	0.137182	+	0.990546i	$0.456196\pi$
$410$	11.6104	0.573395
$411$	−23.9965	−1.18366
$412$	6.47369	0.318936
$413$	10.9873	0.540652
$414$	−1.84425	−0.0906400
$415$	−32.0632	−1.57392
$416$	−2.92714	−0.143515
$417$	0.368856	0.0180629
$418$	−14.9677	−0.732092
$419$	−7.30749	−0.356994	−0.178497	−	0.983940i	$-0.557124\pi$
$419$	−7.30749	−0.356994	−0.178497	+	0.983940i	$0.557124\pi$
$420$	3.68477	0.179798
$421$	−1.20079	−0.0585231	−0.0292616	−	0.999572i	$-0.509316\pi$
$421$	−1.20079	−0.0585231	−0.0292616	+	0.999572i	$0.509316\pi$
$422$	23.5726	1.14750
$423$	18.0552	0.877872
$424$	3.25385	0.158021
$425$	−0.262745	−0.0127450
$426$	6.60433	0.319981
$427$	−2.58674	−0.125181
$428$	−1.65548	−0.0800207
$429$	19.9277	0.962116
$430$	10.1829	0.491061
$431$	25.3299	1.22010	0.610050	−	0.792363i	$-0.291149\pi$
$431$	25.3299	1.22010	0.610050	+	0.792363i	$0.291149\pi$
$432$	5.20785	0.250563
$433$	−7.48240	−0.359581	−0.179790	−	0.983705i	$-0.557542\pi$
$433$	−7.48240	−0.359581	−0.179790	+	0.983705i	$0.557542\pi$
$434$	6.09941	0.292781
$435$	2.61740	0.125494
$436$	17.8784	0.856222
$437$	2.36360	0.113066
$438$	4.98378	0.238134
$439$	12.8269	0.612194	0.306097	−	0.952000i	$-0.400977\pi$
$439$	12.8269	0.612194	0.306097	+	0.952000i	$0.400977\pi$
$440$	−15.4177	−0.735008
$441$	9.25464	0.440697
$442$	0.829162	0.0394392
$443$	−10.8870	−0.517257	−0.258629	−	0.965977i	$-0.583271\pi$
$443$	−10.8870	−0.517257	−0.258629	+	0.965977i	$0.583271\pi$
$444$	−7.27683	−0.345343
$445$	21.5146	1.01989
$446$	20.7377	0.981960
$447$	−3.81517	−0.180451
$448$	1.40780	0.0665123
$449$	24.0051	1.13287	0.566436	−	0.824106i	$-0.308322\pi$
$449$	24.0051	1.13287	0.566436	+	0.824106i	$0.308322\pi$
$450$	1.71064	0.0806405
$451$	−30.1987	−1.42200
$452$	4.83335	0.227341
$453$	3.29659	0.154887
$454$	16.4786	0.773379
$455$	−10.0328	−0.470346
$456$	−2.54100	−0.118993
$457$	0.712242	0.0333173	0.0166586	−	0.999861i	$-0.494697\pi$
$457$	0.712242	0.0333173	0.0166586	+	0.999861i	$0.494697\pi$
$458$	−16.9422	−0.791657
$459$	−1.47521	−0.0688568
$460$	2.43466	0.113516
$461$	20.7434	0.966118	0.483059	−	0.875588i	$-0.339526\pi$
$461$	20.7434	0.966118	0.483059	+	0.875588i	$0.339526\pi$
$462$	−9.58414	−0.445895
$463$	27.6079	1.28305	0.641524	−	0.767103i	$-0.278303\pi$
$463$	27.6079	1.28305	0.641524	+	0.767103i	$0.278303\pi$
$464$	1.00000	0.0464238
$465$	−11.3401	−0.525883
$466$	−23.6202	−1.09418
$467$	−36.7343	−1.69986	−0.849930	−	0.526896i	$-0.823356\pi$
$467$	−36.7343	−1.69986	−0.849930	+	0.526896i	$0.823356\pi$
$468$	−5.39839	−0.249541
$469$	15.2142	0.702527
$470$	−23.8352	−1.09944
$471$	−16.0155	−0.737957
$472$	−7.80462	−0.359237
$473$	−26.4858	−1.21782
$474$	0.521325	0.0239452
$475$	−2.19236	−0.100592
$476$	−0.398783	−0.0182782
$477$	6.00091	0.274763
$478$	−11.5234	−0.527070
$479$	−0.982415	−0.0448877	−0.0224439	−	0.999748i	$-0.507145\pi$
$479$	−0.982415	−0.0448877	−0.0224439	+	0.999748i	$0.507145\pi$
$480$	−2.61740	−0.119467
$481$	19.8132	0.903405
$482$	−13.3151	−0.606484
$483$	1.51347	0.0688651
$484$	29.1015	1.32280
$485$	14.4291	0.655191
$486$	15.5526	0.705481
$487$	26.6766	1.20883	0.604417	−	0.796668i	$-0.293406\pi$
$487$	26.6766	1.20883	0.604417	+	0.796668i	$0.293406\pi$
$488$	1.83744	0.0831768
$489$	6.30581	0.285159
$490$	−12.2173	−0.551923
$491$	36.1524	1.63154	0.815768	−	0.578380i	$-0.196315\pi$
$491$	36.1524	1.63154	0.815768	+	0.578380i	$0.196315\pi$
$492$	−5.12672	−0.231130
$493$	−0.283266	−0.0127577
$494$	6.91859	0.311282
$495$	−28.4340	−1.27802
$496$	−4.33258	−0.194538
$497$	8.64845	0.387936
$498$	14.1580	0.634433
$499$	−3.38486	−0.151527	−0.0757635	−	0.997126i	$-0.524139\pi$
$499$	−3.38486	−0.151527	−0.0757635	+	0.997126i	$0.524139\pi$
$500$	9.91501	0.443413
$501$	10.5983	0.473496
$502$	−24.9345	−1.11288
$503$	7.35598	0.327987	0.163994	−	0.986461i	$-0.447562\pi$
$503$	7.35598	0.327987	0.163994	+	0.986461i	$0.447562\pi$
$504$	2.59634	0.115650
$505$	−20.6655	−0.919601
$506$	−6.33258	−0.281517
$507$	4.76447	0.211597
$508$	−11.2901	−0.500919
$509$	31.6580	1.40321	0.701607	−	0.712564i	$-0.252466\pi$
$509$	31.6580	1.40321	0.701607	+	0.712564i	$0.252466\pi$
$510$	0.741420	0.0328306
$511$	6.52632	0.288708
$512$	−1.00000	−0.0441942
$513$	−12.3092	−0.543467
$514$	−10.3767	−0.457698
$515$	−15.7612	−0.694522
$516$	−4.49639	−0.197942
$517$	61.9957	2.72657
$518$	−9.52910	−0.418685
$519$	8.72963	0.383188
$520$	7.12659	0.312522
$521$	−6.35539	−0.278435	−0.139217	−	0.990262i	$-0.544459\pi$
$521$	−6.35539	−0.278435	−0.139217	+	0.990262i	$0.544459\pi$
$522$	1.84425	0.0807207
$523$	41.7615	1.82610	0.913052	−	0.407843i	$-0.133719\pi$
$523$	41.7615	1.82610	0.913052	+	0.407843i	$0.133719\pi$
$524$	1.86543	0.0814918
$525$	−1.40382	−0.0612678
$526$	23.0340	1.00433
$527$	1.22727	0.0534609
$528$	6.80788	0.296275
$529$	1.00000	0.0434783
$530$	−7.92200	−0.344110
$531$	−14.3937	−0.624633
$532$	−3.32747	−0.144264
$533$	13.9589	0.604628
$534$	−9.50009	−0.411109
$535$	4.03053	0.174255
$536$	−10.8071	−0.466795
$537$	−12.2822	−0.530015
$538$	15.7813	0.680380
$539$	31.7775	1.36875
$540$	−12.6793	−0.545631
$541$	−3.48810	−0.149965	−0.0749826	−	0.997185i	$-0.523890\pi$
$541$	−3.48810	−0.149965	−0.0749826	+	0.997185i	$0.523890\pi$
$542$	13.5883	0.583669
$543$	−5.61097	−0.240790
$544$	0.283266	0.0121449
$545$	−43.5279	−1.86453
$546$	4.43013	0.189592
$547$	11.3433	0.485004	0.242502	−	0.970151i	$-0.422032\pi$
$547$	11.3433	0.485004	0.242502	+	0.970151i	$0.422032\pi$
$548$	22.3212	0.953513
$549$	3.38870	0.144626
$550$	5.87381	0.250460
$551$	−2.36360	−0.100693
$552$	−1.07506	−0.0457575
$553$	0.682681	0.0290306
$554$	18.3380	0.779109
$555$	17.7166	0.752027
$556$	−0.343103	−0.0145508
$557$	22.6479	0.959622	0.479811	−	0.877372i	$-0.340705\pi$
$557$	22.6479	0.959622	0.479811	+	0.877372i	$0.340705\pi$
$558$	−7.99037	−0.338259
$559$	12.2427	0.517810
$560$	−3.42751	−0.144839
$561$	−1.92844	−0.0814190
$562$	−1.24607	−0.0525622
$563$	−27.6650	−1.16594	−0.582969	−	0.812494i	$-0.698109\pi$
$563$	−27.6650	−1.16594	−0.582969	+	0.812494i	$0.698109\pi$
$564$	10.5248	0.443173
$565$	−11.7675	−0.495064
$566$	−19.3796	−0.814584
$567$	−0.0928837	−0.00390075
$568$	−6.14324	−0.257765
$569$	−18.6083	−0.780100	−0.390050	−	0.920794i	$-0.627542\pi$
$569$	−18.6083	−0.780100	−0.390050	+	0.920794i	$0.627542\pi$
$570$	6.18646	0.259122
$571$	−13.0768	−0.547248	−0.273624	−	0.961837i	$-0.588222\pi$
$571$	−13.0768	−0.547248	−0.273624	+	0.961837i	$0.588222\pi$
$572$	−18.5364	−0.775045
$573$	−8.87431	−0.370729
$574$	−6.71350	−0.280216
$575$	−0.927554	−0.0386817
$576$	−1.84425	−0.0768438
$577$	28.7747	1.19791	0.598953	−	0.800784i	$-0.295584\pi$
$577$	28.7747	1.19791	0.598953	+	0.800784i	$0.295584\pi$
$578$	16.9198	0.703769
$579$	15.6131	0.648856
$580$	−2.43466	−0.101094
$581$	18.5400	0.769170
$582$	−6.37137	−0.264102
$583$	20.6052	0.853382
$584$	−4.63583	−0.191832
$585$	13.1432	0.543406
$586$	−12.0036	−0.495863
$587$	10.4943	0.433146	0.216573	−	0.976266i	$-0.430512\pi$
$587$	10.4943	0.433146	0.216573	+	0.976266i	$0.430512\pi$
$588$	5.39474	0.222475
$589$	10.2405	0.421951
$590$	19.0016	0.782282
$591$	0.949127	0.0390419
$592$	6.76879	0.278195
$593$	20.9092	0.858638	0.429319	−	0.903153i	$-0.358754\pi$
$593$	20.9092	0.858638	0.429319	+	0.903153i	$0.358754\pi$
$594$	32.9791	1.35315
$595$	0.970899	0.0398030
$596$	3.54881	0.145365
$597$	11.8146	0.483539
$598$	2.92714	0.119700
$599$	37.7233	1.54133	0.770667	−	0.637239i	$-0.219923\pi$
$599$	37.7233	1.54133	0.770667	+	0.637239i	$0.219923\pi$
$600$	0.997173	0.0407094
$601$	−0.826815	−0.0337265	−0.0168632	−	0.999858i	$-0.505368\pi$
$601$	−0.826815	−0.0337265	−0.0168632	+	0.999858i	$0.505368\pi$
$602$	−5.88807	−0.239980
$603$	−19.9310	−0.811652
$604$	−3.06643	−0.124771
$605$	−70.8523	−2.88055
$606$	9.12513	0.370683
$607$	−27.0782	−1.09907	−0.549536	−	0.835470i	$-0.685195\pi$
$607$	−27.0782	−1.09907	−0.549536	+	0.835470i	$0.685195\pi$
$608$	2.36360	0.0958565
$609$	−1.51347	−0.0613287
$610$	−4.47353	−0.181128
$611$	−28.6566	−1.15932
$612$	0.522415	0.0211174
$613$	−11.4398	−0.462049	−0.231025	−	0.972948i	$-0.574208\pi$
$613$	−11.4398	−0.462049	−0.231025	+	0.972948i	$0.574208\pi$
$614$	−10.2661	−0.414304
$615$	12.4818	0.503315
$616$	8.91501	0.359196
$617$	42.7486	1.72099	0.860496	−	0.509457i	$-0.170154\pi$
$617$	42.7486	1.72099	0.860496	+	0.509457i	$0.170154\pi$
$618$	6.95959	0.279956
$619$	25.7251	1.03398	0.516990	−	0.855992i	$-0.327053\pi$
$619$	25.7251	1.03398	0.516990	+	0.855992i	$0.327053\pi$
$620$	10.5483	0.423632
$621$	−5.20785	−0.208984
$622$	11.9636	0.479698
$623$	−12.4405	−0.498417
$624$	−3.14685	−0.125975
$625$	−28.7774	−1.15110
$626$	1.93119	0.0771860
$627$	−16.0911	−0.642616
$628$	14.8974	0.594470
$629$	−1.91737	−0.0764505
$630$	−6.32119	−0.251842
$631$	40.5079	1.61260	0.806298	−	0.591510i	$-0.201468\pi$
$631$	40.5079	1.61260	0.806298	+	0.591510i	$0.201468\pi$
$632$	−0.484928	−0.0192894
$633$	25.3419	1.00725
$634$	−3.24811	−0.128999
$635$	27.4876	1.09081
$636$	3.49807	0.138708
$637$	−14.6887	−0.581987
$638$	6.33258	0.250709
$639$	−11.3297	−0.448195
$640$	2.43466	0.0962383
$641$	41.4996	1.63913	0.819567	−	0.572983i	$-0.194214\pi$
$641$	41.4996	1.63913	0.819567	+	0.572983i	$0.194214\pi$
$642$	−1.77974	−0.0702406
$643$	46.5714	1.83660	0.918298	−	0.395890i	$-0.129564\pi$
$643$	46.5714	1.83660	0.918298	+	0.395890i	$0.129564\pi$
$644$	−1.40780	−0.0554751
$645$	10.9472	0.431044
$646$	−0.669527	−0.0263422
$647$	37.7841	1.48545	0.742723	−	0.669599i	$-0.233534\pi$
$647$	37.7841	1.48545	0.742723	+	0.669599i	$0.233534\pi$
$648$	0.0659779	0.00259186
$649$	−49.4233	−1.94004
$650$	−2.71508	−0.106494
$651$	6.55721	0.256997
$652$	−5.86556	−0.229713
$653$	−27.4299	−1.07341	−0.536707	−	0.843769i	$-0.680332\pi$
$653$	−27.4299	−1.07341	−0.536707	+	0.843769i	$0.680332\pi$
$654$	19.2204	0.751575
$655$	−4.54169	−0.177458
$656$	4.76879	0.186190
$657$	−8.54964	−0.333553
$658$	13.7823	0.537291
$659$	−20.7617	−0.808762	−0.404381	−	0.914591i	$-0.632513\pi$
$659$	−20.7617	−0.808762	−0.404381	+	0.914591i	$0.632513\pi$
$660$	−16.5749	−0.645176
$661$	24.3202	0.945945	0.472973	−	0.881077i	$-0.343181\pi$
$661$	24.3202	0.945945	0.472973	+	0.881077i	$0.343181\pi$
$662$	9.15255	0.355724
$663$	0.891396	0.0346190
$664$	−13.1695	−0.511076
$665$	8.10125	0.314153
$666$	12.4833	0.483720
$667$	−1.00000	−0.0387202
$668$	−9.85834	−0.381431
$669$	22.2942	0.861945
$670$	26.3115	1.01650
$671$	11.6357	0.449192
$672$	1.51347	0.0583832
$673$	2.76447	0.106562	0.0532812	−	0.998580i	$-0.483032\pi$
$673$	2.76447	0.106562	0.0532812	+	0.998580i	$0.483032\pi$
$674$	−19.8667	−0.765236
$675$	4.83056	0.185928
$676$	−4.43183	−0.170455
$677$	31.0095	1.19179	0.595895	−	0.803062i	$-0.296797\pi$
$677$	31.0095	1.19179	0.595895	+	0.803062i	$0.296797\pi$
$678$	5.19612	0.199556
$679$	−8.34339	−0.320190
$680$	−0.689656	−0.0264471
$681$	17.7154	0.678857
$682$	−27.4364	−1.05059
$683$	−42.0918	−1.61060	−0.805299	−	0.592869i	$-0.797995\pi$
$683$	−42.0918	−1.61060	−0.805299	+	0.592869i	$0.797995\pi$
$684$	4.35907	0.166673
$685$	−54.3444	−2.07639
$686$	16.9191	0.645974
$687$	−18.2138	−0.694901
$688$	4.18246	0.159455
$689$	−9.52448	−0.362854
$690$	2.61740	0.0996425
$691$	−17.3782	−0.661097	−0.330549	−	0.943789i	$-0.607234\pi$
$691$	−17.3782	−0.661097	−0.330549	+	0.943789i	$0.607234\pi$
$692$	−8.12015	−0.308682
$693$	16.4415	0.624562
$694$	25.1380	0.954227
$695$	0.835339	0.0316862
$696$	1.07506	0.0407499
$697$	−1.35084	−0.0511666
$698$	−31.2708	−1.18362
$699$	−25.3930	−0.960453
$700$	1.30581	0.0493550
$701$	−7.64688	−0.288819	−0.144409	−	0.989518i	$-0.546128\pi$
$701$	−7.64688	−0.288819	−0.144409	+	0.989518i	$0.546128\pi$
$702$	−15.2441	−0.575352
$703$	−15.9987	−0.603401
$704$	−6.33258	−0.238668
$705$	−25.6242	−0.965063
$706$	14.1949	0.534232
$707$	11.9495	0.449406
$708$	−8.39041	−0.315331
$709$	48.5670	1.82397	0.911985	−	0.410223i	$-0.134549\pi$
$709$	48.5670	1.82397	0.911985	+	0.410223i	$0.134549\pi$
$710$	14.9567	0.561314
$711$	−0.894329	−0.0335400
$712$	8.83682	0.331174
$713$	4.33258	0.162256
$714$	−0.428714	−0.0160442
$715$	45.1297	1.68775
$716$	11.4247	0.426960
$717$	−12.3884	−0.462652
$718$	20.0879	0.749674
$719$	19.1613	0.714596	0.357298	−	0.933990i	$-0.383698\pi$
$719$	19.1613	0.714596	0.357298	+	0.933990i	$0.383698\pi$
$720$	4.49012	0.167337
$721$	9.11367	0.339411
$722$	13.4134	0.499196
$723$	−14.3145	−0.532360
$724$	5.21923	0.193971
$725$	0.927554	0.0344485
$726$	31.2858	1.16113
$727$	24.3232	0.902096	0.451048	−	0.892500i	$-0.351050\pi$
$727$	24.3232	0.902096	0.451048	+	0.892500i	$0.351050\pi$
$728$	−4.12083	−0.152728
$729$	16.9179	0.626588
$730$	11.2867	0.417738
$731$	−1.18475	−0.0438196
$732$	1.97535	0.0730110
$733$	37.0336	1.36787	0.683934	−	0.729544i	$-0.260268\pi$
$733$	37.0336	1.36787	0.683934	+	0.729544i	$0.260268\pi$
$734$	−12.5724	−0.464054
$735$	−13.1343	−0.484468
$736$	1.00000	0.0368605
$737$	−68.4367	−2.52090
$738$	8.79484	0.323743
$739$	−45.5550	−1.67577	−0.837884	−	0.545848i	$-0.816207\pi$
$739$	−45.5550	−1.67577	−0.837884	+	0.545848i	$0.816207\pi$
$740$	−16.4797	−0.605805
$741$	7.43788	0.273237
$742$	4.58077	0.168165
$743$	−9.43716	−0.346216	−0.173108	−	0.984903i	$-0.555381\pi$
$743$	−9.43716	−0.346216	−0.173108	+	0.984903i	$0.555381\pi$
$744$	−4.65777	−0.170762
$745$	−8.64012	−0.316550
$746$	27.5359	1.00816
$747$	−24.2879	−0.888647
$748$	1.79381	0.0655881
$749$	−2.33059	−0.0851578
$750$	10.6592	0.389219
$751$	−2.72495	−0.0994347	−0.0497174	−	0.998763i	$-0.515832\pi$
$751$	−2.72495	−0.0994347	−0.0497174	+	0.998763i	$0.515832\pi$
$752$	−9.78996	−0.357003
$753$	−26.8060	−0.976865
$754$	−2.92714	−0.106600
$755$	7.46570	0.271705
$756$	7.33161	0.266648
$757$	3.66002	0.133026	0.0665128	−	0.997786i	$-0.478813\pi$
$757$	3.66002	0.133026	0.0665128	+	0.997786i	$0.478813\pi$
$758$	7.71967	0.280391
$759$	−6.80788	−0.247111
$760$	−5.75454	−0.208739
$761$	7.65528	0.277504	0.138752	−	0.990327i	$-0.455691\pi$
$761$	7.65528	0.277504	0.138752	+	0.990327i	$0.455691\pi$
$762$	−12.1375	−0.439697
$763$	25.1693	0.911189
$764$	8.25473	0.298646
$765$	−1.27190	−0.0459857
$766$	29.2577	1.05712
$767$	22.8452	0.824894
$768$	−1.07506	−0.0387928
$769$	−45.6003	−1.64439	−0.822194	−	0.569207i	$-0.807250\pi$
$769$	−45.6003	−1.64439	−0.822194	+	0.569207i	$0.807250\pi$
$770$	−21.7050	−0.782193
$771$	−11.1556	−0.401759
$772$	−14.5230	−0.522694
$773$	−45.3141	−1.62983	−0.814917	−	0.579578i	$-0.803217\pi$
$773$	−45.3141	−1.62983	−0.814917	+	0.579578i	$0.803217\pi$
$774$	7.71352	0.277257
$775$	−4.01870	−0.144356
$776$	5.92654	0.212750
$777$	−10.2443	−0.367513
$778$	22.5251	0.807565
$779$	−11.2715	−0.403843
$780$	7.66149	0.274325
$781$	−38.9025	−1.39204
$782$	−0.283266	−0.0101296
$783$	5.20785	0.186113
$784$	−5.01810	−0.179218
$785$	−36.2700	−1.29453
$786$	2.00545	0.0715319
$787$	−5.24677	−0.187027	−0.0935136	−	0.995618i	$-0.529810\pi$
$787$	−5.24677	−0.187027	−0.0935136	+	0.995618i	$0.529810\pi$
$788$	−0.882862	−0.0314506
$789$	24.7628	0.881581
$790$	1.18063	0.0420050
$791$	6.80439	0.241936
$792$	−11.6789	−0.414991
$793$	−5.37844	−0.190994
$794$	−2.21829	−0.0787243
$795$	−8.51660	−0.302053
$796$	−10.9897	−0.389521
$797$	−1.07622	−0.0381215	−0.0190608	−	0.999818i	$-0.506068\pi$
$797$	−1.07622	−0.0381215	−0.0190608	+	0.999818i	$0.506068\pi$
$798$	−3.57722	−0.126632
$799$	2.77317	0.0981076
$800$	−0.927554	−0.0327940
$801$	16.2973	0.575838
$802$	−22.5036	−0.794628
$803$	−29.3568	−1.03598
$804$	−11.6182	−0.409743
$805$	3.42751	0.120804
$806$	12.6821	0.446707
$807$	16.9658	0.597224
$808$	−8.48804	−0.298608
$809$	−10.4451	−0.367231	−0.183615	−	0.982998i	$-0.558780\pi$
$809$	−10.4451	−0.367231	−0.183615	+	0.982998i	$0.558780\pi$
$810$	−0.160633	−0.00564409
$811$	−19.8905	−0.698452	−0.349226	−	0.937039i	$-0.613555\pi$
$811$	−19.8905	−0.698452	−0.349226	+	0.937039i	$0.613555\pi$
$812$	1.40780	0.0494041
$813$	14.6082	0.512334
$814$	42.8639	1.50238
$815$	14.2806	0.500229
$816$	0.304528	0.0106606
$817$	−9.88565	−0.345855
$818$	−5.54865	−0.194004
$819$	−7.59986	−0.265561
$820$	−11.6104	−0.405451
$821$	14.7808	0.515854	0.257927	−	0.966164i	$-0.416961\pi$
$821$	14.7808	0.515854	0.257927	+	0.966164i	$0.416961\pi$
$822$	23.9965	0.836975
$823$	−10.2504	−0.357306	−0.178653	−	0.983912i	$-0.557174\pi$
$823$	−10.2504	−0.357306	−0.178653	+	0.983912i	$0.557174\pi$
$824$	−6.47369	−0.225522
$825$	6.31468	0.219849
$826$	−10.9873	−0.382299
$827$	−30.6601	−1.06616	−0.533079	−	0.846066i	$-0.678965\pi$
$827$	−30.6601	−1.06616	−0.533079	+	0.846066i	$0.678965\pi$
$828$	1.84425	0.0640922
$829$	−9.76163	−0.339036	−0.169518	−	0.985527i	$-0.554221\pi$
$829$	−9.76163	−0.339036	−0.169518	+	0.985527i	$0.554221\pi$
$830$	32.0632	1.11293
$831$	19.7144	0.683886
$832$	2.92714	0.101480
$833$	1.42146	0.0492506
$834$	−0.368856	−0.0127724
$835$	24.0017	0.830612
$836$	14.9677	0.517667
$837$	−22.5634	−0.779905
$838$	7.30749	0.252433
$839$	21.6571	0.747685	0.373842	−	0.927492i	$-0.378040\pi$
$839$	21.6571	0.747685	0.373842	+	0.927492i	$0.378040\pi$
$840$	−3.68477	−0.127137
$841$	1.00000	0.0344828
$842$	1.20079	0.0413821
$843$	−1.33959	−0.0461380
$844$	−23.5726	−0.811402
$845$	10.7900	0.371187
$846$	−18.0552	−0.620749
$847$	40.9692	1.40772
$848$	−3.25385	−0.111738
$849$	−20.8341	−0.715026
$850$	0.262745	0.00901207
$851$	−6.76879	−0.232031
$852$	−6.60433	−0.226261
$853$	13.1862	0.451488	0.225744	−	0.974187i	$-0.427519\pi$
$853$	13.1862	0.451488	0.225744	+	0.974187i	$0.427519\pi$
$854$	2.58674	0.0885166
$855$	−10.6128	−0.362951
$856$	1.65548	0.0565831
$857$	5.94065	0.202929	0.101464	−	0.994839i	$-0.467647\pi$
$857$	5.94065	0.202929	0.101464	+	0.994839i	$0.467647\pi$
$858$	−19.9277	−0.680319
$859$	51.8223	1.76815	0.884077	−	0.467341i	$-0.154788\pi$
$859$	51.8223	1.76815	0.884077	+	0.467341i	$0.154788\pi$
$860$	−10.1829	−0.347233
$861$	−7.21739	−0.245968
$862$	−25.3299	−0.862741
$863$	0.603904	0.0205571	0.0102786	−	0.999947i	$-0.496728\pi$
$863$	0.603904	0.0205571	0.0102786	+	0.999947i	$0.496728\pi$
$864$	−5.20785	−0.177175
$865$	19.7698	0.672193
$866$	7.48240	0.254262
$867$	18.1897	0.617755
$868$	−6.09941	−0.207027
$869$	−3.07084	−0.104171
$870$	−2.61740	−0.0887380
$871$	31.6339	1.07187
$872$	−17.8784	−0.605441
$873$	10.9300	0.369926
$874$	−2.36360	−0.0799498
$875$	13.9584	0.471879
$876$	−4.98378	−0.168386
$877$	42.1822	1.42439	0.712197	−	0.701980i	$-0.247700\pi$
$877$	42.1822	1.42439	0.712197	+	0.701980i	$0.247700\pi$
$878$	−12.8269	−0.432886
$879$	−12.9045	−0.435258
$880$	15.4177	0.519729
$881$	−21.4138	−0.721450	−0.360725	−	0.932672i	$-0.617471\pi$
$881$	−21.4138	−0.721450	−0.360725	+	0.932672i	$0.617471\pi$
$882$	−9.25464	−0.311620
$883$	−39.1418	−1.31723	−0.658614	−	0.752481i	$-0.728857\pi$
$883$	−39.1418	−1.31723	−0.658614	+	0.752481i	$0.728857\pi$
$884$	−0.829162	−0.0278877
$885$	20.4278	0.686672
$886$	10.8870	0.365756
$887$	1.35330	0.0454392	0.0227196	−	0.999742i	$-0.492768\pi$
$887$	1.35330	0.0454392	0.0227196	+	0.999742i	$0.492768\pi$
$888$	7.27683	0.244194
$889$	−15.8943	−0.533076
$890$	−21.5146	−0.721172
$891$	0.417810	0.0139972
$892$	−20.7377	−0.694351
$893$	23.1395	0.774335
$894$	3.81517	0.127598
$895$	−27.8152	−0.929759
$896$	−1.40780	−0.0470313
$897$	3.14685	0.105070
$898$	−24.0051	−0.801061
$899$	−4.33258	−0.144500
$900$	−1.71064	−0.0570214
$901$	0.921706	0.0307065
$902$	30.1987	1.00551
$903$	−6.33002	−0.210650
$904$	−4.83335	−0.160755
$905$	−12.7070	−0.422396
$906$	−3.29659	−0.109522
$907$	−6.61189	−0.219544	−0.109772	−	0.993957i	$-0.535012\pi$
$907$	−6.61189	−0.219544	−0.109772	+	0.993957i	$0.535012\pi$
$908$	−16.4786	−0.546861
$909$	−15.6541	−0.519213
$910$	10.0328	0.332585
$911$	49.4467	1.63824	0.819121	−	0.573621i	$-0.194462\pi$
$911$	49.4467	1.63824	0.819121	+	0.573621i	$0.194462\pi$
$912$	2.54100	0.0841409
$913$	−83.3969	−2.76003
$914$	−0.712242	−0.0235589
$915$	−4.80930	−0.158990
$916$	16.9422	0.559786
$917$	2.62616	0.0867233
$918$	1.47521	0.0486891
$919$	−27.7304	−0.914743	−0.457371	−	0.889276i	$-0.651209\pi$
$919$	−27.7304	−0.914743	−0.457371	+	0.889276i	$0.651209\pi$
$920$	−2.43466	−0.0802683
$921$	−11.0366	−0.363668
$922$	−20.7434	−0.683149
$923$	17.9821	0.591889
$924$	9.58414	0.315295
$925$	6.27841	0.206433
$926$	−27.6079	−0.907252
$927$	−11.9391	−0.392132
$928$	−1.00000	−0.0328266
$929$	28.5365	0.936251	0.468125	−	0.883662i	$-0.344930\pi$
$929$	28.5365	0.936251	0.468125	+	0.883662i	$0.344930\pi$
$930$	11.3401	0.371856
$931$	11.8608	0.388721
$932$	23.6202	0.773705
$933$	12.8616	0.421069
$934$	36.7343	1.20198
$935$	−4.36730	−0.142826
$936$	5.39839	0.176452
$937$	−39.3430	−1.28528	−0.642640	−	0.766169i	$-0.722161\pi$
$937$	−39.3430	−1.28528	−0.642640	+	0.766169i	$0.722161\pi$
$938$	−15.2142	−0.496762
$939$	2.07614	0.0677524
$940$	23.8352	0.777419
$941$	−26.3134	−0.857791	−0.428896	−	0.903354i	$-0.641097\pi$
$941$	−26.3134	−0.857791	−0.428896	+	0.903354i	$0.641097\pi$
$942$	16.0155	0.521814
$943$	−4.76879	−0.155293
$944$	7.80462	0.254019
$945$	−17.8500	−0.580659
$946$	26.4858	0.861127
$947$	3.64709	0.118515	0.0592573	−	0.998243i	$-0.481127\pi$
$947$	3.64709	0.118515	0.0592573	+	0.998243i	$0.481127\pi$
$948$	−0.521325	−0.0169318
$949$	13.5697	0.440492
$950$	2.19236	0.0711296
$951$	−3.49190	−0.113233
$952$	0.398783	0.0129246
$953$	−54.6502	−1.77029	−0.885147	−	0.465312i	$-0.845942\pi$
$953$	−54.6502	−1.77029	−0.885147	+	0.465312i	$0.845942\pi$
$954$	−6.00091	−0.194287
$955$	−20.0974	−0.650338
$956$	11.5234	0.372695
$957$	6.80788	0.220068
$958$	0.982415	0.0317404
$959$	31.4238	1.01473
$960$	2.61740	0.0844761
$961$	−12.2288	−0.394476
$962$	−19.8132	−0.638804
$963$	3.05312	0.0983855
$964$	13.3151	0.428849
$965$	35.3585	1.13823
$966$	−1.51347	−0.0486950
$967$	−14.5265	−0.467140	−0.233570	−	0.972340i	$-0.575041\pi$
$967$	−14.5265	−0.467140	−0.233570	+	0.972340i	$0.575041\pi$
$968$	−29.1015	−0.935359
$969$	−0.719780	−0.0231227
$970$	−14.4291	−0.463290
$971$	22.3218	0.716342	0.358171	−	0.933656i	$-0.383400\pi$
$971$	22.3218	0.716342	0.358171	+	0.933656i	$0.383400\pi$
$972$	−15.5526	−0.498850
$973$	−0.483021	−0.0154849
$974$	−26.6766	−0.854775
$975$	−2.91887	−0.0934786
$976$	−1.83744	−0.0588149
$977$	57.0816	1.82620	0.913102	−	0.407732i	$-0.133680\pi$
$977$	57.0816	1.82620	0.913102	+	0.407732i	$0.133680\pi$
$978$	−6.30581	−0.201638
$979$	55.9599	1.78849
$980$	12.2173	0.390269
$981$	−32.9724	−1.05273
$982$	−36.1524	−1.15367
$983$	−7.55209	−0.240874	−0.120437	−	0.992721i	$-0.538430\pi$
$983$	−7.55209	−0.240874	−0.120437	+	0.992721i	$0.538430\pi$
$984$	5.12672	0.163434
$985$	2.14947	0.0684877
$986$	0.283266	0.00902104
$987$	14.8168	0.471623
$988$	−6.91859	−0.220110
$989$	−4.18246	−0.132995
$990$	28.4340	0.903693
$991$	3.74352	0.118917	0.0594584	−	0.998231i	$-0.481063\pi$
$991$	3.74352	0.118917	0.0594584	+	0.998231i	$0.481063\pi$
$992$	4.33258	0.137559
$993$	9.83952	0.312248
$994$	−8.64845	−0.274312
$995$	26.7562	0.848230
$996$	−14.1580	−0.448612
$997$	14.7007	0.465575	0.232788	−	0.972528i	$-0.425215\pi$
$997$	14.7007	0.465575	0.232788	+	0.972528i	$0.425215\pi$
$998$	3.38486	0.107146
$999$	35.2508	1.11529

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1334.2.a.j.1.4	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1334.2.a.j.1.4	✓	9	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:	\( N \)	\(=\)	\( 1334 = 2 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1334.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.07506 q^{3} +1.00000 q^{4} -2.43466 q^{5} +1.07506 q^{6} +1.40780 q^{7} -1.00000 q^{8} -1.84425 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.07506 q^{3} +1.00000 q^{4} -2.43466 q^{5} +1.07506 q^{6} +1.40780 q^{7} -1.00000 q^{8} -1.84425 q^{9} +2.43466 q^{10} -6.33258 q^{11} -1.07506 q^{12} +2.92714 q^{13} -1.40780 q^{14} +2.61740 q^{15} +1.00000 q^{16} -0.283266 q^{17} +1.84425 q^{18} -2.36360 q^{19} -2.43466 q^{20} -1.51347 q^{21} +6.33258 q^{22} -1.00000 q^{23} +1.07506 q^{24} +0.927554 q^{25} -2.92714 q^{26} +5.20785 q^{27} +1.40780 q^{28} +1.00000 q^{29} -2.61740 q^{30} -4.33258 q^{31} -1.00000 q^{32} +6.80788 q^{33} +0.283266 q^{34} -3.42751 q^{35} -1.84425 q^{36} +6.76879 q^{37} +2.36360 q^{38} -3.14685 q^{39} +2.43466 q^{40} +4.76879 q^{41} +1.51347 q^{42} +4.18246 q^{43} -6.33258 q^{44} +4.49012 q^{45} +1.00000 q^{46} -9.78996 q^{47} -1.07506 q^{48} -5.01810 q^{49} -0.927554 q^{50} +0.304528 q^{51} +2.92714 q^{52} -3.25385 q^{53} -5.20785 q^{54} +15.4177 q^{55} -1.40780 q^{56} +2.54100 q^{57} -1.00000 q^{58} +7.80462 q^{59} +2.61740 q^{60} -1.83744 q^{61} +4.33258 q^{62} -2.59634 q^{63} +1.00000 q^{64} -7.12659 q^{65} -6.80788 q^{66} +10.8071 q^{67} -0.283266 q^{68} +1.07506 q^{69} +3.42751 q^{70} +6.14324 q^{71} +1.84425 q^{72} +4.63583 q^{73} -6.76879 q^{74} -0.997173 q^{75} -2.36360 q^{76} -8.91501 q^{77} +3.14685 q^{78} +0.484928 q^{79} -2.43466 q^{80} -0.0659779 q^{81} -4.76879 q^{82} +13.1695 q^{83} -1.51347 q^{84} +0.689656 q^{85} -4.18246 q^{86} -1.07506 q^{87} +6.33258 q^{88} -8.83682 q^{89} -4.49012 q^{90} +4.12083 q^{91} -1.00000 q^{92} +4.65777 q^{93} +9.78996 q^{94} +5.75454 q^{95} +1.07506 q^{96} -5.92654 q^{97} +5.01810 q^{98} +11.6789 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 9 q^{2} - 3 q^{3} + 9 q^{4} + 5 q^{5} + 3 q^{6} + 6 q^{7} - 9 q^{8} + 14 q^{9}+O(q^{10}) \) 9 * q - 9 * q^2 - 3 * q^3 + 9 * q^4 + 5 * q^5 + 3 * q^6 + 6 * q^7 - 9 * q^8 + 14 * q^9	\( 9 q - 9 q^{2} - 3 q^{3} + 9 q^{4} + 5 q^{5} + 3 q^{6} + 6 q^{7} - 9 q^{8} + 14 q^{9} - 5 q^{10} - 3 q^{11} - 3 q^{12} + 13 q^{13} - 6 q^{14} - 3 q^{15} + 9 q^{16} + 2 q^{17} - 14 q^{18} + 16 q^{19} + 5 q^{20} + 8 q^{21} + 3 q^{22} - 9 q^{23} + 3 q^{24} + 20 q^{25} - 13 q^{26} - 21 q^{27} + 6 q^{28} + 9 q^{29} + 3 q^{30} + 15 q^{31} - 9 q^{32} + 13 q^{33} - 2 q^{34} + 14 q^{36} + 12 q^{37} - 16 q^{38} - 5 q^{39} - 5 q^{40} - 6 q^{41} - 8 q^{42} - 3 q^{43} - 3 q^{44} + 20 q^{45} + 9 q^{46} - 19 q^{47} - 3 q^{48} + 37 q^{49} - 20 q^{50} + 6 q^{51} + 13 q^{52} + 5 q^{53} + 21 q^{54} + q^{55} - 6 q^{56} - 20 q^{57} - 9 q^{58} + 12 q^{59} - 3 q^{60} + 12 q^{61} - 15 q^{62} + 6 q^{63} + 9 q^{64} + 19 q^{65} - 13 q^{66} - 6 q^{67} + 2 q^{68} + 3 q^{69} + 12 q^{71} - 14 q^{72} - 12 q^{74} + 16 q^{75} + 16 q^{76} + 34 q^{77} + 5 q^{78} + 29 q^{79} + 5 q^{80} + 5 q^{81} + 6 q^{82} + 24 q^{83} + 8 q^{84} + 12 q^{85} + 3 q^{86} - 3 q^{87} + 3 q^{88} - 2 q^{89} - 20 q^{90} + 58 q^{91} - 9 q^{92} + 7 q^{93} + 19 q^{94} + 6 q^{95} + 3 q^{96} + 12 q^{97} - 37 q^{98} - 20 q^{99}+O(q^{100}) \) 9 * q - 9 * q^2 - 3 * q^3 + 9 * q^4 + 5 * q^5 + 3 * q^6 + 6 * q^7 - 9 * q^8 + 14 * q^9 - 5 * q^10 - 3 * q^11 - 3 * q^12 + 13 * q^13 - 6 * q^14 - 3 * q^15 + 9 * q^16 + 2 * q^17 - 14 * q^18 + 16 * q^19 + 5 * q^20 + 8 * q^21 + 3 * q^22 - 9 * q^23 + 3 * q^24 + 20 * q^25 - 13 * q^26 - 21 * q^27 + 6 * q^28 + 9 * q^29 + 3 * q^30 + 15 * q^31 - 9 * q^32 + 13 * q^33 - 2 * q^34 + 14 * q^36 + 12 * q^37 - 16 * q^38 - 5 * q^39 - 5 * q^40 - 6 * q^41 - 8 * q^42 - 3 * q^43 - 3 * q^44 + 20 * q^45 + 9 * q^46 - 19 * q^47 - 3 * q^48 + 37 * q^49 - 20 * q^50 + 6 * q^51 + 13 * q^52 + 5 * q^53 + 21 * q^54 + q^55 - 6 * q^56 - 20 * q^57 - 9 * q^58 + 12 * q^59 - 3 * q^60 + 12 * q^61 - 15 * q^62 + 6 * q^63 + 9 * q^64 + 19 * q^65 - 13 * q^66 - 6 * q^67 + 2 * q^68 + 3 * q^69 + 12 * q^71 - 14 * q^72 - 12 * q^74 + 16 * q^75 + 16 * q^76 + 34 * q^77 + 5 * q^78 + 29 * q^79 + 5 * q^80 + 5 * q^81 + 6 * q^82 + 24 * q^83 + 8 * q^84 + 12 * q^85 + 3 * q^86 - 3 * q^87 + 3 * q^88 - 2 * q^89 - 20 * q^90 + 58 * q^91 - 9 * q^92 + 7 * q^93 + 19 * q^94 + 6 * q^95 + 3 * q^96 + 12 * q^97 - 37 * q^98 - 20 * q^99

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