LMFDB - Embedded newform 431.2.a.e.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [431,2,Mod(1,431)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(431, 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("431.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 431 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	431.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:	$3.44155232712$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.725.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 3x^{2} + x + 1 $ x^4 - x^3 - 3*x^2 + x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.477260$ of defining polynomial
Character	$\chi$	$=$	431.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.477260 q^{2} +1.09529 q^{3} -1.77222 q^{4} -3.09529 q^{5} +0.522740 q^{6} +1.29496 q^{7} -1.80033 q^{8} -1.80033 q^{9} +O(q^{10})$	\(q+0.477260 q^{2} +1.09529 q^{3} -1.77222 q^{4} -3.09529 q^{5} +0.522740 q^{6} +1.29496 q^{7} -1.80033 q^{8} -1.80033 q^{9} -1.47726 q^{10} +1.43574 q^{11} -1.94111 q^{12} -5.00829 q^{13} +0.618034 q^{14} -3.39026 q^{15} +2.68522 q^{16} -2.22137 q^{17} -0.859226 q^{18} -4.20796 q^{19} +5.48555 q^{20} +1.41837 q^{21} +0.685220 q^{22} -2.82599 q^{23} -1.97189 q^{24} +4.58084 q^{25} -2.39026 q^{26} -5.25777 q^{27} -2.29496 q^{28} +1.72674 q^{29} -1.61803 q^{30} -8.09529 q^{31} +4.88221 q^{32} +1.57255 q^{33} -1.06017 q^{34} -4.00829 q^{35} +3.19059 q^{36} +8.45232 q^{37} -2.00829 q^{38} -5.48555 q^{39} +5.57255 q^{40} +6.45349 q^{41} +0.676929 q^{42} +0.162480 q^{43} -2.54445 q^{44} +5.57255 q^{45} -1.34873 q^{46} +9.22454 q^{47} +2.94111 q^{48} -5.32307 q^{49} +2.18625 q^{50} -2.43306 q^{51} +8.87581 q^{52} +11.8707 q^{53} -2.50933 q^{54} -4.44403 q^{55} -2.33136 q^{56} -4.60895 q^{57} +0.824105 q^{58} -0.744115 q^{59} +6.00829 q^{60} -0.916954 q^{61} -3.86356 q^{62} -2.33136 q^{63} -3.04036 q^{64} +15.5021 q^{65} +0.750517 q^{66} -5.18546 q^{67} +3.93677 q^{68} -3.09529 q^{69} -1.91300 q^{70} -2.10754 q^{71} +3.24119 q^{72} -11.7050 q^{73} +4.03395 q^{74} +5.01737 q^{75} +7.45744 q^{76} +1.85923 q^{77} -2.61803 q^{78} +10.2847 q^{79} -8.31154 q^{80} -0.357815 q^{81} +3.07999 q^{82} -13.7580 q^{83} -2.51366 q^{84} +6.87581 q^{85} +0.0775451 q^{86} +1.89129 q^{87} -2.58480 q^{88} -11.0806 q^{89} +2.65956 q^{90} -6.48555 q^{91} +5.00829 q^{92} -8.86673 q^{93} +4.40250 q^{94} +13.0249 q^{95} +5.34746 q^{96} -12.2098 q^{97} -2.54049 q^{98} -2.58480 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{2} - 3 q^{3} - q^{4} - 5 q^{5} + 5 q^{6} + 2 q^{7} - 3 q^{8} - 3 q^{9}+O(q^{10}) $ 4 * q - q^2 - 3 * q^3 - q^4 - 5 * q^5 + 5 * q^6 + 2 * q^7 - 3 * q^8 - 3 * q^9	\( 4 q - q^{2} - 3 q^{3} - q^{4} - 5 q^{5} + 5 q^{6} + 2 q^{7} - 3 q^{8} - 3 q^{9} - 3 q^{10} + q^{11} - 2 q^{12} - 5 q^{13} - 2 q^{14} - 3 q^{15} - 3 q^{16} + 2 q^{17} - 5 q^{18} - 6 q^{19} + 4 q^{20} - 3 q^{21} - 11 q^{22} + 4 q^{23} - 6 q^{24} - 7 q^{25} + q^{26} + 3 q^{27} - 6 q^{28} - 5 q^{29} - 2 q^{30} - 25 q^{31} + 8 q^{32} - 4 q^{33} - 12 q^{34} - q^{35} - 2 q^{36} - q^{37} + 7 q^{38} - 4 q^{39} + 12 q^{40} + 2 q^{41} + 4 q^{42} - 16 q^{43} + 2 q^{44} + 12 q^{45} + 7 q^{46} - 4 q^{47} + 6 q^{48} - 20 q^{49} + 13 q^{50} - 3 q^{51} + 7 q^{52} + 4 q^{53} - 13 q^{54} + 2 q^{55} + 7 q^{56} + 5 q^{57} + 20 q^{58} + 5 q^{59} + 9 q^{60} + 2 q^{62} + 7 q^{63} - 3 q^{64} + 14 q^{65} + 12 q^{66} + 9 q^{67} + 29 q^{68} - 5 q^{69} + 10 q^{71} + 19 q^{72} - 50 q^{73} - 10 q^{74} + 24 q^{75} + 10 q^{76} + 9 q^{77} - 6 q^{78} + 8 q^{79} - 8 q^{81} + 37 q^{82} - 15 q^{83} + 6 q^{84} - q^{85} + 12 q^{86} + 15 q^{87} + 11 q^{88} - 35 q^{89} + 8 q^{90} - 8 q^{91} + 5 q^{92} + 12 q^{93} - 4 q^{94} + 7 q^{95} + 7 q^{96} - 6 q^{97} + 6 q^{98} + 11 q^{99}+O(q^{100}) \) 4 * q - q^2 - 3 * q^3 - q^4 - 5 * q^5 + 5 * q^6 + 2 * q^7 - 3 * q^8 - 3 * q^9 - 3 * q^10 + q^11 - 2 * q^12 - 5 * q^13 - 2 * q^14 - 3 * q^15 - 3 * q^16 + 2 * q^17 - 5 * q^18 - 6 * q^19 + 4 * q^20 - 3 * q^21 - 11 * q^22 + 4 * q^23 - 6 * q^24 - 7 * q^25 + q^26 + 3 * q^27 - 6 * q^28 - 5 * q^29 - 2 * q^30 - 25 * q^31 + 8 * q^32 - 4 * q^33 - 12 * q^34 - q^35 - 2 * q^36 - q^37 + 7 * q^38 - 4 * q^39 + 12 * q^40 + 2 * q^41 + 4 * q^42 - 16 * q^43 + 2 * q^44 + 12 * q^45 + 7 * q^46 - 4 * q^47 + 6 * q^48 - 20 * q^49 + 13 * q^50 - 3 * q^51 + 7 * q^52 + 4 * q^53 - 13 * q^54 + 2 * q^55 + 7 * q^56 + 5 * q^57 + 20 * q^58 + 5 * q^59 + 9 * q^60 + 2 * q^62 + 7 * q^63 - 3 * q^64 + 14 * q^65 + 12 * q^66 + 9 * q^67 + 29 * q^68 - 5 * q^69 + 10 * q^71 + 19 * q^72 - 50 * q^73 - 10 * q^74 + 24 * q^75 + 10 * q^76 + 9 * q^77 - 6 * q^78 + 8 * q^79 - 8 * q^81 + 37 * q^82 - 15 * q^83 + 6 * q^84 - q^85 + 12 * q^86 + 15 * q^87 + 11 * q^88 - 35 * q^89 + 8 * q^90 - 8 * q^91 + 5 * q^92 + 12 * q^93 - 4 * q^94 + 7 * q^95 + 7 * q^96 - 6 * q^97 + 6 * q^98 + 11 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0.477260	0.337474	0.168737	−	0.985661i	$-0.446031\pi$
$2$	0.477260	0.337474	0.168737	+	0.985661i	$0.446031\pi$
$3$	1.09529	0.632368	0.316184	−	0.948698i	$-0.397598\pi$
$3$	1.09529	0.632368	0.316184	+	0.948698i	$0.397598\pi$
$4$	−1.77222	−0.886111
$5$	−3.09529	−1.38426	−0.692129	−	0.721774i	$-0.743327\pi$
$5$	−3.09529	−1.38426	−0.692129	+	0.721774i	$0.743327\pi$
$6$	0.522740	0.213408
$7$	1.29496	0.489450	0.244725	−	0.969593i	$-0.421302\pi$
$7$	1.29496	0.489450	0.244725	+	0.969593i	$0.421302\pi$
$8$	−1.80033	−0.636513
$9$	−1.80033	−0.600110
$10$	−1.47726	−0.467151
$11$	1.43574	0.432891	0.216445	−	0.976295i	$-0.430554\pi$
$11$	1.43574	0.432891	0.216445	+	0.976295i	$0.430554\pi$
$12$	−1.94111	−0.560349
$13$	−5.00829	−1.38905	−0.694525	−	0.719469i	$-0.744385\pi$
$13$	−5.00829	−1.38905	−0.694525	+	0.719469i	$0.744385\pi$
$14$	0.618034	0.165177
$15$	−3.39026	−0.875361
$16$	2.68522	0.671305
$17$	−2.22137	−0.538763	−0.269381	−	0.963034i	$-0.586819\pi$
$17$	−2.22137	−0.538763	−0.269381	+	0.963034i	$0.586819\pi$
$18$	−0.859226	−0.202522
$19$	−4.20796	−0.965372	−0.482686	−	0.875793i	$-0.660339\pi$
$19$	−4.20796	−0.965372	−0.482686	+	0.875793i	$0.660339\pi$
$20$	5.48555	1.22661
$21$	1.41837	0.309513
$22$	0.685220	0.146089
$23$	−2.82599	−0.589260	−0.294630	−	0.955611i	$-0.595196\pi$
$23$	−2.82599	−0.589260	−0.294630	+	0.955611i	$0.595196\pi$
$24$	−1.97189	−0.402511
$25$	4.58084	0.916169
$26$	−2.39026	−0.468768
$27$	−5.25777	−1.01186
$28$	−2.29496	−0.433707
$29$	1.72674	0.320648	0.160324	−	0.987064i	$-0.448746\pi$
$29$	1.72674	0.320648	0.160324	+	0.987064i	$0.448746\pi$
$30$	−1.61803	−0.295411
$31$	−8.09529	−1.45396	−0.726979	−	0.686660i	$-0.759076\pi$
$31$	−8.09529	−1.45396	−0.726979	+	0.686660i	$0.759076\pi$
$32$	4.88221	0.863061
$33$	1.57255	0.273747
$34$	−1.06017	−0.181818
$35$	−4.00829	−0.677525
$36$	3.19059	0.531765
$37$	8.45232	1.38955	0.694776	−	0.719226i	$-0.255503\pi$
$37$	8.45232	1.38955	0.694776	+	0.719226i	$0.255503\pi$
$38$	−2.00829	−0.325788
$39$	−5.48555	−0.878391
$40$	5.57255	0.881098
$41$	6.45349	1.00787	0.503933	−	0.863743i	$-0.331886\pi$
$41$	6.45349	1.00787	0.503933	+	0.863743i	$0.331886\pi$
$42$	0.676929	0.104452
$43$	0.162480	0.0247779	0.0123890	−	0.999923i	$-0.496056\pi$
$43$	0.162480	0.0247779	0.0123890	+	0.999923i	$0.496056\pi$
$44$	−2.54445	−0.383590
$45$	5.57255	0.830707
$46$	−1.34873	−0.198860
$47$	9.22454	1.34554	0.672769	−	0.739853i	$-0.265105\pi$
$47$	9.22454	1.34554	0.672769	+	0.739853i	$0.265105\pi$
$48$	2.94111	0.424512
$49$	−5.32307	−0.760439
$50$	2.18625	0.309183
$51$	−2.43306	−0.340696
$52$	8.87581	1.23085
$53$	11.8707	1.63056	0.815282	−	0.579064i	$-0.196582\pi$
$53$	11.8707	1.63056	0.815282	+	0.579064i	$0.196582\pi$
$54$	−2.50933	−0.341476
$55$	−4.44403	−0.599233
$56$	−2.33136	−0.311541
$57$	−4.60895	−0.610471
$58$	0.824105	0.108210
$59$	−0.744115	−0.0968755	−0.0484377	−	0.998826i	$-0.515424\pi$
$59$	−0.744115	−0.0968755	−0.0484377	+	0.998826i	$0.515424\pi$
$60$	6.00829	0.775667
$61$	−0.916954	−0.117404	−0.0587020	−	0.998276i	$-0.518696\pi$
$61$	−0.916954	−0.117404	−0.0587020	+	0.998276i	$0.518696\pi$
$62$	−3.86356	−0.490673
$63$	−2.33136	−0.293724
$64$	−3.04036	−0.380044
$65$	15.5021	1.92280
$66$	0.750517	0.0923823
$67$	−5.18546	−0.633505	−0.316753	−	0.948508i	$-0.602593\pi$
$67$	−5.18546	−0.633505	−0.316753	+	0.948508i	$0.602593\pi$
$68$	3.93677	0.477404
$69$	−3.09529	−0.372630
$70$	−1.91300	−0.228647
$71$	−2.10754	−0.250119	−0.125060	−	0.992149i	$-0.539912\pi$
$71$	−2.10754	−0.250119	−0.125060	+	0.992149i	$0.539912\pi$
$72$	3.24119	0.381978
$73$	−11.7050	−1.36997	−0.684985	−	0.728557i	$-0.740191\pi$
$73$	−11.7050	−1.36997	−0.684985	+	0.728557i	$0.740191\pi$
$74$	4.03395	0.468938
$75$	5.01737	0.579356
$76$	7.45744	0.855427
$77$	1.85923	0.211878
$78$	−2.61803	−0.296434
$79$	10.2847	1.15712	0.578560	−	0.815640i	$-0.303615\pi$
$79$	10.2847	1.15712	0.578560	+	0.815640i	$0.303615\pi$
$80$	−8.31154	−0.929259
$81$	−0.357815	−0.0397572
$82$	3.07999	0.340128
$83$	−13.7580	−1.51014	−0.755069	−	0.655645i	$-0.772397\pi$
$83$	−13.7580	−1.51014	−0.755069	+	0.655645i	$0.772397\pi$
$84$	−2.51366	−0.274263
$85$	6.87581	0.745786
$86$	0.0775451	0.00836190
$87$	1.89129	0.202768
$88$	−2.58480	−0.275541
$89$	−11.0806	−1.17454	−0.587271	−	0.809391i	$-0.699798\pi$
$89$	−11.0806	−1.17454	−0.587271	+	0.809391i	$0.699798\pi$
$90$	2.65956	0.280342
$91$	−6.48555	−0.679870
$92$	5.00829	0.522150
$93$	−8.86673	−0.919437
$94$	4.40250	0.454084
$95$	13.0249	1.33632
$96$	5.34746	0.545772
$97$	−12.2098	−1.23972	−0.619861	−	0.784712i	$-0.712811\pi$
$97$	−12.2098	−1.23972	−0.619861	+	0.784712i	$0.712811\pi$
$98$	−2.54049	−0.256628
$99$	−2.58480	−0.259782
$100$	−8.11828	−0.811828
$101$	2.77618	0.276240	0.138120	−	0.990415i	$-0.455894\pi$
$101$	2.77618	0.276240	0.138120	+	0.990415i	$0.455894\pi$
$102$	−1.16120	−0.114976
$103$	−0.870195	−0.0857429	−0.0428715	−	0.999081i	$-0.513651\pi$
$103$	−0.870195	−0.0857429	−0.0428715	+	0.999081i	$0.513651\pi$
$104$	9.01658	0.884149
$105$	−4.39026	−0.428445
$106$	5.66540	0.550273
$107$	−7.28076	−0.703857	−0.351929	−	0.936027i	$-0.614474\pi$
$107$	−7.28076	−0.703857	−0.351929	+	0.936027i	$0.614474\pi$
$108$	9.31795	0.896620
$109$	3.88538	0.372152	0.186076	−	0.982535i	$-0.440423\pi$
$109$	3.88538	0.372152	0.186076	+	0.982535i	$0.440423\pi$
$110$	−2.12096	−0.202225
$111$	9.25777	0.878709
$112$	3.47726	0.328570
$113$	−16.0222	−1.50724	−0.753621	−	0.657310i	$-0.771694\pi$
$113$	−16.0222	−1.50724	−0.753621	+	0.657310i	$0.771694\pi$
$114$	−2.19967	−0.206018
$115$	8.74728	0.815688
$116$	−3.06017	−0.284130
$117$	9.01658	0.833583
$118$	−0.355136	−0.0326929
$119$	−2.87660	−0.263697
$120$	6.10358	0.557179
$121$	−8.93866	−0.812605
$122$	−0.437625	−0.0396207
$123$	7.06846	0.637342
$124$	14.3467	1.28837
$125$	1.29741	0.116044
$126$	−1.11267	−0.0991241
$127$	−11.3702	−1.00894	−0.504471	−	0.863428i	$-0.668313\pi$
$127$	−11.3702	−1.00894	−0.504471	+	0.863428i	$0.668313\pi$
$128$	−11.2155	−0.991316
$129$	0.177963	0.0156688
$130$	7.39855	0.648896
$131$	16.4139	1.43408	0.717042	−	0.697030i	$-0.245495\pi$
$131$	16.4139	1.43408	0.717042	+	0.697030i	$0.245495\pi$
$132$	−2.78692	−0.242570
$133$	−5.44915	−0.472501
$134$	−2.47481	−0.213791
$135$	16.2744	1.40067
$136$	3.99921	0.342929
$137$	14.6590	1.25240	0.626201	−	0.779661i	$-0.284609\pi$
$137$	14.6590	1.25240	0.626201	+	0.779661i	$0.284609\pi$
$138$	−1.47726	−0.125753
$139$	11.0115	0.933980	0.466990	−	0.884263i	$-0.345338\pi$
$139$	11.0115	0.933980	0.466990	+	0.884263i	$0.345338\pi$
$140$	7.10358	0.600362
$141$	10.1036	0.850875
$142$	−1.00585	−0.0844087
$143$	−7.19059	−0.601307
$144$	−4.83428	−0.402857
$145$	−5.34478	−0.443860
$146$	−5.58635	−0.462329
$147$	−5.83033	−0.480877
$148$	−14.9794	−1.23130
$149$	−8.68955	−0.711876	−0.355938	−	0.934510i	$-0.615839\pi$
$149$	−8.68955	−0.711876	−0.355938	+	0.934510i	$0.615839\pi$
$150$	2.39459	0.195518
$151$	10.7284	0.873065	0.436532	−	0.899689i	$-0.356206\pi$
$151$	10.7284	0.873065	0.436532	+	0.899689i	$0.356206\pi$
$152$	7.57572	0.614472
$153$	3.99921	0.323317
$154$	0.887334	0.0715034
$155$	25.0573	2.01265
$156$	9.72162	0.778352
$157$	−13.2432	−1.05692	−0.528461	−	0.848958i	$-0.677231\pi$
$157$	−13.2432	−1.05692	−0.528461	+	0.848958i	$0.677231\pi$
$158$	4.90848	0.390498
$159$	13.0019	1.03112
$160$	−15.1119	−1.19470
$161$	−3.65956	−0.288414
$162$	−0.170771	−0.0134170
$163$	9.28550	0.727297	0.363648	−	0.931536i	$-0.381531\pi$
$163$	9.28550	0.727297	0.363648	+	0.931536i	$0.381531\pi$
$164$	−11.4370	−0.893081
$165$	−4.86752	−0.378936
$166$	−6.56615	−0.509632
$167$	−12.4187	−0.960991	−0.480496	−	0.876997i	$-0.659543\pi$
$167$	−12.4187	−0.960991	−0.480496	+	0.876997i	$0.659543\pi$
$168$	−2.55353	−0.197009
$169$	12.0830	0.929460
$170$	3.28155	0.251683
$171$	7.57572	0.579330
$172$	−0.287950	−0.0219560
$173$	12.4029	0.942974	0.471487	−	0.881873i	$-0.343717\pi$
$173$	12.4029	0.942974	0.471487	+	0.881873i	$0.343717\pi$
$174$	0.902638	0.0684288
$175$	5.93202	0.448419
$176$	3.85527	0.290602
$177$	−0.815024	−0.0612610
$178$	−5.28833	−0.396377
$179$	−14.4997	−1.08376	−0.541879	−	0.840457i	$-0.682287\pi$
$179$	−14.4997	−1.08376	−0.541879	+	0.840457i	$0.682287\pi$
$180$	−9.87581	−0.736099
$181$	−13.2712	−0.986440	−0.493220	−	0.869905i	$-0.664180\pi$
$181$	−13.2712	−0.986440	−0.493220	+	0.869905i	$0.664180\pi$
$182$	−3.09529	−0.229438
$183$	−1.00433	−0.0742425
$184$	5.08772	0.375072
$185$	−26.1624	−1.92350
$186$	−4.23173	−0.310286
$187$	−3.18931	−0.233225
$188$	−16.3479	−1.19230
$189$	−6.80862	−0.495254
$190$	6.21625	0.450974
$191$	−22.4400	−1.62370	−0.811851	−	0.583865i	$-0.801540\pi$
$191$	−22.4400	−1.62370	−0.811851	+	0.583865i	$0.801540\pi$
$192$	−3.33008	−0.240328
$193$	0.475982	0.0342619	0.0171310	−	0.999853i	$-0.494547\pi$
$193$	0.475982	0.0342619	0.0171310	+	0.999853i	$0.494547\pi$
$194$	−5.82727	−0.418374
$195$	16.9794	1.21592
$196$	9.43367	0.673833
$197$	15.4029	1.09741	0.548705	−	0.836016i	$-0.315121\pi$
$197$	15.4029	1.09741	0.548705	+	0.836016i	$0.315121\pi$
$198$	−1.23362	−0.0876697
$199$	−6.74784	−0.478342	−0.239171	−	0.970978i	$-0.576876\pi$
$199$	−6.74784	−0.478342	−0.239171	+	0.970978i	$0.576876\pi$
$200$	−8.24704	−0.583154
$201$	−5.67961	−0.400609
$202$	1.32496	0.0932238
$203$	2.23607	0.156941
$204$	4.31192	0.301895
$205$	−19.9754	−1.39514
$206$	−0.415309	−0.0289360
$207$	5.08772	0.353621
$208$	−13.4484	−0.932476
$209$	−6.04152	−0.417901
$210$	−2.09529	−0.144589
$211$	−27.6453	−1.90318	−0.951590	−	0.307371i	$-0.900551\pi$
$211$	−27.6453	−1.90318	−0.951590	+	0.307371i	$0.900551\pi$
$212$	−21.0375	−1.44486
$213$	−2.30838	−0.158167
$214$	−3.47481	−0.237533
$215$	−0.502923	−0.0342990
$216$	9.46573	0.644062
$217$	−10.4831	−0.711640
$218$	1.85434	0.125591
$219$	−12.8205	−0.866326
$220$	7.87581	0.530987
$221$	11.1253	0.748368
$222$	4.41837	0.296541
$223$	−5.04059	−0.337543	−0.168771	−	0.985655i	$-0.553980\pi$
$223$	−5.04059	−0.337543	−0.168771	+	0.985655i	$0.553980\pi$
$224$	6.32228	0.422425
$225$	−8.24704	−0.549802
$226$	−7.64675	−0.508654
$227$	13.6801	0.907980	0.453990	−	0.891007i	$-0.350000\pi$
$227$	13.6801	0.907980	0.453990	+	0.891007i	$0.350000\pi$
$228$	8.16809	0.540945
$229$	5.38970	0.356161	0.178081	−	0.984016i	$-0.443011\pi$
$229$	5.38970	0.356161	0.178081	+	0.984016i	$0.443011\pi$
$230$	4.17473	0.275273
$231$	2.03640	0.133985
$232$	−3.10871	−0.204097
$233$	−3.98263	−0.260911	−0.130455	−	0.991454i	$-0.541644\pi$
$233$	−3.98263	−0.260911	−0.130455	+	0.991454i	$0.541644\pi$
$234$	4.30325	0.281312
$235$	−28.5527	−1.86257
$236$	1.31874	0.0858425
$237$	11.2648	0.731726
$238$	−1.37289	−0.0889909
$239$	8.88782	0.574905	0.287453	−	0.957795i	$-0.407192\pi$
$239$	8.88782	0.574905	0.287453	+	0.957795i	$0.407192\pi$
$240$	−9.10358	−0.587634
$241$	−10.3727	−0.668161	−0.334081	−	0.942545i	$-0.608426\pi$
$241$	−10.3727	−0.668161	−0.334081	+	0.942545i	$0.608426\pi$
$242$	−4.26606	−0.274233
$243$	15.3814	0.986718
$244$	1.62505	0.104033
$245$	16.4765	1.05264
$246$	3.37350	0.215086
$247$	21.0747	1.34095
$248$	14.5742	0.925463
$249$	−15.0691	−0.954964
$250$	0.619201	0.0391617
$251$	9.91221	0.625653	0.312826	−	0.949810i	$-0.398724\pi$
$251$	9.91221	0.625653	0.312826	+	0.949810i	$0.398724\pi$
$252$	4.13169	0.260272
$253$	−4.05738	−0.255086
$254$	−5.42654	−0.340492
$255$	7.53103	0.471611
$256$	0.728021	0.0455013
$257$	25.5148	1.59157	0.795783	−	0.605581i	$-0.207059\pi$
$257$	25.5148	1.59157	0.795783	+	0.605581i	$0.207059\pi$
$258$	0.0849347	0.00528780
$259$	10.9454	0.680116
$260$	−27.4732	−1.70382
$261$	−3.10871	−0.192424
$262$	7.83367	0.483966
$263$	−3.83885	−0.236714	−0.118357	−	0.992971i	$-0.537763\pi$
$263$	−3.83885	−0.236714	−0.118357	+	0.992971i	$0.537763\pi$
$264$	−2.83112	−0.174243
$265$	−36.7433	−2.25712
$266$	−2.60066	−0.159457
$267$	−12.1365	−0.742743
$268$	9.18980	0.561356
$269$	31.8861	1.94413	0.972065	−	0.234712i	$-0.0754148\pi$
$269$	31.8861	1.94413	0.972065	+	0.234712i	$0.0754148\pi$
$270$	7.76710	0.472691
$271$	−0.486829	−0.0295728	−0.0147864	−	0.999891i	$-0.504707\pi$
$271$	−0.486829	−0.0295728	−0.0147864	+	0.999891i	$0.504707\pi$
$272$	−5.96488	−0.361674
$273$	−7.10358	−0.429929
$274$	6.99615	0.422653
$275$	6.57689	0.396601
$276$	5.48555	0.330191
$277$	−22.8207	−1.37116	−0.685581	−	0.727996i	$-0.740452\pi$
$277$	−22.8207	−1.37116	−0.685581	+	0.727996i	$0.740452\pi$
$278$	5.25533	0.315194
$279$	14.5742	0.872535
$280$	7.21625	0.431253
$281$	−8.37635	−0.499691	−0.249846	−	0.968286i	$-0.580380\pi$
$281$	−8.37635	−0.499691	−0.249846	+	0.968286i	$0.580380\pi$
$282$	4.82204	0.287148
$283$	−22.2252	−1.32115	−0.660574	−	0.750761i	$-0.729687\pi$
$283$	−22.2252	−1.32115	−0.660574	+	0.750761i	$0.729687\pi$
$284$	3.73503	0.221633
$285$	14.2661	0.845049
$286$	−3.43178	−0.202925
$287$	8.35702	0.493300
$288$	−8.78959	−0.517932
$289$	−12.0655	−0.709735
$290$	−2.55085	−0.149791
$291$	−13.3734	−0.783961
$292$	20.7439	1.21395
$293$	−22.4034	−1.30882	−0.654409	−	0.756140i	$-0.727083\pi$
$293$	−22.4034	−1.30882	−0.654409	+	0.756140i	$0.727083\pi$
$294$	−2.78258	−0.162283
$295$	2.30325	0.134101
$296$	−15.2170	−0.884469
$297$	−7.54878	−0.438025
$298$	−4.14718	−0.240239
$299$	14.1534	0.818512
$300$	−8.89190	−0.513374
$301$	0.210405	0.0121276
$302$	5.12024	0.294636
$303$	3.04073	0.174686
$304$	−11.2993	−0.648059
$305$	2.83824	0.162517
$306$	1.90866	0.109111
$307$	−12.1360	−0.692640	−0.346320	−	0.938117i	$-0.612569\pi$
$307$	−12.1360	−0.692640	−0.346320	+	0.938117i	$0.612569\pi$
$308$	−3.29496	−0.187748
$309$	−0.953120	−0.0542211
$310$	11.9589	0.679217
$311$	0.106496	0.00603886	0.00301943	−	0.999995i	$-0.499039\pi$
$311$	0.106496	0.00603886	0.00301943	+	0.999995i	$0.499039\pi$
$312$	9.87581	0.559108
$313$	13.0209	0.735986	0.367993	−	0.929829i	$-0.380045\pi$
$313$	13.0209	0.735986	0.367993	+	0.929829i	$0.380045\pi$
$314$	−6.32045	−0.356683
$315$	7.21625	0.406590
$316$	−18.2268	−1.02534
$317$	4.19918	0.235849	0.117925	−	0.993023i	$-0.462376\pi$
$317$	4.19918	0.235849	0.117925	+	0.993023i	$0.462376\pi$
$318$	6.20528	0.347975
$319$	2.47915	0.138806
$320$	9.41080	0.526079
$321$	−7.97457	−0.445097
$322$	−1.74656	−0.0973320
$323$	9.34746	0.520106
$324$	0.634127	0.0352293
$325$	−22.9422	−1.27260
$326$	4.43160	0.245444
$327$	4.25563	0.235337
$328$	−11.6184	−0.641519
$329$	11.9454	0.658573
$330$	−2.32307	−0.127881
$331$	32.1475	1.76699	0.883493	−	0.468445i	$-0.155186\pi$
$331$	32.1475	1.76699	0.883493	+	0.468445i	$0.155186\pi$
$332$	24.3823	1.33815
$333$	−15.2170	−0.833885
$334$	−5.92697	−0.324309
$335$	16.0505	0.876934
$336$	3.80862	0.207777
$337$	30.0178	1.63517	0.817587	−	0.575806i	$-0.195311\pi$
$337$	30.0178	1.63517	0.817587	+	0.575806i	$0.195311\pi$
$338$	5.76672	0.313668
$339$	−17.5490	−0.953132
$340$	−12.1855	−0.660850
$341$	−11.6227	−0.629405
$342$	3.61559	0.195509
$343$	−15.9579	−0.861647
$344$	−0.292517	−0.0157715
$345$	9.58084	0.515815
$346$	5.91940	0.318229
$347$	14.3903	0.772509	0.386255	−	0.922392i	$-0.373769\pi$
$347$	14.3903	0.772509	0.386255	+	0.922392i	$0.373769\pi$
$348$	−3.35179	−0.179675
$349$	−22.0811	−1.18197	−0.590987	−	0.806681i	$-0.701261\pi$
$349$	−22.0811	−1.18197	−0.590987	+	0.806681i	$0.701261\pi$
$350$	2.83112	0.151330
$351$	26.3325	1.40552
$352$	7.00957	0.373611
$353$	1.27270	0.0677390	0.0338695	−	0.999426i	$-0.489217\pi$
$353$	1.27270	0.0677390	0.0338695	+	0.999426i	$0.489217\pi$
$354$	−0.388979	−0.0206740
$355$	6.52346	0.346229
$356$	19.6373	1.04077
$357$	−3.15072	−0.166754
$358$	−6.92012	−0.365740
$359$	−1.26953	−0.0670034	−0.0335017	−	0.999439i	$-0.510666\pi$
$359$	−1.26953	−0.0670034	−0.0335017	+	0.999439i	$0.510666\pi$
$360$	−10.0324	−0.528756
$361$	−1.29307	−0.0680565
$362$	−6.33381	−0.332898
$363$	−9.79046	−0.513866
$364$	11.4938	0.602441
$365$	36.2305	1.89639
$366$	−0.479328	−0.0250549
$367$	−14.6622	−0.765359	−0.382679	−	0.923881i	$-0.624999\pi$
$367$	−14.6622	−0.765359	−0.382679	+	0.923881i	$0.624999\pi$
$368$	−7.58841	−0.395573
$369$	−11.6184	−0.604830
$370$	−12.4863	−0.649130
$371$	15.3721	0.798079
$372$	15.7138	0.814723
$373$	23.0283	1.19236	0.596181	−	0.802850i	$-0.296684\pi$
$373$	23.0283	1.19236	0.596181	+	0.802850i	$0.296684\pi$
$374$	−1.52213	−0.0787075
$375$	1.42104	0.0733824
$376$	−16.6072	−0.856452
$377$	−8.64803	−0.445396
$378$	−3.24948	−0.167135
$379$	−1.56207	−0.0802383	−0.0401191	−	0.999195i	$-0.512774\pi$
$379$	−1.56207	−0.0802383	−0.0401191	+	0.999195i	$0.512774\pi$
$380$	−23.0830	−1.18413
$381$	−12.4537	−0.638023
$382$	−10.7097	−0.547957
$383$	4.91416	0.251102	0.125551	−	0.992087i	$-0.459930\pi$
$383$	4.91416	0.251102	0.125551	+	0.992087i	$0.459930\pi$
$384$	−12.2842	−0.626877
$385$	−5.75485	−0.293294
$386$	0.227167	0.0115625
$387$	−0.292517	−0.0148695
$388$	21.6386	1.09853
$389$	−27.7001	−1.40445	−0.702225	−	0.711955i	$-0.747810\pi$
$389$	−27.7001	−1.40445	−0.702225	+	0.711955i	$0.747810\pi$
$390$	8.10358	0.410341
$391$	6.27759	0.317471
$392$	9.58329	0.484029
$393$	17.9780	0.906870
$394$	7.35118	0.370347
$395$	−31.8342	−1.60175
$396$	4.58084	0.230196
$397$	29.6372	1.48745	0.743725	−	0.668486i	$-0.233057\pi$
$397$	29.6372	1.48745	0.743725	+	0.668486i	$0.233057\pi$
$398$	−3.22047	−0.161428
$399$	−5.96842	−0.298795
$400$	12.3006	0.615029
$401$	5.07065	0.253216	0.126608	−	0.991953i	$-0.459591\pi$
$401$	5.07065	0.253216	0.126608	+	0.991953i	$0.459591\pi$
$402$	−2.71065	−0.135195
$403$	40.5436	2.01962
$404$	−4.92001	−0.244780
$405$	1.10754	0.0550342
$406$	1.06719	0.0529635
$407$	12.1353	0.601525
$408$	4.38031	0.216858
$409$	−33.9070	−1.67660	−0.838298	−	0.545213i	$-0.816449\pi$
$409$	−33.9070	−1.67660	−0.838298	+	0.545213i	$0.816449\pi$
$410$	−9.53348	−0.470825
$411$	16.0559	0.791980
$412$	1.54218	0.0759778
$413$	−0.963601	−0.0474157
$414$	2.42817	0.119338
$415$	42.5851	2.09042
$416$	−24.4515	−1.19883
$417$	12.0608	0.590619
$418$	−2.88338	−0.141031
$419$	10.9721	0.536023	0.268012	−	0.963416i	$-0.413633\pi$
$419$	10.9721	0.536023	0.268012	+	0.963416i	$0.413633\pi$
$420$	7.78051	0.379650
$421$	20.1666	0.982859	0.491430	−	0.870917i	$-0.336475\pi$
$421$	20.1666	0.982859	0.491430	+	0.870917i	$0.336475\pi$
$422$	−13.1940	−0.642273
$423$	−16.6072	−0.807471
$424$	−21.3712	−1.03788
$425$	−10.1758	−0.493597
$426$	−1.10170	−0.0533774
$427$	−1.18742	−0.0574633
$428$	12.9031	0.623696
$429$	−7.87581	−0.380248
$430$	−0.240025	−0.0115750
$431$	−1.00000	−0.0481683
$431$	−1.00000	−0.0481683
$432$	−14.1183	−0.679266
$433$	27.3562	1.31466	0.657328	−	0.753604i	$-0.271686\pi$
$433$	27.3562	1.31466	0.657328	+	0.753604i	$0.271686\pi$
$434$	−5.00317	−0.240160
$435$	−5.85410	−0.280683
$436$	−6.88575	−0.329768
$437$	11.8917	0.568856
$438$	−6.11869	−0.292362
$439$	3.78974	0.180874	0.0904372	−	0.995902i	$-0.471174\pi$
$439$	3.78974	0.180874	0.0904372	+	0.995902i	$0.471174\pi$
$440$	8.00072	0.381419
$441$	9.58329	0.456347
$442$	5.30966	0.252555
$443$	5.80607	0.275854	0.137927	−	0.990442i	$-0.455956\pi$
$443$	5.80607	0.275854	0.137927	+	0.990442i	$0.455956\pi$
$444$	−16.4068	−0.778634
$445$	34.2977	1.62587
$446$	−2.40567	−0.113912
$447$	−9.51762	−0.450168
$448$	−3.93715	−0.186013
$449$	−15.9466	−0.752569	−0.376284	−	0.926504i	$-0.622798\pi$
$449$	−15.9466	−0.752569	−0.376284	+	0.926504i	$0.622798\pi$
$450$	−3.93598	−0.185544
$451$	9.26551	0.436296
$452$	28.3949	1.33558
$453$	11.7508	0.552098
$454$	6.52896	0.306419
$455$	20.0747	0.941116
$456$	8.29764	0.388573
$457$	−17.6848	−0.827260	−0.413630	−	0.910445i	$-0.635739\pi$
$457$	−17.6848	−0.827260	−0.413630	+	0.910445i	$0.635739\pi$
$458$	2.57229	0.120195
$459$	11.6795	0.545152
$460$	−15.5021	−0.722791
$461$	−5.01870	−0.233744	−0.116872	−	0.993147i	$-0.537287\pi$
$461$	−5.01870	−0.233744	−0.116872	+	0.993147i	$0.537287\pi$
$462$	0.971892	0.0452165
$463$	−25.2131	−1.17175	−0.585877	−	0.810400i	$-0.699250\pi$
$463$	−25.2131	−1.17175	−0.585877	+	0.810400i	$0.699250\pi$
$464$	4.63668	0.215253
$465$	27.4451	1.27274
$466$	−1.90075	−0.0880505
$467$	25.4620	1.17824	0.589121	−	0.808045i	$-0.299474\pi$
$467$	25.4620	1.17824	0.589121	+	0.808045i	$0.299474\pi$
$468$	−15.9794	−0.738648
$469$	−6.71498	−0.310069
$470$	−13.6270	−0.628569
$471$	−14.5052	−0.668364
$472$	1.33965	0.0616625
$473$	0.233278	0.0107261
$474$	5.37623	0.246939
$475$	−19.2760	−0.884444
$476$	5.09797	0.233665
$477$	−21.3712	−0.978518
$478$	4.24180	0.194016
$479$	−5.25747	−0.240220	−0.120110	−	0.992761i	$-0.538325\pi$
$479$	−5.25747	−0.240220	−0.120110	+	0.992761i	$0.538325\pi$
$480$	−16.5519	−0.755490
$481$	−42.3317	−1.93016
$482$	−4.95045	−0.225487
$483$	−4.00829	−0.182384
$484$	15.8413	0.720059
$485$	37.7931	1.71609
$486$	7.34093	0.332991
$487$	−5.69584	−0.258103	−0.129052	−	0.991638i	$-0.541193\pi$
$487$	−5.69584	−0.258103	−0.129052	+	0.991638i	$0.541193\pi$
$488$	1.65082	0.0747291
$489$	10.1704	0.459919
$490$	7.86356	0.355239
$491$	−18.5280	−0.836158	−0.418079	−	0.908411i	$-0.637297\pi$
$491$	−18.5280	−0.836158	−0.418079	+	0.908411i	$0.637297\pi$
$492$	−12.5269	−0.564756
$493$	−3.83574	−0.172753
$494$	10.0581	0.452536
$495$	8.00072	0.359606
$496$	−21.7376	−0.976049
$497$	−2.72919	−0.122421
$498$	−7.19187	−0.322275
$499$	12.6043	0.564246	0.282123	−	0.959378i	$-0.408961\pi$
$499$	12.6043	0.564246	0.282123	+	0.959378i	$0.408961\pi$
$500$	−2.29930	−0.102828
$501$	−13.6022	−0.607700
$502$	4.73070	0.211141
$503$	37.3499	1.66535	0.832675	−	0.553762i	$-0.186808\pi$
$503$	37.3499	1.66535	0.832675	+	0.553762i	$0.186808\pi$
$504$	4.19722	0.186959
$505$	−8.59309	−0.382388
$506$	−1.93643	−0.0860847
$507$	13.2344	0.587761
$508$	20.1505	0.894036
$509$	−1.95950	−0.0868533	−0.0434267	−	0.999057i	$-0.513827\pi$
$509$	−1.95950	−0.0868533	−0.0434267	+	0.999057i	$0.513827\pi$
$510$	3.59426	0.159156
$511$	−15.1576	−0.670532
$512$	22.7784	1.00667
$513$	22.1245	0.976821
$514$	12.1772	0.537112
$515$	2.69351	0.118690
$516$	−0.315390	−0.0138843
$517$	13.2440	0.582471
$518$	5.22382	0.229521
$519$	13.5848	0.596307
$520$	−27.9090	−1.22389
$521$	−10.7327	−0.470206	−0.235103	−	0.971970i	$-0.575543\pi$
$521$	−10.7327	−0.470206	−0.235103	+	0.971970i	$0.575543\pi$
$522$	−1.48366	−0.0649381
$523$	31.1813	1.36346	0.681732	−	0.731602i	$-0.261227\pi$
$523$	31.1813	1.36346	0.681732	+	0.731602i	$0.261227\pi$
$524$	−29.0890	−1.27076
$525$	6.49731	0.283566
$526$	−1.83213	−0.0798847
$527$	17.9827	0.783338
$528$	4.22265	0.183767
$529$	−15.0138	−0.652772
$530$	−17.5361	−0.761719
$531$	1.33965	0.0581360
$532$	9.65711	0.418689
$533$	−32.3209	−1.39998
$534$	−5.79227	−0.250656
$535$	22.5361	0.974320
$536$	9.33555	0.403234
$537$	−15.8814	−0.685334
$538$	15.2180	0.656093
$539$	−7.64253	−0.329187
$540$	−28.8418	−1.24115
$541$	35.9196	1.54431	0.772153	−	0.635437i	$-0.219180\pi$
$541$	35.9196	1.54431	0.772153	+	0.635437i	$0.219180\pi$
$542$	−0.232344	−0.00998003
$543$	−14.5359	−0.623793
$544$	−10.8452	−0.464985
$545$	−12.0264	−0.515154
$546$	−3.39026	−0.145090
$547$	27.4678	1.17444	0.587218	−	0.809429i	$-0.300223\pi$
$547$	27.4678	1.17444	0.587218	+	0.809429i	$0.300223\pi$
$548$	−25.9790	−1.10977
$549$	1.65082	0.0704553
$550$	3.13889	0.133843
$551$	−7.26606	−0.309545
$552$	5.57255	0.237184
$553$	13.3183	0.566353
$554$	−10.8914	−0.462731
$555$	−28.6555	−1.21636
$556$	−19.5148	−0.827610
$557$	−20.2364	−0.857446	−0.428723	−	0.903436i	$-0.641036\pi$
$557$	−20.2364	−0.857446	−0.428723	+	0.903436i	$0.641036\pi$
$558$	6.95569	0.294458
$559$	−0.813746	−0.0344178
$560$	−10.7631	−0.454826
$561$	−3.49323	−0.147484
$562$	−3.99770	−0.168633
$563$	−9.49490	−0.400162	−0.200081	−	0.979779i	$-0.564121\pi$
$563$	−9.49490	−0.400162	−0.200081	+	0.979779i	$0.564121\pi$
$564$	−17.9058	−0.753970
$565$	49.5934	2.08641
$566$	−10.6072	−0.445853
$567$	−0.463357	−0.0194592
$568$	3.79427	0.159204
$569$	−20.7759	−0.870970	−0.435485	−	0.900196i	$-0.643423\pi$
$569$	−20.7759	−0.870970	−0.435485	+	0.900196i	$0.643423\pi$
$570$	6.80862	0.285182
$571$	33.8909	1.41829	0.709145	−	0.705062i	$-0.249081\pi$
$571$	33.8909	1.41829	0.709145	+	0.705062i	$0.249081\pi$
$572$	12.7433	0.532825
$573$	−24.5784	−1.02678
$574$	3.98847	0.166476
$575$	−12.9454	−0.539862
$576$	5.47365	0.228069
$577$	−40.6805	−1.69355	−0.846777	−	0.531949i	$-0.821460\pi$
$577$	−40.6805	−1.69355	−0.846777	+	0.531949i	$0.821460\pi$
$578$	−5.75838	−0.239517
$579$	0.521340	0.0216661
$580$	9.47214	0.393309
$581$	−17.8161	−0.739137
$582$	−6.38258	−0.264566
$583$	17.0432	0.705856
$584$	21.0729	0.872005
$585$	−27.9090	−1.15389
$586$	−10.6922	−0.441692
$587$	−36.3866	−1.50184	−0.750919	−	0.660395i	$-0.770389\pi$
$587$	−36.3866	−1.50184	−0.750919	+	0.660395i	$0.770389\pi$
$588$	10.3326	0.426111
$589$	34.0647	1.40361
$590$	1.09925	0.0452554
$591$	16.8707	0.693967
$592$	22.6963	0.932814
$593$	9.82069	0.403287	0.201644	−	0.979459i	$-0.435372\pi$
$593$	9.82069	0.403287	0.201644	+	0.979459i	$0.435372\pi$
$594$	−3.60273	−0.147822
$595$	8.90392	0.365025
$596$	15.3998	0.630801
$597$	−7.39087	−0.302488
$598$	6.75485	0.276226
$599$	33.7544	1.37917	0.689584	−	0.724206i	$-0.257793\pi$
$599$	33.7544	1.37917	0.689584	+	0.724206i	$0.257793\pi$
$600$	−9.03293	−0.368768
$601$	14.9086	0.608133	0.304066	−	0.952651i	$-0.401656\pi$
$601$	14.9086	0.608133	0.304066	+	0.952651i	$0.401656\pi$
$602$	0.100418	0.00409273
$603$	9.33555	0.380173
$604$	−19.0131	−0.773633
$605$	27.6678	1.12486
$606$	1.45122	0.0589518
$607$	−15.4888	−0.628673	−0.314336	−	0.949312i	$-0.601782\pi$
$607$	−15.4888	−0.628673	−0.314336	+	0.949312i	$0.601782\pi$
$608$	−20.5441	−0.833175
$609$	2.44915	0.0992446
$610$	1.35458	0.0548453
$611$	−46.1992	−1.86902
$612$	−7.08749	−0.286495
$613$	23.5544	0.951352	0.475676	−	0.879621i	$-0.342204\pi$
$613$	23.5544	0.951352	0.475676	+	0.879621i	$0.342204\pi$
$614$	−5.79204	−0.233748
$615$	−21.8790	−0.882245
$616$	−3.34722	−0.134863
$617$	−3.27002	−0.131646	−0.0658231	−	0.997831i	$-0.520967\pi$
$617$	−3.27002	−0.131646	−0.0658231	+	0.997831i	$0.520967\pi$
$618$	−0.454886	−0.0182982
$619$	3.83394	0.154099	0.0770495	−	0.997027i	$-0.475450\pi$
$619$	3.83394	0.154099	0.0770495	+	0.997027i	$0.475450\pi$
$620$	−44.4071	−1.78343
$621$	14.8584	0.596248
$622$	0.0508265	0.00203796
$623$	−14.3490	−0.574879
$624$	−14.7299	−0.589668
$625$	−26.9201	−1.07680
$626$	6.21436	0.248376
$627$	−6.61724	−0.264267
$628$	23.4699	0.936550
$629$	−18.7758	−0.748639
$630$	3.44403	0.137213
$631$	44.2779	1.76268	0.881338	−	0.472487i	$-0.156644\pi$
$631$	44.2779	1.76268	0.881338	+	0.472487i	$0.156644\pi$
$632$	−18.5159	−0.736523
$633$	−30.2797	−1.20351
$634$	2.00410	0.0795930
$635$	35.1941	1.39664
$636$	−23.0422	−0.913684
$637$	26.6595	1.05629
$638$	1.18320	0.0468433
$639$	3.79427	0.150099
$640$	34.7151	1.37224
$641$	19.2255	0.759362	0.379681	−	0.925117i	$-0.376034\pi$
$641$	19.2255	0.759362	0.379681	+	0.925117i	$0.376034\pi$
$642$	−3.80594	−0.150209
$643$	−23.5565	−0.928976	−0.464488	−	0.885579i	$-0.653762\pi$
$643$	−23.5565	−0.928976	−0.464488	+	0.885579i	$0.653762\pi$
$644$	6.48555	0.255567
$645$	−0.550848	−0.0216896
$646$	4.46117	0.175522
$647$	14.3630	0.564667	0.282334	−	0.959316i	$-0.408892\pi$
$647$	14.3630	0.564667	0.282334	+	0.959316i	$0.408892\pi$
$648$	0.644185	0.0253060
$649$	−1.06835	−0.0419365
$650$	−10.9494	−0.429471
$651$	−11.4821	−0.450018
$652$	−16.4560	−0.644466
$653$	13.1518	0.514668	0.257334	−	0.966322i	$-0.417156\pi$
$653$	13.1518	0.514668	0.257334	+	0.966322i	$0.417156\pi$
$654$	2.03104	0.0794200
$655$	−50.8057	−1.98514
$656$	17.3290	0.676585
$657$	21.0729	0.822134
$658$	5.70108	0.222251
$659$	−50.4645	−1.96582	−0.982910	−	0.184089i	$-0.941066\pi$
$659$	−50.4645	−1.96582	−0.982910	+	0.184089i	$0.941066\pi$
$660$	8.62632	0.335779
$661$	−28.9230	−1.12497	−0.562487	−	0.826806i	$-0.690155\pi$
$661$	−28.9230	−1.12497	−0.562487	+	0.826806i	$0.690155\pi$
$662$	15.3427	0.596311
$663$	12.1855	0.473244
$664$	24.7690	0.961223
$665$	16.8667	0.654064
$666$	−7.26245	−0.281414
$667$	−4.87976	−0.188945
$668$	22.0088	0.851545
$669$	−5.52093	−0.213451
$670$	7.66028	0.295942
$671$	−1.31650	−0.0508231
$672$	6.92476	0.267128
$673$	−8.13184	−0.313459	−0.156730	−	0.987642i	$-0.550095\pi$
$673$	−8.13184	−0.313459	−0.156730	+	0.987642i	$0.550095\pi$
$674$	14.3263	0.551828
$675$	−24.0850	−0.927034
$676$	−21.4137	−0.823605
$677$	−43.3049	−1.66434	−0.832171	−	0.554519i	$-0.812902\pi$
$677$	−43.3049	−1.66434	−0.832171	+	0.554519i	$0.812902\pi$
$678$	−8.37544	−0.321657
$679$	−15.8113	−0.606782
$680$	−12.3787	−0.474703
$681$	14.9837	0.574178
$682$	−5.54706	−0.212408
$683$	−1.14827	−0.0439375	−0.0219688	−	0.999759i	$-0.506993\pi$
$683$	−1.14827	−0.0439375	−0.0219688	+	0.999759i	$0.506993\pi$
$684$	−13.4259	−0.513351
$685$	−45.3739	−1.73365
$686$	−7.61608	−0.290783
$687$	5.90331	0.225225
$688$	0.436294	0.0166335
$689$	−59.4518	−2.26493
$690$	4.57255	0.174074
$691$	−38.2292	−1.45431	−0.727153	−	0.686475i	$-0.759157\pi$
$691$	−38.2292	−1.45431	−0.727153	+	0.686475i	$0.759157\pi$
$692$	−21.9807	−0.835580
$693$	−3.34722	−0.127150
$694$	6.86789	0.260702
$695$	−34.0837	−1.29287
$696$	−3.40495	−0.129064
$697$	−14.3356	−0.543000
$698$	−10.5384	−0.398885
$699$	−4.36215	−0.164992
$700$	−10.5129	−0.397349
$701$	24.3912	0.921243	0.460622	−	0.887597i	$-0.347627\pi$
$701$	24.3912	0.921243	0.460622	+	0.887597i	$0.347627\pi$
$702$	12.5674	0.474327
$703$	−35.5670	−1.34144
$704$	−4.36515	−0.164518
$705$	−31.2736	−1.17783
$706$	0.607409	0.0228601
$707$	3.59505	0.135206
$708$	1.44440	0.0542841
$709$	−8.81747	−0.331147	−0.165574	−	0.986197i	$-0.552948\pi$
$709$	−8.81747	−0.331147	−0.165574	+	0.986197i	$0.552948\pi$
$710$	3.11339	0.116843
$711$	−18.5159	−0.694400
$712$	19.9487	0.747611
$713$	22.8773	0.856760
$714$	−1.50371	−0.0562750
$715$	22.2570	0.832364
$716$	25.6967	0.960330
$717$	9.73478	0.363552
$718$	−0.605897	−0.0226119
$719$	−21.2427	−0.792219	−0.396110	−	0.918203i	$-0.629640\pi$
$719$	−21.2427	−0.792219	−0.396110	+	0.918203i	$0.629640\pi$
$720$	14.9635	0.557658
$721$	−1.12687	−0.0419669
$722$	−0.617133	−0.0229673
$723$	−11.3611	−0.422524
$724$	23.5195	0.874095
$725$	7.90994	0.293768
$726$	−4.67259	−0.173416
$727$	32.3992	1.20162	0.600811	−	0.799391i	$-0.294845\pi$
$727$	32.3992	1.20162	0.600811	+	0.799391i	$0.294845\pi$
$728$	11.6761	0.432746
$729$	17.9206	0.663726
$730$	17.2914	0.639983
$731$	−0.360928	−0.0133494
$732$	1.77990	0.0657871
$733$	−12.0632	−0.445564	−0.222782	−	0.974868i	$-0.571514\pi$
$733$	−12.0632	−0.445564	−0.222782	+	0.974868i	$0.571514\pi$
$734$	−6.99767	−0.258289
$735$	18.0466	0.665658
$736$	−13.7971	−0.508568
$737$	−7.44496	−0.274239
$738$	−5.54500	−0.204114
$739$	−50.4147	−1.85453	−0.927267	−	0.374401i	$-0.877848\pi$
$739$	−50.4147	−1.85453	−0.927267	+	0.374401i	$0.877848\pi$
$740$	46.3656	1.70443
$741$	23.0830	0.847974
$742$	7.33649	0.269331
$743$	−15.4343	−0.566231	−0.283115	−	0.959086i	$-0.591368\pi$
$743$	−15.4343	−0.566231	−0.283115	+	0.959086i	$0.591368\pi$
$744$	15.9630	0.585234
$745$	26.8967	0.985420
$746$	10.9905	0.402391
$747$	24.7690	0.906250
$748$	5.65217	0.206664
$749$	−9.42831	−0.344503
$750$	0.678207	0.0247646
$751$	−4.49983	−0.164201	−0.0821006	−	0.996624i	$-0.526163\pi$
$751$	−4.49983	−0.164201	−0.0821006	+	0.996624i	$0.526163\pi$
$752$	24.7699	0.903266
$753$	10.8568	0.395643
$754$	−4.12736	−0.150310
$755$	−33.2075	−1.20855
$756$	12.0664	0.438851
$757$	−20.5386	−0.746488	−0.373244	−	0.927733i	$-0.621754\pi$
$757$	−20.5386	−0.746488	−0.373244	+	0.927733i	$0.621754\pi$
$758$	−0.745515	−0.0270783
$759$	−4.44403	−0.161308
$760$	−23.4491	−0.850588
$761$	7.16236	0.259635	0.129818	−	0.991538i	$-0.458561\pi$
$761$	7.16236	0.259635	0.129818	+	0.991538i	$0.458561\pi$
$762$	−5.94366	−0.215316
$763$	5.03142	0.182150
$764$	39.7687	1.43878
$765$	−12.3787	−0.447554
$766$	2.34533	0.0847404
$767$	3.72674	0.134565
$768$	0.797397	0.0287736
$769$	−13.2650	−0.478347	−0.239173	−	0.970977i	$-0.576876\pi$
$769$	−13.2650	−0.478347	−0.239173	+	0.970977i	$0.576876\pi$
$770$	−2.74656	−0.0989792
$771$	27.9462	1.00646
$772$	−0.843546	−0.0303599
$773$	10.8387	0.389839	0.194920	−	0.980819i	$-0.437555\pi$
$773$	10.8387	0.389839	0.194920	+	0.980819i	$0.437555\pi$
$774$	−0.139607	−0.00501806
$775$	−37.0833	−1.33207
$776$	21.9818	0.789100
$777$	11.9885	0.430084
$778$	−13.2201	−0.473965
$779$	−27.1560	−0.972965
$780$	−30.0913	−1.07744
$781$	−3.02588	−0.108274
$782$	2.99604	0.107138
$783$	−9.07882	−0.324451
$784$	−14.2936	−0.510486
$785$	40.9916	1.46305
$786$	8.58018	0.306045
$787$	3.29270	0.117372	0.0586860	−	0.998276i	$-0.481309\pi$
$787$	3.29270	0.117372	0.0586860	+	0.998276i	$0.481309\pi$
$788$	−27.2973	−0.972427
$789$	−4.20467	−0.149690
$790$	−15.1932	−0.540550
$791$	−20.7481	−0.737719
$792$	4.65350	0.165355
$793$	4.59237	0.163080
$794$	14.1447	0.501975
$795$	−40.2447	−1.42733
$796$	11.9587	0.423864
$797$	11.7296	0.415483	0.207741	−	0.978184i	$-0.433389\pi$
$797$	11.7296	0.415483	0.207741	+	0.978184i	$0.433389\pi$
$798$	−2.84849	−0.100835
$799$	−20.4912	−0.724925
$800$	22.3646	0.790710
$801$	19.9487	0.704854
$802$	2.42002	0.0854539
$803$	−16.8054	−0.593048
$804$	10.0655	0.354984
$805$	11.3274	0.399239
$806$	19.3498	0.681569
$807$	34.9247	1.22941
$808$	−4.99804	−0.175831
$809$	−22.6521	−0.796404	−0.398202	−	0.917298i	$-0.630366\pi$
$809$	−22.6521	−0.796404	−0.398202	+	0.917298i	$0.630366\pi$
$810$	0.528585	0.0185726
$811$	20.4965	0.719728	0.359864	−	0.933005i	$-0.382823\pi$
$811$	20.4965	0.719728	0.359864	+	0.933005i	$0.382823\pi$
$812$	−3.96281	−0.139067
$813$	−0.533221	−0.0187009
$814$	5.79170	0.202999
$815$	−28.7414	−1.00677
$816$	−6.53330	−0.228711
$817$	−0.683708	−0.0239199
$818$	−16.1825	−0.565807
$819$	11.6761	0.407997
$820$	35.4009	1.23625
$821$	−9.24297	−0.322582	−0.161291	−	0.986907i	$-0.551566\pi$
$821$	−9.24297	−0.322582	−0.161291	+	0.986907i	$0.551566\pi$
$822$	7.66285	0.267272
$823$	56.4625	1.96816	0.984080	−	0.177728i	$-0.0568748\pi$
$823$	56.4625	1.96816	0.984080	+	0.177728i	$0.0568748\pi$
$824$	1.56664	0.0545765
$825$	7.20363	0.250798
$826$	−0.459888	−0.0160016
$827$	45.9082	1.59638	0.798192	−	0.602403i	$-0.205790\pi$
$827$	45.9082	1.59638	0.798192	+	0.602403i	$0.205790\pi$
$828$	−9.01658	−0.313348
$829$	−50.3644	−1.74923	−0.874614	−	0.484819i	$-0.838885\pi$
$829$	−50.3644	−1.74923	−0.874614	+	0.484819i	$0.838885\pi$
$830$	20.3242	0.705462
$831$	−24.9954	−0.867080
$832$	15.2270	0.527901
$833$	11.8245	0.409696
$834$	5.75613	0.199318
$835$	38.4397	1.33026
$836$	10.7069	0.370307
$837$	42.5632	1.47120
$838$	5.23656	0.180894
$839$	51.0331	1.76186	0.880930	−	0.473247i	$-0.156918\pi$
$839$	51.0331	1.76186	0.880930	+	0.473247i	$0.156918\pi$
$840$	7.90392	0.272711
$841$	−26.0184	−0.897185
$842$	9.62470	0.331689
$843$	−9.17457	−0.315989
$844$	48.9936	1.68643
$845$	−37.4004	−1.28661
$846$	−7.92597	−0.272500
$847$	−11.5752	−0.397730
$848$	31.8754	1.09461
$849$	−24.3431	−0.835452
$850$	−4.85649	−0.166576
$851$	−23.8862	−0.818808
$852$	4.09096	0.140154
$853$	37.1003	1.27029	0.635144	−	0.772393i	$-0.280941\pi$
$853$	37.1003	1.27029	0.635144	+	0.772393i	$0.280941\pi$
$854$	−0.566709	−0.0193924
$855$	−23.4491	−0.801942
$856$	13.1078	0.448015
$857$	−24.3171	−0.830655	−0.415328	−	0.909672i	$-0.636333\pi$
$857$	−24.3171	−0.830655	−0.415328	+	0.909672i	$0.636333\pi$
$858$	−3.75881	−0.128324
$859$	37.0740	1.26495	0.632475	−	0.774581i	$-0.282039\pi$
$859$	37.0740	1.26495	0.632475	+	0.774581i	$0.282039\pi$
$860$	0.891291	0.0303928
$861$	9.15340	0.311947
$862$	−0.477260	−0.0162555
$863$	−50.4014	−1.71568	−0.857842	−	0.513913i	$-0.828195\pi$
$863$	−50.4014	−1.71568	−0.857842	+	0.513913i	$0.828195\pi$
$864$	−25.6696	−0.873296
$865$	−38.3906	−1.30532
$866$	13.0560	0.443662
$867$	−13.2153	−0.448814
$868$	18.5784	0.630592
$869$	14.7661	0.500907
$870$	−2.79393	−0.0947231
$871$	25.9703	0.879970
$872$	−6.99496	−0.236879
$873$	21.9818	0.743970
$874$	5.67542	0.191974
$875$	1.68010	0.0567976
$876$	22.7207	0.767662
$877$	27.1574	0.917041	0.458521	−	0.888684i	$-0.348380\pi$
$877$	27.1574	0.917041	0.458521	+	0.888684i	$0.348380\pi$
$878$	1.80869	0.0610404
$879$	−24.5383	−0.827656
$880$	−11.9332	−0.402268
$881$	26.5077	0.893069	0.446534	−	0.894767i	$-0.352658\pi$
$881$	26.5077	0.893069	0.446534	+	0.894767i	$0.352658\pi$
$882$	4.57372	0.154005
$883$	38.1048	1.28233	0.641165	−	0.767403i	$-0.278452\pi$
$883$	38.1048	1.28233	0.641165	+	0.767403i	$0.278452\pi$
$884$	−19.7165	−0.663137
$885$	2.52274	0.0848010
$886$	2.77100	0.0930937
$887$	−3.48408	−0.116984	−0.0584920	−	0.998288i	$-0.518629\pi$
$887$	−3.48408	−0.116984	−0.0584920	+	0.998288i	$0.518629\pi$
$888$	−16.6671	−0.559310
$889$	−14.7240	−0.493827
$890$	16.3689	0.548688
$891$	−0.513728	−0.0172105
$892$	8.93305	0.299101
$893$	−38.8165	−1.29894
$894$	−4.54238	−0.151920
$895$	44.8808	1.50020
$896$	−14.5236	−0.485200
$897$	15.5021	0.517601
$898$	−7.61070	−0.253972
$899$	−13.9785	−0.466209
$900$	14.6156	0.487186
$901$	−26.3692	−0.878487
$902$	4.42206	0.147238
$903$	0.230456	0.00766908
$904$	28.8453	0.959379
$905$	41.0782	1.36549
$906$	5.60816	0.186319
$907$	−31.7875	−1.05549	−0.527743	−	0.849404i	$-0.676962\pi$
$907$	−31.7875	−1.05549	−0.527743	+	0.849404i	$0.676962\pi$
$908$	−24.2442	−0.804571
$909$	−4.99804	−0.165775
$910$	9.58084	0.317602
$911$	4.42820	0.146713	0.0733565	−	0.997306i	$-0.476629\pi$
$911$	4.42820	0.146713	0.0733565	+	0.997306i	$0.476629\pi$
$912$	−12.3761	−0.409812
$913$	−19.7529	−0.653725
$914$	−8.44025	−0.279179
$915$	3.10871	0.102771
$916$	−9.55175	−0.315599
$917$	21.2553	0.701913
$918$	5.57415	0.183974
$919$	25.7385	0.849035	0.424517	−	0.905420i	$-0.360444\pi$
$919$	25.7385	0.849035	0.424517	+	0.905420i	$0.360444\pi$
$920$	−15.7480	−0.519196
$921$	−13.2925	−0.438003
$922$	−2.39523	−0.0788826
$923$	10.5552	0.347428
$924$	−3.60895	−0.118726
$925$	38.7188	1.27306
$926$	−12.0332	−0.395436
$927$	1.56664	0.0514552
$928$	8.43032	0.276739
$929$	29.3724	0.963678	0.481839	−	0.876260i	$-0.339969\pi$
$929$	29.3724	0.963678	0.481839	+	0.876260i	$0.339969\pi$
$930$	13.0985	0.429515
$931$	22.3993	0.734106
$932$	7.05810	0.231196
$933$	0.116645	0.00381878
$934$	12.1520	0.397626
$935$	9.87185	0.322844
$936$	−16.2328	−0.530587
$937$	2.88384	0.0942111	0.0471055	−	0.998890i	$-0.485000\pi$
$937$	2.88384	0.0942111	0.0471055	+	0.998890i	$0.485000\pi$
$938$	−3.20479	−0.104640
$939$	14.2617	0.465414
$940$	50.6017	1.65045
$941$	39.7737	1.29659	0.648293	−	0.761391i	$-0.275483\pi$
$941$	39.7737	1.29659	0.648293	+	0.761391i	$0.275483\pi$
$942$	−6.92275	−0.225555
$943$	−18.2375	−0.593895
$944$	−1.99811	−0.0650330
$945$	21.0747	0.685560
$946$	0.111334	0.00361979
$947$	−23.9973	−0.779806	−0.389903	−	0.920856i	$-0.627492\pi$
$947$	−23.9973	−0.779806	−0.389903	+	0.920856i	$0.627492\pi$
$948$	−19.9637	−0.648391
$949$	58.6222	1.90296
$950$	−9.19967	−0.298477
$951$	4.59934	0.149144
$952$	5.17883	0.167847
$953$	22.8631	0.740608	0.370304	−	0.928911i	$-0.379254\pi$
$953$	22.8631	0.740608	0.370304	+	0.928911i	$0.379254\pi$
$954$	−10.1996	−0.330224
$955$	69.4584	2.24762
$956$	−15.7512	−0.509430
$957$	2.71540	0.0877763
$958$	−2.50918	−0.0810679
$959$	18.9829	0.612989
$960$	10.3076	0.332676
$961$	34.5338	1.11399
$962$	−20.2032	−0.651378
$963$	13.1078	0.422392
$964$	18.3827	0.592065
$965$	−1.47330	−0.0474273
$966$	−1.91300	−0.0615497
$967$	5.39896	0.173619	0.0868094	−	0.996225i	$-0.472333\pi$
$967$	5.39896	0.173619	0.0868094	+	0.996225i	$0.472333\pi$
$968$	16.0925	0.517234
$969$	10.2382	0.328899
$970$	18.0371	0.579137
$971$	−50.8436	−1.63165	−0.815824	−	0.578301i	$-0.803716\pi$
$971$	−50.8436	−1.63165	−0.815824	+	0.578301i	$0.803716\pi$
$972$	−27.2593	−0.874342
$973$	14.2594	0.457136
$974$	−2.71840	−0.0871031
$975$	−25.1285	−0.804755
$976$	−2.46222	−0.0788138
$977$	−32.3004	−1.03338	−0.516691	−	0.856172i	$-0.672836\pi$
$977$	−32.3004	−1.03338	−0.516691	+	0.856172i	$0.672836\pi$
$978$	4.85390	0.155211
$979$	−15.9088	−0.508448
$980$	−29.2000	−0.932759
$981$	−6.99496	−0.223332
$982$	−8.84269	−0.282181
$983$	−22.7019	−0.724077	−0.362039	−	0.932163i	$-0.617919\pi$
$983$	−22.7019	−0.724077	−0.362039	+	0.932163i	$0.617919\pi$
$984$	−12.7256	−0.405677
$985$	−47.6764	−1.51910
$986$	−1.83065	−0.0582997
$987$	13.0838	0.416461
$988$	−37.3490	−1.18823
$989$	−0.459167	−0.0146007
$990$	3.81842	0.121357
$991$	−0.0305178	−0.000969429 0	−0.000484714	−	1.00000i	$-0.500154\pi$
$991$	−0.0305178	−0.000969429 0	−0.000484714		1.00000i	$0.500154\pi$
$992$	−39.5229	−1.25485
$993$	35.2109	1.11739
$994$	−1.30253	−0.0413138
$995$	20.8865	0.662148
$996$	26.7058	0.846204
$997$	7.20114	0.228063	0.114031	−	0.993477i	$-0.463624\pi$
$997$	7.20114	0.228063	0.114031	+	0.993477i	$0.463624\pi$
$998$	6.01553	0.190418
$999$	−44.4404	−1.40603

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	431.2.a.e.1.3	✓	4
3.2	odd	2		3879.2.a.m.1.2		4
4.3	odd	2		6896.2.a.o.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
431.2.a.e.1.3	✓	4	1.1	even	1	trivial
3879.2.a.m.1.2		4	3.2	odd	2
6896.2.a.o.1.1		4	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 431 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	431.a (trivial)

\(f(q)\)	\(=\)	\(q+0.477260 q^{2} +1.09529 q^{3} -1.77222 q^{4} -3.09529 q^{5} +0.522740 q^{6} +1.29496 q^{7} -1.80033 q^{8} -1.80033 q^{9} +O(q^{10})\)	\(q+0.477260 q^{2} +1.09529 q^{3} -1.77222 q^{4} -3.09529 q^{5} +0.522740 q^{6} +1.29496 q^{7} -1.80033 q^{8} -1.80033 q^{9} -1.47726 q^{10} +1.43574 q^{11} -1.94111 q^{12} -5.00829 q^{13} +0.618034 q^{14} -3.39026 q^{15} +2.68522 q^{16} -2.22137 q^{17} -0.859226 q^{18} -4.20796 q^{19} +5.48555 q^{20} +1.41837 q^{21} +0.685220 q^{22} -2.82599 q^{23} -1.97189 q^{24} +4.58084 q^{25} -2.39026 q^{26} -5.25777 q^{27} -2.29496 q^{28} +1.72674 q^{29} -1.61803 q^{30} -8.09529 q^{31} +4.88221 q^{32} +1.57255 q^{33} -1.06017 q^{34} -4.00829 q^{35} +3.19059 q^{36} +8.45232 q^{37} -2.00829 q^{38} -5.48555 q^{39} +5.57255 q^{40} +6.45349 q^{41} +0.676929 q^{42} +0.162480 q^{43} -2.54445 q^{44} +5.57255 q^{45} -1.34873 q^{46} +9.22454 q^{47} +2.94111 q^{48} -5.32307 q^{49} +2.18625 q^{50} -2.43306 q^{51} +8.87581 q^{52} +11.8707 q^{53} -2.50933 q^{54} -4.44403 q^{55} -2.33136 q^{56} -4.60895 q^{57} +0.824105 q^{58} -0.744115 q^{59} +6.00829 q^{60} -0.916954 q^{61} -3.86356 q^{62} -2.33136 q^{63} -3.04036 q^{64} +15.5021 q^{65} +0.750517 q^{66} -5.18546 q^{67} +3.93677 q^{68} -3.09529 q^{69} -1.91300 q^{70} -2.10754 q^{71} +3.24119 q^{72} -11.7050 q^{73} +4.03395 q^{74} +5.01737 q^{75} +7.45744 q^{76} +1.85923 q^{77} -2.61803 q^{78} +10.2847 q^{79} -8.31154 q^{80} -0.357815 q^{81} +3.07999 q^{82} -13.7580 q^{83} -2.51366 q^{84} +6.87581 q^{85} +0.0775451 q^{86} +1.89129 q^{87} -2.58480 q^{88} -11.0806 q^{89} +2.65956 q^{90} -6.48555 q^{91} +5.00829 q^{92} -8.86673 q^{93} +4.40250 q^{94} +13.0249 q^{95} +5.34746 q^{96} -12.2098 q^{97} -2.54049 q^{98} -2.58480 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{2} - 3 q^{3} - q^{4} - 5 q^{5} + 5 q^{6} + 2 q^{7} - 3 q^{8} - 3 q^{9}+O(q^{10}) \) 4 * q - q^2 - 3 * q^3 - q^4 - 5 * q^5 + 5 * q^6 + 2 * q^7 - 3 * q^8 - 3 * q^9	\( 4 q - q^{2} - 3 q^{3} - q^{4} - 5 q^{5} + 5 q^{6} + 2 q^{7} - 3 q^{8} - 3 q^{9} - 3 q^{10} + q^{11} - 2 q^{12} - 5 q^{13} - 2 q^{14} - 3 q^{15} - 3 q^{16} + 2 q^{17} - 5 q^{18} - 6 q^{19} + 4 q^{20} - 3 q^{21} - 11 q^{22} + 4 q^{23} - 6 q^{24} - 7 q^{25} + q^{26} + 3 q^{27} - 6 q^{28} - 5 q^{29} - 2 q^{30} - 25 q^{31} + 8 q^{32} - 4 q^{33} - 12 q^{34} - q^{35} - 2 q^{36} - q^{37} + 7 q^{38} - 4 q^{39} + 12 q^{40} + 2 q^{41} + 4 q^{42} - 16 q^{43} + 2 q^{44} + 12 q^{45} + 7 q^{46} - 4 q^{47} + 6 q^{48} - 20 q^{49} + 13 q^{50} - 3 q^{51} + 7 q^{52} + 4 q^{53} - 13 q^{54} + 2 q^{55} + 7 q^{56} + 5 q^{57} + 20 q^{58} + 5 q^{59} + 9 q^{60} + 2 q^{62} + 7 q^{63} - 3 q^{64} + 14 q^{65} + 12 q^{66} + 9 q^{67} + 29 q^{68} - 5 q^{69} + 10 q^{71} + 19 q^{72} - 50 q^{73} - 10 q^{74} + 24 q^{75} + 10 q^{76} + 9 q^{77} - 6 q^{78} + 8 q^{79} - 8 q^{81} + 37 q^{82} - 15 q^{83} + 6 q^{84} - q^{85} + 12 q^{86} + 15 q^{87} + 11 q^{88} - 35 q^{89} + 8 q^{90} - 8 q^{91} + 5 q^{92} + 12 q^{93} - 4 q^{94} + 7 q^{95} + 7 q^{96} - 6 q^{97} + 6 q^{98} + 11 q^{99}+O(q^{100}) \) 4 * q - q^2 - 3 * q^3 - q^4 - 5 * q^5 + 5 * q^6 + 2 * q^7 - 3 * q^8 - 3 * q^9 - 3 * q^10 + q^11 - 2 * q^12 - 5 * q^13 - 2 * q^14 - 3 * q^15 - 3 * q^16 + 2 * q^17 - 5 * q^18 - 6 * q^19 + 4 * q^20 - 3 * q^21 - 11 * q^22 + 4 * q^23 - 6 * q^24 - 7 * q^25 + q^26 + 3 * q^27 - 6 * q^28 - 5 * q^29 - 2 * q^30 - 25 * q^31 + 8 * q^32 - 4 * q^33 - 12 * q^34 - q^35 - 2 * q^36 - q^37 + 7 * q^38 - 4 * q^39 + 12 * q^40 + 2 * q^41 + 4 * q^42 - 16 * q^43 + 2 * q^44 + 12 * q^45 + 7 * q^46 - 4 * q^47 + 6 * q^48 - 20 * q^49 + 13 * q^50 - 3 * q^51 + 7 * q^52 + 4 * q^53 - 13 * q^54 + 2 * q^55 + 7 * q^56 + 5 * q^57 + 20 * q^58 + 5 * q^59 + 9 * q^60 + 2 * q^62 + 7 * q^63 - 3 * q^64 + 14 * q^65 + 12 * q^66 + 9 * q^67 + 29 * q^68 - 5 * q^69 + 10 * q^71 + 19 * q^72 - 50 * q^73 - 10 * q^74 + 24 * q^75 + 10 * q^76 + 9 * q^77 - 6 * q^78 + 8 * q^79 - 8 * q^81 + 37 * q^82 - 15 * q^83 + 6 * q^84 - q^85 + 12 * q^86 + 15 * q^87 + 11 * q^88 - 35 * q^89 + 8 * q^90 - 8 * q^91 + 5 * q^92 + 12 * q^93 - 4 * q^94 + 7 * q^95 + 7 * q^96 - 6 * q^97 + 6 * q^98 + 11 * q^99

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