LMFDB - Embedded newform 4334.2.a.b.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4334,2,Mod(1,4334)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4334, 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("4334.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4334 = 2 \cdot 11 \cdot 197 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4334.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:	$34.6071642360$
Analytic rank:	$1$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 6 x^{14} - 8 x^{13} + 94 x^{12} - 13 x^{11} - 582 x^{10} + 295 x^{9} + 1814 x^{8} - 1056 x^{7} + \cdots - 45 $ x^15 - 6x^14 - 8x^13 + 94x^12 - 13x^11 - 582x^10 + 295x^9 + 1814x^8 - 1056x^7 - 2966x^6 + 1621x^5 + 2328x^4 - 1160x^3 - 589x^2 + 364x - 45
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.15333$ of defining polynomial
Character	$\chi$	$=$	4334.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -2.15333 q^{3} +1.00000 q^{4} -3.80220 q^{5} -2.15333 q^{6} +3.88018 q^{7} +1.00000 q^{8} +1.63685 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.15333 q^{3} +1.00000 q^{4} -3.80220 q^{5} -2.15333 q^{6} +3.88018 q^{7} +1.00000 q^{8} +1.63685 q^{9} -3.80220 q^{10} +1.00000 q^{11} -2.15333 q^{12} -2.68872 q^{13} +3.88018 q^{14} +8.18741 q^{15} +1.00000 q^{16} +0.563498 q^{17} +1.63685 q^{18} -5.44288 q^{19} -3.80220 q^{20} -8.35532 q^{21} +1.00000 q^{22} +6.33890 q^{23} -2.15333 q^{24} +9.45673 q^{25} -2.68872 q^{26} +2.93532 q^{27} +3.88018 q^{28} -6.39585 q^{29} +8.18741 q^{30} -4.11453 q^{31} +1.00000 q^{32} -2.15333 q^{33} +0.563498 q^{34} -14.7532 q^{35} +1.63685 q^{36} -0.728015 q^{37} -5.44288 q^{38} +5.78972 q^{39} -3.80220 q^{40} +9.65393 q^{41} -8.35532 q^{42} -0.443676 q^{43} +1.00000 q^{44} -6.22363 q^{45} +6.33890 q^{46} -0.446833 q^{47} -2.15333 q^{48} +8.05577 q^{49} +9.45673 q^{50} -1.21340 q^{51} -2.68872 q^{52} -9.29648 q^{53} +2.93532 q^{54} -3.80220 q^{55} +3.88018 q^{56} +11.7203 q^{57} -6.39585 q^{58} +12.5695 q^{59} +8.18741 q^{60} -6.09043 q^{61} -4.11453 q^{62} +6.35126 q^{63} +1.00000 q^{64} +10.2231 q^{65} -2.15333 q^{66} +16.0033 q^{67} +0.563498 q^{68} -13.6498 q^{69} -14.7532 q^{70} +9.48287 q^{71} +1.63685 q^{72} -4.56247 q^{73} -0.728015 q^{74} -20.3635 q^{75} -5.44288 q^{76} +3.88018 q^{77} +5.78972 q^{78} +7.83601 q^{79} -3.80220 q^{80} -11.2313 q^{81} +9.65393 q^{82} +1.69925 q^{83} -8.35532 q^{84} -2.14253 q^{85} -0.443676 q^{86} +13.7724 q^{87} +1.00000 q^{88} -17.3736 q^{89} -6.22363 q^{90} -10.4327 q^{91} +6.33890 q^{92} +8.85995 q^{93} -0.446833 q^{94} +20.6949 q^{95} -2.15333 q^{96} -12.8995 q^{97} +8.05577 q^{98} +1.63685 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + 15 q^{2} - 9 q^{3} + 15 q^{4} - 11 q^{5} - 9 q^{6} - 11 q^{7} + 15 q^{8} + 10 q^{9}+O(q^{10}) $ 15 * q + 15 * q^2 - 9 * q^3 + 15 * q^4 - 11 * q^5 - 9 * q^6 - 11 * q^7 + 15 * q^8 + 10 * q^9	\( 15 q + 15 q^{2} - 9 q^{3} + 15 q^{4} - 11 q^{5} - 9 q^{6} - 11 q^{7} + 15 q^{8} + 10 q^{9} - 11 q^{10} + 15 q^{11} - 9 q^{12} - 21 q^{13} - 11 q^{14} - 2 q^{15} + 15 q^{16} - 4 q^{17} + 10 q^{18} - 22 q^{19} - 11 q^{20} - 13 q^{21} + 15 q^{22} - 16 q^{23} - 9 q^{24} + 6 q^{25} - 21 q^{26} - 21 q^{27} - 11 q^{28} - 8 q^{29} - 2 q^{30} - 33 q^{31} + 15 q^{32} - 9 q^{33} - 4 q^{34} - 2 q^{35} + 10 q^{36} - q^{37} - 22 q^{38} + q^{39} - 11 q^{40} - 10 q^{41} - 13 q^{42} - 8 q^{43} + 15 q^{44} - 10 q^{45} - 16 q^{46} - 31 q^{47} - 9 q^{48} + 2 q^{49} + 6 q^{50} + 2 q^{51} - 21 q^{52} - 18 q^{53} - 21 q^{54} - 11 q^{55} - 11 q^{56} + 16 q^{57} - 8 q^{58} - 37 q^{59} - 2 q^{60} - 31 q^{61} - 33 q^{62} - 20 q^{63} + 15 q^{64} - 13 q^{65} - 9 q^{66} + q^{67} - 4 q^{68} - 25 q^{69} - 2 q^{70} - 28 q^{71} + 10 q^{72} - 20 q^{73} - q^{74} - 9 q^{75} - 22 q^{76} - 11 q^{77} + q^{78} - 6 q^{79} - 11 q^{80} + 3 q^{81} - 10 q^{82} - 15 q^{83} - 13 q^{84} - 31 q^{85} - 8 q^{86} - 16 q^{87} + 15 q^{88} - 17 q^{89} - 10 q^{90} - 21 q^{91} - 16 q^{92} + 10 q^{93} - 31 q^{94} - 3 q^{95} - 9 q^{96} - 9 q^{97} + 2 q^{98} + 10 q^{99}+O(q^{100}) \) 15 * q + 15 * q^2 - 9 * q^3 + 15 * q^4 - 11 * q^5 - 9 * q^6 - 11 * q^7 + 15 * q^8 + 10 * q^9 - 11 * q^10 + 15 * q^11 - 9 * q^12 - 21 * q^13 - 11 * q^14 - 2 * q^15 + 15 * q^16 - 4 * q^17 + 10 * q^18 - 22 * q^19 - 11 * q^20 - 13 * q^21 + 15 * q^22 - 16 * q^23 - 9 * q^24 + 6 * q^25 - 21 * q^26 - 21 * q^27 - 11 * q^28 - 8 * q^29 - 2 * q^30 - 33 * q^31 + 15 * q^32 - 9 * q^33 - 4 * q^34 - 2 * q^35 + 10 * q^36 - q^37 - 22 * q^38 + q^39 - 11 * q^40 - 10 * q^41 - 13 * q^42 - 8 * q^43 + 15 * q^44 - 10 * q^45 - 16 * q^46 - 31 * q^47 - 9 * q^48 + 2 * q^49 + 6 * q^50 + 2 * q^51 - 21 * q^52 - 18 * q^53 - 21 * q^54 - 11 * q^55 - 11 * q^56 + 16 * q^57 - 8 * q^58 - 37 * q^59 - 2 * q^60 - 31 * q^61 - 33 * q^62 - 20 * q^63 + 15 * q^64 - 13 * q^65 - 9 * q^66 + q^67 - 4 * q^68 - 25 * q^69 - 2 * q^70 - 28 * q^71 + 10 * q^72 - 20 * q^73 - q^74 - 9 * q^75 - 22 * q^76 - 11 * q^77 + q^78 - 6 * q^79 - 11 * q^80 + 3 * q^81 - 10 * q^82 - 15 * q^83 - 13 * q^84 - 31 * q^85 - 8 * q^86 - 16 * q^87 + 15 * q^88 - 17 * q^89 - 10 * q^90 - 21 * q^91 - 16 * q^92 + 10 * q^93 - 31 * q^94 - 3 * q^95 - 9 * q^96 - 9 * q^97 + 2 * q^98 + 10 * 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
$3$	−2.15333	−1.24323	−0.621614	−	0.783324i	$-0.713523\pi$
$3$	−2.15333	−1.24323	−0.621614	+	0.783324i	$0.713523\pi$
$4$	1.00000	0.500000
$5$	−3.80220	−1.70040	−0.850198	−	0.526463i	$-0.823518\pi$
$5$	−3.80220	−1.70040	−0.850198	+	0.526463i	$0.823518\pi$
$6$	−2.15333	−0.879095
$7$	3.88018	1.46657	0.733284	−	0.679922i	$-0.237986\pi$
$7$	3.88018	1.46657	0.733284	+	0.679922i	$0.237986\pi$
$8$	1.00000	0.353553
$9$	1.63685	0.545617
$10$	−3.80220	−1.20236
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−2.15333	−0.621614
$13$	−2.68872	−0.745717	−0.372858	−	0.927888i	$-0.621622\pi$
$13$	−2.68872	−0.745717	−0.372858	+	0.927888i	$0.621622\pi$
$14$	3.88018	1.03702
$15$	8.18741	2.11398
$16$	1.00000	0.250000
$17$	0.563498	0.136668	0.0683342	−	0.997662i	$-0.478232\pi$
$17$	0.563498	0.136668	0.0683342	+	0.997662i	$0.478232\pi$
$18$	1.63685	0.385809
$19$	−5.44288	−1.24868	−0.624341	−	0.781152i	$-0.714633\pi$
$19$	−5.44288	−1.24868	−0.624341	+	0.781152i	$0.714633\pi$
$20$	−3.80220	−0.850198
$21$	−8.35532	−1.82328
$22$	1.00000	0.213201
$23$	6.33890	1.32175	0.660876	−	0.750495i	$-0.270185\pi$
$23$	6.33890	1.32175	0.660876	+	0.750495i	$0.270185\pi$
$24$	−2.15333	−0.439548
$25$	9.45673	1.89135
$26$	−2.68872	−0.527302
$27$	2.93532	0.564902
$28$	3.88018	0.733284
$29$	−6.39585	−1.18768	−0.593839	−	0.804584i	$-0.702389\pi$
$29$	−6.39585	−1.18768	−0.593839	+	0.804584i	$0.702389\pi$
$30$	8.18741	1.49481
$31$	−4.11453	−0.738991	−0.369495	−	0.929233i	$-0.620469\pi$
$31$	−4.11453	−0.738991	−0.369495	+	0.929233i	$0.620469\pi$
$32$	1.00000	0.176777
$33$	−2.15333	−0.374847
$34$	0.563498	0.0966391
$35$	−14.7532	−2.49375
$36$	1.63685	0.272808
$37$	−0.728015	−0.119685	−0.0598424	−	0.998208i	$-0.519060\pi$
$37$	−0.728015	−0.119685	−0.0598424	+	0.998208i	$0.519060\pi$
$38$	−5.44288	−0.882952
$39$	5.78972	0.927096
$40$	−3.80220	−0.601181
$41$	9.65393	1.50769	0.753846	−	0.657052i	$-0.228197\pi$
$41$	9.65393	1.50769	0.753846	+	0.657052i	$0.228197\pi$
$42$	−8.35532	−1.28925
$43$	−0.443676	−0.0676600	−0.0338300	−	0.999428i	$-0.510770\pi$
$43$	−0.443676	−0.0676600	−0.0338300	+	0.999428i	$0.510770\pi$
$44$	1.00000	0.150756
$45$	−6.22363	−0.927764
$46$	6.33890	0.934620
$47$	−0.446833	−0.0651773	−0.0325886	−	0.999469i	$-0.510375\pi$
$47$	−0.446833	−0.0651773	−0.0325886	+	0.999469i	$0.510375\pi$
$48$	−2.15333	−0.310807
$49$	8.05577	1.15082
$50$	9.45673	1.33738
$51$	−1.21340	−0.169910
$52$	−2.68872	−0.372858
$53$	−9.29648	−1.27697	−0.638485	−	0.769634i	$-0.720439\pi$
$53$	−9.29648	−1.27697	−0.638485	+	0.769634i	$0.720439\pi$
$54$	2.93532	0.399446
$55$	−3.80220	−0.512689
$56$	3.88018	0.518510
$57$	11.7203	1.55240
$58$	−6.39585	−0.839816
$59$	12.5695	1.63641	0.818204	−	0.574928i	$-0.194970\pi$
$59$	12.5695	1.63641	0.818204	+	0.574928i	$0.194970\pi$
$60$	8.18741	1.05699
$61$	−6.09043	−0.779800	−0.389900	−	0.920857i	$-0.627490\pi$
$61$	−6.09043	−0.779800	−0.389900	+	0.920857i	$0.627490\pi$
$62$	−4.11453	−0.522545
$63$	6.35126	0.800184
$64$	1.00000	0.125000
$65$	10.2231	1.26801
$66$	−2.15333	−0.265057
$67$	16.0033	1.95511	0.977557	−	0.210672i	$-0.0675653\pi$
$67$	16.0033	1.95511	0.977557	+	0.210672i	$0.0675653\pi$
$68$	0.563498	0.0683342
$69$	−13.6498	−1.64324
$70$	−14.7532	−1.76335
$71$	9.48287	1.12541	0.562705	−	0.826658i	$-0.309761\pi$
$71$	9.48287	1.12541	0.562705	+	0.826658i	$0.309761\pi$
$72$	1.63685	0.192905
$73$	−4.56247	−0.533996	−0.266998	−	0.963697i	$-0.586032\pi$
$73$	−4.56247	−0.533996	−0.266998	+	0.963697i	$0.586032\pi$
$74$	−0.728015	−0.0846300
$75$	−20.3635	−2.35137
$76$	−5.44288	−0.624341
$77$	3.88018	0.442187
$78$	5.78972	0.655556
$79$	7.83601	0.881620	0.440810	−	0.897601i	$-0.354691\pi$
$79$	7.83601	0.881620	0.440810	+	0.897601i	$0.354691\pi$
$80$	−3.80220	−0.425099
$81$	−11.2313	−1.24792
$82$	9.65393	1.06610
$83$	1.69925	0.186517	0.0932586	−	0.995642i	$-0.470272\pi$
$83$	1.69925	0.186517	0.0932586	+	0.995642i	$0.470272\pi$
$84$	−8.35532	−0.911640
$85$	−2.14253	−0.232390
$86$	−0.443676	−0.0478428
$87$	13.7724	1.47656
$88$	1.00000	0.106600
$89$	−17.3736	−1.84160	−0.920799	−	0.390036i	$-0.872462\pi$
$89$	−17.3736	−1.84160	−0.920799	+	0.390036i	$0.872462\pi$
$90$	−6.22363	−0.656028
$91$	−10.4327	−1.09365
$92$	6.33890	0.660876
$93$	8.85995	0.918734
$94$	−0.446833	−0.0460873
$95$	20.6949	2.12326
$96$	−2.15333	−0.219774
$97$	−12.8995	−1.30975	−0.654874	−	0.755738i	$-0.727278\pi$
$97$	−12.8995	−1.30975	−0.654874	+	0.755738i	$0.727278\pi$
$98$	8.05577	0.813756
$99$	1.63685	0.164510
$100$	9.45673	0.945673
$101$	2.60524	0.259231	0.129616	−	0.991564i	$-0.458626\pi$
$101$	2.60524	0.259231	0.129616	+	0.991564i	$0.458626\pi$
$102$	−1.21340	−0.120144
$103$	−11.8912	−1.17167	−0.585835	−	0.810430i	$-0.699234\pi$
$103$	−11.8912	−1.17167	−0.585835	+	0.810430i	$0.699234\pi$
$104$	−2.68872	−0.263651
$105$	31.7686	3.10030
$106$	−9.29648	−0.902954
$107$	−14.1112	−1.36419	−0.682093	−	0.731266i	$-0.738930\pi$
$107$	−14.1112	−1.36419	−0.682093	+	0.731266i	$0.738930\pi$
$108$	2.93532	0.282451
$109$	−19.5892	−1.87631	−0.938154	−	0.346219i	$-0.887465\pi$
$109$	−19.5892	−1.87631	−0.938154	+	0.346219i	$0.887465\pi$
$110$	−3.80220	−0.362526
$111$	1.56766	0.148796
$112$	3.88018	0.366642
$113$	−16.9320	−1.59283	−0.796415	−	0.604751i	$-0.793273\pi$
$113$	−16.9320	−1.59283	−0.796415	+	0.604751i	$0.793273\pi$
$114$	11.7203	1.09771
$115$	−24.1018	−2.24750
$116$	−6.39585	−0.593839
$117$	−4.40103	−0.406875
$118$	12.5695	1.15712
$119$	2.18647	0.200434
$120$	8.18741	0.747405
$121$	1.00000	0.0909091
$122$	−6.09043	−0.551402
$123$	−20.7881	−1.87440
$124$	−4.11453	−0.369495
$125$	−16.9454	−1.51564
$126$	6.35126	0.565816
$127$	−14.6425	−1.29931	−0.649657	−	0.760227i	$-0.725088\pi$
$127$	−14.6425	−1.29931	−0.649657	+	0.760227i	$0.725088\pi$
$128$	1.00000	0.0883883
$129$	0.955383	0.0841168
$130$	10.2231	0.896621
$131$	5.32404	0.465164	0.232582	−	0.972577i	$-0.425283\pi$
$131$	5.32404	0.465164	0.232582	+	0.972577i	$0.425283\pi$
$132$	−2.15333	−0.187424
$133$	−21.1193	−1.83128
$134$	16.0033	1.38247
$135$	−11.1607	−0.960558
$136$	0.563498	0.0483196
$137$	5.98353	0.511208	0.255604	−	0.966782i	$-0.417726\pi$
$137$	5.98353	0.511208	0.255604	+	0.966782i	$0.417726\pi$
$138$	−13.6498	−1.16195
$139$	−9.71273	−0.823823	−0.411911	−	0.911224i	$-0.635139\pi$
$139$	−9.71273	−0.823823	−0.411911	+	0.911224i	$0.635139\pi$
$140$	−14.7532	−1.24687
$141$	0.962181	0.0810303
$142$	9.48287	0.795785
$143$	−2.68872	−0.224842
$144$	1.63685	0.136404
$145$	24.3183	2.01952
$146$	−4.56247	−0.377592
$147$	−17.3468	−1.43074
$148$	−0.728015	−0.0598424
$149$	8.23936	0.674994	0.337497	−	0.941327i	$-0.390420\pi$
$149$	8.23936	0.674994	0.337497	+	0.941327i	$0.390420\pi$
$150$	−20.3635	−1.66267
$151$	7.20832	0.586605	0.293302	−	0.956020i	$-0.405246\pi$
$151$	7.20832	0.586605	0.293302	+	0.956020i	$0.405246\pi$
$152$	−5.44288	−0.441476
$153$	0.922361	0.0745685
$154$	3.88018	0.312674
$155$	15.6443	1.25658
$156$	5.78972	0.463548
$157$	−13.3421	−1.06482	−0.532408	−	0.846488i	$-0.678713\pi$
$157$	−13.3421	−1.06482	−0.532408	+	0.846488i	$0.678713\pi$
$158$	7.83601	0.623399
$159$	20.0184	1.58757
$160$	−3.80220	−0.300590
$161$	24.5961	1.93844
$162$	−11.2313	−0.882412
$163$	−0.0637832	−0.00499589	−0.00249794	−	0.999997i	$-0.500795\pi$
$163$	−0.0637832	−0.00499589	−0.00249794	+	0.999997i	$0.500795\pi$
$164$	9.65393	0.753846
$165$	8.18741	0.637389
$166$	1.69925	0.131888
$167$	−20.7930	−1.60901	−0.804505	−	0.593946i	$-0.797569\pi$
$167$	−20.7930	−1.60901	−0.804505	+	0.593946i	$0.797569\pi$
$168$	−8.35532	−0.644627
$169$	−5.77078	−0.443906
$170$	−2.14253	−0.164325
$171$	−8.90918	−0.681302
$172$	−0.443676	−0.0338300
$173$	−8.85822	−0.673478	−0.336739	−	0.941598i	$-0.609324\pi$
$173$	−8.85822	−0.673478	−0.336739	+	0.941598i	$0.609324\pi$
$174$	13.7724	1.04408
$175$	36.6938	2.77379
$176$	1.00000	0.0753778
$177$	−27.0663	−2.03443
$178$	−17.3736	−1.30221
$179$	7.90804	0.591075	0.295537	−	0.955331i	$-0.404501\pi$
$179$	7.90804	0.591075	0.295537	+	0.955331i	$0.404501\pi$
$180$	−6.22363	−0.463882
$181$	−2.52215	−0.187470	−0.0937351	−	0.995597i	$-0.529881\pi$
$181$	−2.52215	−0.187470	−0.0937351	+	0.995597i	$0.529881\pi$
$182$	−10.4327	−0.773324
$183$	13.1147	0.969469
$184$	6.33890	0.467310
$185$	2.76806	0.203512
$186$	8.85995	0.649643
$187$	0.563498	0.0412071
$188$	−0.446833	−0.0325886
$189$	11.3896	0.828468
$190$	20.6949	1.50137
$191$	−21.3401	−1.54412	−0.772058	−	0.635552i	$-0.780773\pi$
$191$	−21.3401	−1.54412	−0.772058	+	0.635552i	$0.780773\pi$
$192$	−2.15333	−0.155404
$193$	−15.9416	−1.14750	−0.573752	−	0.819029i	$-0.694513\pi$
$193$	−15.9416	−1.14750	−0.573752	+	0.819029i	$0.694513\pi$
$194$	−12.8995	−0.926132
$195$	−22.0137	−1.57643
$196$	8.05577	0.575412
$197$	−1.00000	−0.0712470
$197$	−1.00000	−0.0712470
$198$	1.63685	0.116326
$199$	17.5782	1.24608	0.623041	−	0.782189i	$-0.285897\pi$
$199$	17.5782	1.24608	0.623041	+	0.782189i	$0.285897\pi$
$200$	9.45673	0.668692
$201$	−34.4604	−2.43065
$202$	2.60524	0.183304
$203$	−24.8170	−1.74181
$204$	−1.21340	−0.0849550
$205$	−36.7062	−2.56367
$206$	−11.8912	−0.828496
$207$	10.3758	0.721170
$208$	−2.68872	−0.186429
$209$	−5.44288	−0.376492
$210$	31.7686	2.19224
$211$	23.9774	1.65067	0.825335	−	0.564643i	$-0.190986\pi$
$211$	23.9774	1.65067	0.825335	+	0.564643i	$0.190986\pi$
$212$	−9.29648	−0.638485
$213$	−20.4198	−1.39914
$214$	−14.1112	−0.964625
$215$	1.68695	0.115049
$216$	2.93532	0.199723
$217$	−15.9651	−1.08378
$218$	−19.5892	−1.32675
$219$	9.82451	0.663879
$220$	−3.80220	−0.256344
$221$	−1.51509	−0.101916
$222$	1.56766	0.105214
$223$	−26.1996	−1.75446	−0.877228	−	0.480074i	$-0.840610\pi$
$223$	−26.1996	−1.75446	−0.877228	+	0.480074i	$0.840610\pi$
$224$	3.88018	0.259255
$225$	15.4792	1.03195
$226$	−16.9320	−1.12630
$227$	7.09480	0.470899	0.235449	−	0.971887i	$-0.424344\pi$
$227$	7.09480	0.470899	0.235449	+	0.971887i	$0.424344\pi$
$228$	11.7203	0.776199
$229$	−16.0012	−1.05739	−0.528696	−	0.848811i	$-0.677319\pi$
$229$	−16.0012	−1.05739	−0.528696	+	0.848811i	$0.677319\pi$
$230$	−24.1018	−1.58922
$231$	−8.35532	−0.549740
$232$	−6.39585	−0.419908
$233$	8.23044	0.539194	0.269597	−	0.962973i	$-0.413110\pi$
$233$	8.23044	0.539194	0.269597	+	0.962973i	$0.413110\pi$
$234$	−4.40103	−0.287704
$235$	1.69895	0.110827
$236$	12.5695	0.818204
$237$	−16.8735	−1.09605
$238$	2.18647	0.141728
$239$	−2.74395	−0.177491	−0.0887456	−	0.996054i	$-0.528286\pi$
$239$	−2.74395	−0.177491	−0.0887456	+	0.996054i	$0.528286\pi$
$240$	8.18741	0.528495
$241$	7.64908	0.492720	0.246360	−	0.969178i	$-0.420765\pi$
$241$	7.64908	0.492720	0.246360	+	0.969178i	$0.420765\pi$
$242$	1.00000	0.0642824
$243$	15.3787	0.986546
$244$	−6.09043	−0.389900
$245$	−30.6296	−1.95686
$246$	−20.7881	−1.32540
$247$	14.6344	0.931164
$248$	−4.11453	−0.261273
$249$	−3.65906	−0.231883
$250$	−16.9454	−1.07172
$251$	−24.2051	−1.52781	−0.763906	−	0.645327i	$-0.776721\pi$
$251$	−24.2051	−1.52781	−0.763906	+	0.645327i	$0.776721\pi$
$252$	6.35126	0.400092
$253$	6.33890	0.398523
$254$	−14.6425	−0.918754
$255$	4.61359	0.288914
$256$	1.00000	0.0625000
$257$	26.1736	1.63267	0.816333	−	0.577582i	$-0.196004\pi$
$257$	26.1736	1.63267	0.816333	+	0.577582i	$0.196004\pi$
$258$	0.955383	0.0594796
$259$	−2.82482	−0.175526
$260$	10.2231	0.634007
$261$	−10.4690	−0.648017
$262$	5.32404	0.328921
$263$	−4.86364	−0.299905	−0.149952	−	0.988693i	$-0.547912\pi$
$263$	−4.86364	−0.299905	−0.149952	+	0.988693i	$0.547912\pi$
$264$	−2.15333	−0.132529
$265$	35.3471	2.17135
$266$	−21.1193	−1.29491
$267$	37.4112	2.28953
$268$	16.0033	0.977557
$269$	−25.9696	−1.58339	−0.791696	−	0.610915i	$-0.790802\pi$
$269$	−25.9696	−1.58339	−0.791696	+	0.610915i	$0.790802\pi$
$270$	−11.1607	−0.679217
$271$	20.7493	1.26043	0.630215	−	0.776420i	$-0.282967\pi$
$271$	20.7493	1.26043	0.630215	+	0.776420i	$0.282967\pi$
$272$	0.563498	0.0341671
$273$	22.4651	1.35965
$274$	5.98353	0.361479
$275$	9.45673	0.570262
$276$	−13.6498	−0.821620
$277$	−25.0990	−1.50805	−0.754027	−	0.656843i	$-0.771891\pi$
$277$	−25.0990	−1.50805	−0.754027	+	0.656843i	$0.771891\pi$
$278$	−9.71273	−0.582531
$279$	−6.73486	−0.403206
$280$	−14.7532	−0.881673
$281$	27.2598	1.62618	0.813092	−	0.582135i	$-0.197782\pi$
$281$	27.2598	1.62618	0.813092	+	0.582135i	$0.197782\pi$
$282$	0.962181	0.0572970
$283$	3.41998	0.203297	0.101648	−	0.994820i	$-0.467588\pi$
$283$	3.41998	0.203297	0.101648	+	0.994820i	$0.467588\pi$
$284$	9.48287	0.562705
$285$	−44.5631	−2.63969
$286$	−2.68872	−0.158987
$287$	37.4590	2.21113
$288$	1.63685	0.0964523
$289$	−16.6825	−0.981322
$290$	24.3183	1.42802
$291$	27.7770	1.62832
$292$	−4.56247	−0.266998
$293$	16.8697	0.985538	0.492769	−	0.870160i	$-0.335985\pi$
$293$	16.8697	0.985538	0.492769	+	0.870160i	$0.335985\pi$
$294$	−17.3468	−1.01168
$295$	−47.7917	−2.78254
$296$	−0.728015	−0.0423150
$297$	2.93532	0.170324
$298$	8.23936	0.477293
$299$	−17.0435	−0.985653
$300$	−20.3635	−1.17569
$301$	−1.72154	−0.0992280
$302$	7.20832	0.414792
$303$	−5.60996	−0.322284
$304$	−5.44288	−0.312171
$305$	23.1570	1.32597
$306$	0.922361	0.0527279
$307$	2.91428	0.166327	0.0831635	−	0.996536i	$-0.473498\pi$
$307$	2.91428	0.166327	0.0831635	+	0.996536i	$0.473498\pi$
$308$	3.88018	0.221094
$309$	25.6056	1.45665
$310$	15.6443	0.888534
$311$	−27.8975	−1.58192	−0.790961	−	0.611867i	$-0.790419\pi$
$311$	−27.8975	−1.58192	−0.790961	+	0.611867i	$0.790419\pi$
$312$	5.78972	0.327778
$313$	−25.9112	−1.46459	−0.732293	−	0.680990i	$-0.761550\pi$
$313$	−25.9112	−1.46459	−0.732293	+	0.680990i	$0.761550\pi$
$314$	−13.3421	−0.752938
$315$	−24.1488	−1.36063
$316$	7.83601	0.440810
$317$	14.5345	0.816338	0.408169	−	0.912906i	$-0.366167\pi$
$317$	14.5345	0.816338	0.408169	+	0.912906i	$0.366167\pi$
$318$	20.0184	1.12258
$319$	−6.39585	−0.358099
$320$	−3.80220	−0.212549
$321$	30.3862	1.69599
$322$	24.5961	1.37068
$323$	−3.06705	−0.170655
$324$	−11.2313	−0.623960
$325$	−25.4265	−1.41041
$326$	−0.0637832	−0.00353263
$327$	42.1822	2.33268
$328$	9.65393	0.533049
$329$	−1.73379	−0.0955870
$330$	8.18741	0.450702
$331$	19.3716	1.06476	0.532380	−	0.846506i	$-0.321298\pi$
$331$	19.3716	1.06476	0.532380	+	0.846506i	$0.321298\pi$
$332$	1.69925	0.0932586
$333$	−1.19165	−0.0653020
$334$	−20.7930	−1.13774
$335$	−60.8477	−3.32447
$336$	−8.35532	−0.455820
$337$	19.4074	1.05719	0.528593	−	0.848875i	$-0.322720\pi$
$337$	19.4074	1.05719	0.528593	+	0.848875i	$0.322720\pi$
$338$	−5.77078	−0.313889
$339$	36.4603	1.98025
$340$	−2.14253	−0.116195
$341$	−4.11453	−0.222814
$342$	−8.90918	−0.481753
$343$	4.09657	0.221194
$344$	−0.443676	−0.0239214
$345$	51.8992	2.79416
$346$	−8.85822	−0.476221
$347$	−11.1501	−0.598568	−0.299284	−	0.954164i	$-0.596748\pi$
$347$	−11.1501	−0.598568	−0.299284	+	0.954164i	$0.596748\pi$
$348$	13.7724	0.738278
$349$	−35.2488	−1.88683	−0.943413	−	0.331619i	$-0.892405\pi$
$349$	−35.2488	−1.88683	−0.943413	+	0.331619i	$0.892405\pi$
$350$	36.6938	1.96137
$351$	−7.89225	−0.421257
$352$	1.00000	0.0533002
$353$	−4.54455	−0.241882	−0.120941	−	0.992660i	$-0.538591\pi$
$353$	−4.54455	−0.241882	−0.120941	+	0.992660i	$0.538591\pi$
$354$	−27.0663	−1.43856
$355$	−36.0558	−1.91364
$356$	−17.3736	−0.920799
$357$	−4.70820	−0.249185
$358$	7.90804	0.417953
$359$	−28.4886	−1.50357	−0.751786	−	0.659407i	$-0.770808\pi$
$359$	−28.4886	−1.50357	−0.751786	+	0.659407i	$0.770808\pi$
$360$	−6.22363	−0.328014
$361$	10.6250	0.559209
$362$	−2.52215	−0.132561
$363$	−2.15333	−0.113021
$364$	−10.4327	−0.546823
$365$	17.3474	0.908005
$366$	13.1147	0.685518
$367$	16.7053	0.872012	0.436006	−	0.899944i	$-0.356393\pi$
$367$	16.7053	0.872012	0.436006	+	0.899944i	$0.356393\pi$
$368$	6.33890	0.330438
$369$	15.8020	0.822621
$370$	2.76806	0.143904
$371$	−36.0720	−1.87276
$372$	8.85995	0.459367
$373$	0.0363801	0.00188369	0.000941844	−	1.00000i	$-0.499700\pi$
$373$	0.0363801	0.00188369	0.000941844		1.00000i	$0.499700\pi$
$374$	0.563498	0.0291378
$375$	36.4891	1.88429
$376$	−0.446833	−0.0230437
$377$	17.1966	0.885672
$378$	11.3896	0.585816
$379$	−2.07942	−0.106813	−0.0534063	−	0.998573i	$-0.517008\pi$
$379$	−2.07942	−0.106813	−0.0534063	+	0.998573i	$0.517008\pi$
$380$	20.6949	1.06163
$381$	31.5303	1.61534
$382$	−21.3401	−1.09186
$383$	6.62722	0.338635	0.169317	−	0.985562i	$-0.445844\pi$
$383$	6.62722	0.338635	0.169317	+	0.985562i	$0.445844\pi$
$384$	−2.15333	−0.109887
$385$	−14.7532	−0.751893
$386$	−15.9416	−0.811408
$387$	−0.726231	−0.0369164
$388$	−12.8995	−0.654874
$389$	5.47664	0.277677	0.138838	−	0.990315i	$-0.455663\pi$
$389$	5.47664	0.277677	0.138838	+	0.990315i	$0.455663\pi$
$390$	−22.0137	−1.11470
$391$	3.57196	0.180642
$392$	8.05577	0.406878
$393$	−11.4644	−0.578305
$394$	−1.00000	−0.0503793
$395$	−29.7941	−1.49910
$396$	1.63685	0.0822548
$397$	10.3316	0.518528	0.259264	−	0.965806i	$-0.416520\pi$
$397$	10.3316	0.518528	0.259264	+	0.965806i	$0.416520\pi$
$398$	17.5782	0.881113
$399$	45.4770	2.27670
$400$	9.45673	0.472836
$401$	1.86641	0.0932042	0.0466021	−	0.998914i	$-0.485161\pi$
$401$	1.86641	0.0932042	0.0466021	+	0.998914i	$0.485161\pi$
$402$	−34.4604	−1.71873
$403$	11.0628	0.551078
$404$	2.60524	0.129616
$405$	42.7036	2.12196
$406$	−24.8170	−1.23165
$407$	−0.728015	−0.0360863
$408$	−1.21340	−0.0600722
$409$	0.780681	0.0386022	0.0193011	−	0.999814i	$-0.493856\pi$
$409$	0.780681	0.0386022	0.0193011	+	0.999814i	$0.493856\pi$
$410$	−36.7062	−1.81279
$411$	−12.8846	−0.635548
$412$	−11.8912	−0.585835
$413$	48.7718	2.39991
$414$	10.3758	0.509944
$415$	−6.46090	−0.317153
$416$	−2.68872	−0.131825
$417$	20.9148	1.02420
$418$	−5.44288	−0.266220
$419$	−7.79385	−0.380755	−0.190377	−	0.981711i	$-0.560971\pi$
$419$	−7.79385	−0.380755	−0.190377	+	0.981711i	$0.560971\pi$
$420$	31.7686	1.55015
$421$	29.4313	1.43439	0.717196	−	0.696872i	$-0.245425\pi$
$421$	29.4313	1.43439	0.717196	+	0.696872i	$0.245425\pi$
$422$	23.9774	1.16720
$423$	−0.731398	−0.0355618
$424$	−9.29648	−0.451477
$425$	5.32885	0.258487
$426$	−20.4198	−0.989342
$427$	−23.6319	−1.14363
$428$	−14.1112	−0.682093
$429$	5.78972	0.279530
$430$	1.68695	0.0813517
$431$	31.5949	1.52187	0.760936	−	0.648827i	$-0.224740\pi$
$431$	31.5949	1.52187	0.760936	+	0.648827i	$0.224740\pi$
$432$	2.93532	0.141226
$433$	−22.5636	−1.08434	−0.542169	−	0.840269i	$-0.682397\pi$
$433$	−22.5636	−1.08434	−0.542169	+	0.840269i	$0.682397\pi$
$434$	−15.9651	−0.766349
$435$	−52.3654	−2.51073
$436$	−19.5892	−0.938154
$437$	−34.5019	−1.65045
$438$	9.82451	0.469433
$439$	−25.3879	−1.21170	−0.605848	−	0.795580i	$-0.707166\pi$
$439$	−25.3879	−1.21170	−0.605848	+	0.795580i	$0.707166\pi$
$440$	−3.80220	−0.181263
$441$	13.1861	0.627909
$442$	−1.51509	−0.0720654
$443$	−6.92706	−0.329114	−0.164557	−	0.986368i	$-0.552620\pi$
$443$	−6.92706	−0.329114	−0.164557	+	0.986368i	$0.552620\pi$
$444$	1.56766	0.0743978
$445$	66.0580	3.13145
$446$	−26.1996	−1.24059
$447$	−17.7421	−0.839172
$448$	3.88018	0.183321
$449$	1.60767	0.0758707	0.0379353	−	0.999280i	$-0.487922\pi$
$449$	1.60767	0.0758707	0.0379353	+	0.999280i	$0.487922\pi$
$450$	15.4792	0.729699
$451$	9.65393	0.454586
$452$	−16.9320	−0.796415
$453$	−15.5219	−0.729284
$454$	7.09480	0.332976
$455$	39.6673	1.85963
$456$	11.7203	0.548855
$457$	−25.4706	−1.19146	−0.595732	−	0.803183i	$-0.703138\pi$
$457$	−25.4706	−1.19146	−0.595732	+	0.803183i	$0.703138\pi$
$458$	−16.0012	−0.747689
$459$	1.65405	0.0772043
$460$	−24.1018	−1.12375
$461$	−9.10573	−0.424096	−0.212048	−	0.977259i	$-0.568013\pi$
$461$	−9.10573	−0.424096	−0.212048	+	0.977259i	$0.568013\pi$
$462$	−8.35532	−0.388725
$463$	24.4655	1.13701	0.568504	−	0.822681i	$-0.307522\pi$
$463$	24.4655	1.13701	0.568504	+	0.822681i	$0.307522\pi$
$464$	−6.39585	−0.296920
$465$	−33.6873	−1.56221
$466$	8.23044	0.381268
$467$	27.7745	1.28525	0.642626	−	0.766180i	$-0.277845\pi$
$467$	27.7745	1.28525	0.642626	+	0.766180i	$0.277845\pi$
$468$	−4.40103	−0.203438
$469$	62.0956	2.86731
$470$	1.69895	0.0783667
$471$	28.7300	1.32381
$472$	12.5695	0.578558
$473$	−0.443676	−0.0204003
$474$	−16.8735	−0.775027
$475$	−51.4719	−2.36169
$476$	2.18647	0.100217
$477$	−15.2169	−0.696736
$478$	−2.74395	−0.125505
$479$	31.0522	1.41881	0.709405	−	0.704801i	$-0.248964\pi$
$479$	31.0522	1.41881	0.709405	+	0.704801i	$0.248964\pi$
$480$	8.18741	0.373702
$481$	1.95743	0.0892510
$482$	7.64908	0.348406
$483$	−52.9635	−2.40992
$484$	1.00000	0.0454545
$485$	49.0466	2.22709
$486$	15.3787	0.697593
$487$	−28.6694	−1.29913	−0.649567	−	0.760305i	$-0.725050\pi$
$487$	−28.6694	−1.29913	−0.649567	+	0.760305i	$0.725050\pi$
$488$	−6.09043	−0.275701
$489$	0.137347	0.00621103
$490$	−30.6296	−1.38371
$491$	−36.9246	−1.66638	−0.833192	−	0.552984i	$-0.813489\pi$
$491$	−36.9246	−1.66638	−0.833192	+	0.552984i	$0.813489\pi$
$492$	−20.7881	−0.937202
$493$	−3.60405	−0.162318
$494$	14.6344	0.658432
$495$	−6.22363	−0.279731
$496$	−4.11453	−0.184748
$497$	36.7952	1.65049
$498$	−3.65906	−0.163966
$499$	−35.1112	−1.57179	−0.785896	−	0.618359i	$-0.787798\pi$
$499$	−35.1112	−1.57179	−0.785896	+	0.618359i	$0.787798\pi$
$500$	−16.9454	−0.757820
$501$	44.7742	2.00037
$502$	−24.2051	−1.08033
$503$	19.4413	0.866845	0.433422	−	0.901191i	$-0.357306\pi$
$503$	19.4413	0.866845	0.433422	+	0.901191i	$0.357306\pi$
$504$	6.35126	0.282908
$505$	−9.90565	−0.440796
$506$	6.33890	0.281799
$507$	12.4264	0.551877
$508$	−14.6425	−0.649657
$509$	16.1921	0.717702	0.358851	−	0.933395i	$-0.383169\pi$
$509$	16.1921	0.717702	0.358851	+	0.933395i	$0.383169\pi$
$510$	4.61359	0.204293
$511$	−17.7032	−0.783142
$512$	1.00000	0.0441942
$513$	−15.9766	−0.705384
$514$	26.1736	1.15447
$515$	45.2126	1.99230
$516$	0.955383	0.0420584
$517$	−0.446833	−0.0196517
$518$	−2.82482	−0.124116
$519$	19.0747	0.837287
$520$	10.2231	0.448311
$521$	−8.14589	−0.356878	−0.178439	−	0.983951i	$-0.557105\pi$
$521$	−8.14589	−0.356878	−0.178439	+	0.983951i	$0.557105\pi$
$522$	−10.4690	−0.458217
$523$	−21.2085	−0.927384	−0.463692	−	0.885997i	$-0.653475\pi$
$523$	−21.2085	−0.927384	−0.463692	+	0.885997i	$0.653475\pi$
$524$	5.32404	0.232582
$525$	−79.0140	−3.44845
$526$	−4.86364	−0.212065
$527$	−2.31853	−0.100997
$528$	−2.15333	−0.0937119
$529$	17.1817	0.747029
$530$	35.3471	1.53538
$531$	20.5744	0.892851
$532$	−21.1193	−0.915640
$533$	−25.9567	−1.12431
$534$	37.4112	1.61894
$535$	53.6538	2.31966
$536$	16.0033	0.691237
$537$	−17.0287	−0.734841
$538$	−25.9696	−1.11963
$539$	8.05577	0.346987
$540$	−11.1607	−0.480279
$541$	−38.5903	−1.65913	−0.829563	−	0.558413i	$-0.811410\pi$
$541$	−38.5903	−1.65913	−0.829563	+	0.558413i	$0.811410\pi$
$542$	20.7493	0.891259
$543$	5.43104	0.233068
$544$	0.563498	0.0241598
$545$	74.4822	3.19047
$546$	22.4651	0.961418
$547$	−26.2767	−1.12351	−0.561755	−	0.827304i	$-0.689874\pi$
$547$	−26.2767	−1.12351	−0.561755	+	0.827304i	$0.689874\pi$
$548$	5.98353	0.255604
$549$	−9.96912	−0.425472
$550$	9.45673	0.403236
$551$	34.8118	1.48303
$552$	−13.6498	−0.580973
$553$	30.4051	1.29296
$554$	−25.0990	−1.06636
$555$	−5.96055	−0.253011
$556$	−9.71273	−0.411911
$557$	32.0927	1.35981	0.679905	−	0.733300i	$-0.262021\pi$
$557$	32.0927	1.35981	0.679905	+	0.733300i	$0.262021\pi$
$558$	−6.73486	−0.285109
$559$	1.19292	0.0504552
$560$	−14.7532	−0.623437
$561$	−1.21340	−0.0512298
$562$	27.2598	1.14989
$563$	6.64976	0.280254	0.140127	−	0.990134i	$-0.455249\pi$
$563$	6.64976	0.280254	0.140127	+	0.990134i	$0.455249\pi$
$564$	0.962181	0.0405151
$565$	64.3789	2.70844
$566$	3.41998	0.143753
$567$	−43.5793	−1.83016
$568$	9.48287	0.397893
$569$	4.02131	0.168582	0.0842911	−	0.996441i	$-0.473137\pi$
$569$	4.02131	0.168582	0.0842911	+	0.996441i	$0.473137\pi$
$570$	−44.5631	−1.86654
$571$	−14.1123	−0.590583	−0.295291	−	0.955407i	$-0.595417\pi$
$571$	−14.1123	−0.590583	−0.295291	+	0.955407i	$0.595417\pi$
$572$	−2.68872	−0.112421
$573$	45.9524	1.91969
$574$	37.4590	1.56351
$575$	59.9453	2.49989
$576$	1.63685	0.0682021
$577$	42.2665	1.75958	0.879790	−	0.475363i	$-0.157683\pi$
$577$	42.2665	1.75958	0.879790	+	0.475363i	$0.157683\pi$
$578$	−16.6825	−0.693899
$579$	34.3277	1.42661
$580$	24.3183	1.00976
$581$	6.59340	0.273540
$582$	27.7770	1.15139
$583$	−9.29648	−0.385021
$584$	−4.56247	−0.188796
$585$	16.7336	0.691849
$586$	16.8697	0.696881
$587$	26.3356	1.08699	0.543493	−	0.839414i	$-0.317101\pi$
$587$	26.3356	1.08699	0.543493	+	0.839414i	$0.317101\pi$
$588$	−17.3468	−0.715369
$589$	22.3949	0.922765
$590$	−47.7917	−1.96755
$591$	2.15333	0.0885763
$592$	−0.728015	−0.0299212
$593$	28.1439	1.15573	0.577866	−	0.816132i	$-0.303886\pi$
$593$	28.1439	1.15573	0.577866	+	0.816132i	$0.303886\pi$
$594$	2.93532	0.120438
$595$	−8.31340	−0.340816
$596$	8.23936	0.337497
$597$	−37.8517	−1.54917
$598$	−17.0435	−0.696962
$599$	20.0936	0.821001	0.410501	−	0.911860i	$-0.365354\pi$
$599$	20.0936	0.821001	0.410501	+	0.911860i	$0.365354\pi$
$600$	−20.3635	−0.831337
$601$	−31.5850	−1.28838	−0.644189	−	0.764866i	$-0.722805\pi$
$601$	−31.5850	−1.28838	−0.644189	+	0.764866i	$0.722805\pi$
$602$	−1.72154	−0.0701648
$603$	26.1950	1.06674
$604$	7.20832	0.293302
$605$	−3.80220	−0.154581
$606$	−5.60996	−0.227889
$607$	18.1606	0.737117	0.368559	−	0.929605i	$-0.379851\pi$
$607$	18.1606	0.737117	0.368559	+	0.929605i	$0.379851\pi$
$608$	−5.44288	−0.220738
$609$	53.4393	2.16547
$610$	23.1570	0.937601
$611$	1.20141	0.0486038
$612$	0.922361	0.0372842
$613$	−5.27768	−0.213164	−0.106582	−	0.994304i	$-0.533991\pi$
$613$	−5.27768	−0.213164	−0.106582	+	0.994304i	$0.533991\pi$
$614$	2.91428	0.117611
$615$	79.0407	3.18723
$616$	3.88018	0.156337
$617$	18.2434	0.734450	0.367225	−	0.930132i	$-0.380308\pi$
$617$	18.2434	0.734450	0.367225	+	0.930132i	$0.380308\pi$
$618$	25.6056	1.03001
$619$	−26.9660	−1.08386	−0.541928	−	0.840425i	$-0.682305\pi$
$619$	−26.9660	−1.08386	−0.541928	+	0.840425i	$0.682305\pi$
$620$	15.6443	0.628289
$621$	18.6067	0.746661
$622$	−27.8975	−1.11859
$623$	−67.4127	−2.70083
$624$	5.78972	0.231774
$625$	17.1461	0.685844
$626$	−25.9112	−1.03562
$627$	11.7203	0.468066
$628$	−13.3421	−0.532408
$629$	−0.410235	−0.0163571
$630$	−24.1488	−0.962111
$631$	−18.8252	−0.749419	−0.374710	−	0.927142i	$-0.622258\pi$
$631$	−18.8252	−0.749419	−0.374710	+	0.927142i	$0.622258\pi$
$632$	7.83601	0.311700
$633$	−51.6313	−2.05216
$634$	14.5345	0.577238
$635$	55.6738	2.20935
$636$	20.0184	0.793783
$637$	−21.6597	−0.858189
$638$	−6.39585	−0.253214
$639$	15.5220	0.614042
$640$	−3.80220	−0.150295
$641$	34.3141	1.35532	0.677662	−	0.735373i	$-0.262993\pi$
$641$	34.3141	1.35532	0.677662	+	0.735373i	$0.262993\pi$
$642$	30.3862	1.19925
$643$	33.6326	1.32634	0.663170	−	0.748469i	$-0.269211\pi$
$643$	33.6326	1.32634	0.663170	+	0.748469i	$0.269211\pi$
$644$	24.5961	0.969220
$645$	−3.63256	−0.143032
$646$	−3.06705	−0.120672
$647$	−19.7859	−0.777863	−0.388932	−	0.921267i	$-0.627156\pi$
$647$	−19.7859	−0.777863	−0.388932	+	0.921267i	$0.627156\pi$
$648$	−11.2313	−0.441206
$649$	12.5695	0.493396
$650$	−25.4265	−0.997310
$651$	34.3782	1.34739
$652$	−0.0637832	−0.00249794
$653$	14.3537	0.561702	0.280851	−	0.959751i	$-0.409383\pi$
$653$	14.3537	0.561702	0.280851	+	0.959751i	$0.409383\pi$
$654$	42.1822	1.64945
$655$	−20.2431	−0.790963
$656$	9.65393	0.376923
$657$	−7.46807	−0.291357
$658$	−1.73379	−0.0675902
$659$	31.7827	1.23808	0.619039	−	0.785360i	$-0.287522\pi$
$659$	31.7827	1.23808	0.619039	+	0.785360i	$0.287522\pi$
$660$	8.18741	0.318694
$661$	6.55332	0.254894	0.127447	−	0.991845i	$-0.459322\pi$
$661$	6.55332	0.254894	0.127447	+	0.991845i	$0.459322\pi$
$662$	19.3716	0.752898
$663$	3.26249	0.126705
$664$	1.69925	0.0659438
$665$	80.3000	3.11390
$666$	−1.19165	−0.0461755
$667$	−40.5426	−1.56982
$668$	−20.7930	−0.804505
$669$	56.4165	2.18119
$670$	−60.8477	−2.35075
$671$	−6.09043	−0.235118
$672$	−8.35532	−0.322313
$673$	6.86269	0.264537	0.132269	−	0.991214i	$-0.457774\pi$
$673$	6.86269	0.264537	0.132269	+	0.991214i	$0.457774\pi$
$674$	19.4074	0.747544
$675$	27.7585	1.06843
$676$	−5.77078	−0.221953
$677$	5.04769	0.193999	0.0969993	−	0.995284i	$-0.469076\pi$
$677$	5.04769	0.193999	0.0969993	+	0.995284i	$0.469076\pi$
$678$	36.4603	1.40025
$679$	−50.0524	−1.92084
$680$	−2.14253	−0.0821624
$681$	−15.2775	−0.585434
$682$	−4.11453	−0.157553
$683$	9.87636	0.377909	0.188954	−	0.981986i	$-0.439490\pi$
$683$	9.87636	0.377909	0.188954	+	0.981986i	$0.439490\pi$
$684$	−8.90918	−0.340651
$685$	−22.7506	−0.869256
$686$	4.09657	0.156408
$687$	34.4560	1.31458
$688$	−0.443676	−0.0169150
$689$	24.9956	0.952258
$690$	51.8992	1.97577
$691$	−22.4253	−0.853099	−0.426550	−	0.904464i	$-0.640271\pi$
$691$	−22.4253	−0.853099	−0.426550	+	0.904464i	$0.640271\pi$
$692$	−8.85822	−0.336739
$693$	6.35126	0.241265
$694$	−11.1501	−0.423252
$695$	36.9297	1.40082
$696$	13.7724	0.522041
$697$	5.43997	0.206054
$698$	−35.2488	−1.33419
$699$	−17.7229	−0.670341
$700$	36.6938	1.38689
$701$	−6.75576	−0.255161	−0.127581	−	0.991828i	$-0.540721\pi$
$701$	−6.75576	−0.255161	−0.127581	+	0.991828i	$0.540721\pi$
$702$	−7.89225	−0.297874
$703$	3.96250	0.149448
$704$	1.00000	0.0376889
$705$	−3.65841	−0.137784
$706$	−4.54455	−0.171036
$707$	10.1088	0.380180
$708$	−27.0663	−1.01721
$709$	30.1800	1.13343	0.566716	−	0.823913i	$-0.308214\pi$
$709$	30.1800	1.13343	0.566716	+	0.823913i	$0.308214\pi$
$710$	−36.0558	−1.35315
$711$	12.8264	0.481026
$712$	−17.3736	−0.651104
$713$	−26.0816	−0.976763
$714$	−4.70820	−0.176200
$715$	10.2231	0.382321
$716$	7.90804	0.295537
$717$	5.90863	0.220662
$718$	−28.4886	−1.06319
$719$	−22.9575	−0.856171	−0.428086	−	0.903738i	$-0.640812\pi$
$719$	−22.9575	−0.856171	−0.428086	+	0.903738i	$0.640812\pi$
$720$	−6.22363	−0.231941
$721$	−46.1398	−1.71834
$722$	10.6250	0.395420
$723$	−16.4710	−0.612564
$724$	−2.52215	−0.0937351
$725$	−60.4838	−2.24631
$726$	−2.15333	−0.0799177
$727$	−24.3974	−0.904851	−0.452425	−	0.891802i	$-0.649441\pi$
$727$	−24.3974	−0.904851	−0.452425	+	0.891802i	$0.649441\pi$
$728$	−10.4327	−0.386662
$729$	0.578269	0.0214174
$730$	17.3474	0.642056
$731$	−0.250011	−0.00924698
$732$	13.1147	0.484735
$733$	−6.97759	−0.257723	−0.128861	−	0.991663i	$-0.541132\pi$
$733$	−6.97759	−0.257723	−0.128861	+	0.991663i	$0.541132\pi$
$734$	16.7053	0.616606
$735$	65.9559	2.43282
$736$	6.33890	0.233655
$737$	16.0033	0.589489
$738$	15.8020	0.581681
$739$	−18.9320	−0.696425	−0.348213	−	0.937416i	$-0.613211\pi$
$739$	−18.9320	−0.696425	−0.348213	+	0.937416i	$0.613211\pi$
$740$	2.76806	0.101756
$741$	−31.5127	−1.15765
$742$	−36.0720	−1.32424
$743$	−14.2519	−0.522852	−0.261426	−	0.965224i	$-0.584193\pi$
$743$	−14.2519	−0.522852	−0.261426	+	0.965224i	$0.584193\pi$
$744$	8.85995	0.324822
$745$	−31.3277	−1.14776
$746$	0.0363801	0.00133197
$747$	2.78142	0.101767
$748$	0.563498	0.0206035
$749$	−54.7541	−2.00067
$750$	36.4891	1.33239
$751$	16.7787	0.612264	0.306132	−	0.951989i	$-0.400965\pi$
$751$	16.7787	0.612264	0.306132	+	0.951989i	$0.400965\pi$
$752$	−0.446833	−0.0162943
$753$	52.1217	1.89942
$754$	17.1966	0.626265
$755$	−27.4075	−0.997461
$756$	11.3896	0.414234
$757$	6.50806	0.236539	0.118270	−	0.992982i	$-0.462265\pi$
$757$	6.50806	0.236539	0.118270	+	0.992982i	$0.462265\pi$
$758$	−2.07942	−0.0755279
$759$	−13.6498	−0.495455
$760$	20.6949	0.750684
$761$	−21.2259	−0.769438	−0.384719	−	0.923034i	$-0.625702\pi$
$761$	−21.2259	−0.769438	−0.384719	+	0.923034i	$0.625702\pi$
$762$	31.5303	1.14222
$763$	−76.0096	−2.75173
$764$	−21.3401	−0.772058
$765$	−3.50700	−0.126796
$766$	6.62722	0.239451
$767$	−33.7959	−1.22030
$768$	−2.15333	−0.0777018
$769$	−31.1149	−1.12203	−0.561015	−	0.827805i	$-0.689589\pi$
$769$	−31.1149	−1.12203	−0.561015	+	0.827805i	$0.689589\pi$
$770$	−14.7532	−0.531669
$771$	−56.3606	−2.02978
$772$	−15.9416	−0.573752
$773$	22.2010	0.798514	0.399257	−	0.916839i	$-0.369268\pi$
$773$	22.2010	0.798514	0.399257	+	0.916839i	$0.369268\pi$
$774$	−0.726231	−0.0261038
$775$	−38.9100	−1.39769
$776$	−12.8995	−0.463066
$777$	6.08279	0.218219
$778$	5.47664	0.196347
$779$	−52.5452	−1.88263
$780$	−22.0137	−0.788215
$781$	9.48287	0.339324
$782$	3.57196	0.127733
$783$	−18.7738	−0.670923
$784$	8.05577	0.287706
$785$	50.7293	1.81061
$786$	−11.4644	−0.408923
$787$	46.6180	1.66175	0.830875	−	0.556459i	$-0.187840\pi$
$787$	46.6180	1.66175	0.830875	+	0.556459i	$0.187840\pi$
$788$	−1.00000	−0.0356235
$789$	10.4730	0.372850
$790$	−29.7941	−1.06003
$791$	−65.6992	−2.33599
$792$	1.63685	0.0581629
$793$	16.3755	0.581510
$794$	10.3316	0.366655
$795$	−76.1141	−2.69949
$796$	17.5782	0.623041
$797$	24.8705	0.880960	0.440480	−	0.897762i	$-0.354808\pi$
$797$	24.8705	0.880960	0.440480	+	0.897762i	$0.354808\pi$
$798$	45.4770	1.60987
$799$	−0.251790	−0.00890767
$800$	9.45673	0.334346
$801$	−28.4380	−1.00481
$802$	1.86641	0.0659053
$803$	−4.56247	−0.161006
$804$	−34.4604	−1.21533
$805$	−93.5191	−3.29612
$806$	11.0628	0.389671
$807$	55.9212	1.96852
$808$	2.60524	0.0916521
$809$	−3.84598	−0.135217	−0.0676087	−	0.997712i	$-0.521537\pi$
$809$	−3.84598	−0.135217	−0.0676087	+	0.997712i	$0.521537\pi$
$810$	42.7036	1.50045
$811$	−16.5602	−0.581508	−0.290754	−	0.956798i	$-0.593906\pi$
$811$	−16.5602	−0.581508	−0.290754	+	0.956798i	$0.593906\pi$
$812$	−24.8170	−0.870906
$813$	−44.6802	−1.56700
$814$	−0.728015	−0.0255169
$815$	0.242517	0.00849499
$816$	−1.21340	−0.0424775
$817$	2.41488	0.0844858
$818$	0.780681	0.0272959
$819$	−17.0768	−0.596711
$820$	−36.7062	−1.28184
$821$	29.4918	1.02927	0.514636	−	0.857409i	$-0.327927\pi$
$821$	29.4918	1.02927	0.514636	+	0.857409i	$0.327927\pi$
$822$	−12.8846	−0.449400
$823$	−29.4938	−1.02809	−0.514045	−	0.857763i	$-0.671854\pi$
$823$	−29.4938	−1.02809	−0.514045	+	0.857763i	$0.671854\pi$
$824$	−11.8912	−0.414248
$825$	−20.3635	−0.708966
$826$	48.7718	1.69699
$827$	21.0141	0.730732	0.365366	−	0.930864i	$-0.380944\pi$
$827$	21.0141	0.730732	0.365366	+	0.930864i	$0.380944\pi$
$828$	10.3758	0.360585
$829$	−0.615721	−0.0213849	−0.0106924	−	0.999943i	$-0.503404\pi$
$829$	−0.615721	−0.0213849	−0.0106924	+	0.999943i	$0.503404\pi$
$830$	−6.46090	−0.224261
$831$	54.0466	1.87486
$832$	−2.68872	−0.0932146
$833$	4.53941	0.157281
$834$	20.9148	0.724219
$835$	79.0591	2.73595
$836$	−5.44288	−0.188246
$837$	−12.0774	−0.417458
$838$	−7.79385	−0.269234
$839$	−9.21553	−0.318155	−0.159078	−	0.987266i	$-0.550852\pi$
$839$	−9.21553	−0.318155	−0.159078	+	0.987266i	$0.550852\pi$
$840$	31.7686	1.09612
$841$	11.9068	0.410581
$842$	29.4313	1.01427
$843$	−58.6995	−2.02172
$844$	23.9774	0.825335
$845$	21.9417	0.754816
$846$	−0.731398	−0.0251460
$847$	3.88018	0.133324
$848$	−9.29648	−0.319243
$849$	−7.36436	−0.252744
$850$	5.32885	0.182778
$851$	−4.61481	−0.158194
$852$	−20.4198	−0.699571
$853$	−22.0781	−0.755940	−0.377970	−	0.925818i	$-0.623378\pi$
$853$	−22.0781	−0.755940	−0.377970	+	0.925818i	$0.623378\pi$
$854$	−23.6319	−0.808669
$855$	33.8745	1.15848
$856$	−14.1112	−0.482312
$857$	−19.1747	−0.654994	−0.327497	−	0.944852i	$-0.606205\pi$
$857$	−19.1747	−0.654994	−0.327497	+	0.944852i	$0.606205\pi$
$858$	5.78972	0.197658
$859$	28.8956	0.985904	0.492952	−	0.870056i	$-0.335918\pi$
$859$	28.8956	0.985904	0.492952	+	0.870056i	$0.335918\pi$
$860$	1.68695	0.0575244
$861$	−80.6617	−2.74894
$862$	31.5949	1.07613
$863$	10.2037	0.347337	0.173669	−	0.984804i	$-0.444438\pi$
$863$	10.2037	0.347337	0.173669	+	0.984804i	$0.444438\pi$
$864$	2.93532	0.0998616
$865$	33.6807	1.14518
$866$	−22.5636	−0.766743
$867$	35.9229	1.22001
$868$	−15.9651	−0.541891
$869$	7.83601	0.265818
$870$	−52.3654	−1.77535
$871$	−43.0284	−1.45796
$872$	−19.5892	−0.663375
$873$	−21.1146	−0.714620
$874$	−34.5019	−1.16704
$875$	−65.7511	−2.22279
$876$	9.82451	0.331940
$877$	−2.96698	−0.100188	−0.0500939	−	0.998745i	$-0.515952\pi$
$877$	−2.96698	−0.100188	−0.0500939	+	0.998745i	$0.515952\pi$
$878$	−25.3879	−0.856798
$879$	−36.3261	−1.22525
$880$	−3.80220	−0.128172
$881$	−37.0640	−1.24872	−0.624359	−	0.781138i	$-0.714640\pi$
$881$	−37.0640	−1.24872	−0.624359	+	0.781138i	$0.714640\pi$
$882$	13.1861	0.443998
$883$	−9.31762	−0.313563	−0.156782	−	0.987633i	$-0.550112\pi$
$883$	−9.31762	−0.313563	−0.156782	+	0.987633i	$0.550112\pi$
$884$	−1.51509	−0.0509579
$885$	102.912	3.45933
$886$	−6.92706	−0.232719
$887$	53.3534	1.79143	0.895716	−	0.444626i	$-0.146663\pi$
$887$	53.3534	1.79143	0.895716	+	0.444626i	$0.146663\pi$
$888$	1.56766	0.0526072
$889$	−56.8156	−1.90553
$890$	66.0580	2.21427
$891$	−11.2313	−0.376262
$892$	−26.1996	−0.877228
$893$	2.43206	0.0813858
$894$	−17.7421	−0.593384
$895$	−30.0680	−1.00506
$896$	3.88018	0.129628
$897$	36.7004	1.22539
$898$	1.60767	0.0536487
$899$	26.3159	0.877684
$900$	15.4792	0.515975
$901$	−5.23855	−0.174521
$902$	9.65393	0.321441
$903$	3.70705	0.123363
$904$	−16.9320	−0.563150
$905$	9.58973	0.318774
$906$	−15.5219	−0.515682
$907$	−27.3750	−0.908971	−0.454485	−	0.890754i	$-0.650177\pi$
$907$	−27.3750	−0.908971	−0.454485	+	0.890754i	$0.650177\pi$
$908$	7.09480	0.235449
$909$	4.26439	0.141441
$910$	39.6673	1.31496
$911$	−33.0620	−1.09539	−0.547696	−	0.836677i	$-0.684495\pi$
$911$	−33.0620	−1.09539	−0.547696	+	0.836677i	$0.684495\pi$
$912$	11.7203	0.388099
$913$	1.69925	0.0562370
$914$	−25.4706	−0.842493
$915$	−49.8649	−1.64848
$916$	−16.0012	−0.528696
$917$	20.6582	0.682195
$918$	1.65405	0.0545917
$919$	0.0602297	0.00198680	0.000993398	−	1.00000i	$-0.499684\pi$
$919$	0.0602297	0.00198680	0.000993398		1.00000i	$0.499684\pi$
$920$	−24.1018	−0.794612
$921$	−6.27543	−0.206782
$922$	−9.10573	−0.299881
$923$	−25.4968	−0.839237
$924$	−8.35532	−0.274870
$925$	−6.88464	−0.226365
$926$	24.4655	0.803986
$927$	−19.4640	−0.639283
$928$	−6.39585	−0.209954
$929$	−25.6981	−0.843129	−0.421564	−	0.906798i	$-0.638519\pi$
$929$	−25.6981	−0.843129	−0.421564	+	0.906798i	$0.638519\pi$
$930$	−33.6873	−1.10465
$931$	−43.8466	−1.43701
$932$	8.23044	0.269597
$933$	60.0726	1.96669
$934$	27.7745	0.908811
$935$	−2.14253	−0.0700683
$936$	−4.40103	−0.143852
$937$	−26.5434	−0.867135	−0.433568	−	0.901121i	$-0.642745\pi$
$937$	−26.5434	−0.867135	−0.433568	+	0.901121i	$0.642745\pi$
$938$	62.0956	2.02749
$939$	55.7954	1.82081
$940$	1.69895	0.0554136
$941$	4.15031	0.135296	0.0676481	−	0.997709i	$-0.478450\pi$
$941$	4.15031	0.135296	0.0676481	+	0.997709i	$0.478450\pi$
$942$	28.7300	0.936074
$943$	61.1953	1.99279
$944$	12.5695	0.409102
$945$	−43.3054	−1.40872
$946$	−0.443676	−0.0144252
$947$	26.7210	0.868317	0.434158	−	0.900837i	$-0.357046\pi$
$947$	26.7210	0.868317	0.434158	+	0.900837i	$0.357046\pi$
$948$	−16.8735	−0.548027
$949$	12.2672	0.398210
$950$	−51.4719	−1.66997
$951$	−31.2976	−1.01489
$952$	2.18647	0.0708639
$953$	−48.8470	−1.58231	−0.791156	−	0.611615i	$-0.790520\pi$
$953$	−48.8470	−1.58231	−0.791156	+	0.611615i	$0.790520\pi$
$954$	−15.2169	−0.492667
$955$	81.1394	2.62561
$956$	−2.74395	−0.0887456
$957$	13.7724	0.445198
$958$	31.0522	1.00325
$959$	23.2172	0.749722
$960$	8.18741	0.264248
$961$	−14.0707	−0.453892
$962$	1.95743	0.0631100
$963$	−23.0980	−0.744322
$964$	7.64908	0.246360
$965$	60.6133	1.95121
$966$	−52.9635	−1.70407
$967$	−46.8843	−1.50770	−0.753848	−	0.657049i	$-0.771805\pi$
$967$	−46.8843	−1.50770	−0.753848	+	0.657049i	$0.771805\pi$
$968$	1.00000	0.0321412
$969$	6.60439	0.212164
$970$	49.0466	1.57479
$971$	−6.43244	−0.206427	−0.103213	−	0.994659i	$-0.532912\pi$
$971$	−6.43244	−0.206427	−0.103213	+	0.994659i	$0.532912\pi$
$972$	15.3787	0.493273
$973$	−37.6871	−1.20819
$974$	−28.6694	−0.918626
$975$	54.7518	1.75346
$976$	−6.09043	−0.194950
$977$	3.31262	0.105980	0.0529900	−	0.998595i	$-0.483125\pi$
$977$	3.31262	0.105980	0.0529900	+	0.998595i	$0.483125\pi$
$978$	0.137347	0.00439186
$979$	−17.3736	−0.555263
$980$	−30.6296	−0.978428
$981$	−32.0646	−1.02374
$982$	−36.9246	−1.17831
$983$	−1.03594	−0.0330412	−0.0165206	−	0.999864i	$-0.505259\pi$
$983$	−1.03594	−0.0330412	−0.0165206	+	0.999864i	$0.505259\pi$
$984$	−20.7881	−0.662702
$985$	3.80220	0.121148
$986$	−3.60405	−0.114776
$987$	3.73343	0.118836
$988$	14.6344	0.465582
$989$	−2.81242	−0.0894297
$990$	−6.22363	−0.197800
$991$	−1.34192	−0.0426275	−0.0213137	−	0.999773i	$-0.506785\pi$
$991$	−1.34192	−0.0426275	−0.0213137	+	0.999773i	$0.506785\pi$
$992$	−4.11453	−0.130636
$993$	−41.7135	−1.32374
$994$	36.7952	1.16707
$995$	−66.8357	−2.11883
$996$	−3.65906	−0.115942
$997$	41.7800	1.32319	0.661593	−	0.749863i	$-0.269881\pi$
$997$	41.7800	1.32319	0.661593	+	0.749863i	$0.269881\pi$
$998$	−35.1112	−1.11142
$999$	−2.13695	−0.0676103

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4334.2.a.b.1.5	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4334.2.a.b.1.5	✓	15	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:	\( N \)	\(=\)	\( 4334 = 2 \cdot 11 \cdot 197 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4334.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.15333 q^{3} +1.00000 q^{4} -3.80220 q^{5} -2.15333 q^{6} +3.88018 q^{7} +1.00000 q^{8} +1.63685 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.15333 q^{3} +1.00000 q^{4} -3.80220 q^{5} -2.15333 q^{6} +3.88018 q^{7} +1.00000 q^{8} +1.63685 q^{9} -3.80220 q^{10} +1.00000 q^{11} -2.15333 q^{12} -2.68872 q^{13} +3.88018 q^{14} +8.18741 q^{15} +1.00000 q^{16} +0.563498 q^{17} +1.63685 q^{18} -5.44288 q^{19} -3.80220 q^{20} -8.35532 q^{21} +1.00000 q^{22} +6.33890 q^{23} -2.15333 q^{24} +9.45673 q^{25} -2.68872 q^{26} +2.93532 q^{27} +3.88018 q^{28} -6.39585 q^{29} +8.18741 q^{30} -4.11453 q^{31} +1.00000 q^{32} -2.15333 q^{33} +0.563498 q^{34} -14.7532 q^{35} +1.63685 q^{36} -0.728015 q^{37} -5.44288 q^{38} +5.78972 q^{39} -3.80220 q^{40} +9.65393 q^{41} -8.35532 q^{42} -0.443676 q^{43} +1.00000 q^{44} -6.22363 q^{45} +6.33890 q^{46} -0.446833 q^{47} -2.15333 q^{48} +8.05577 q^{49} +9.45673 q^{50} -1.21340 q^{51} -2.68872 q^{52} -9.29648 q^{53} +2.93532 q^{54} -3.80220 q^{55} +3.88018 q^{56} +11.7203 q^{57} -6.39585 q^{58} +12.5695 q^{59} +8.18741 q^{60} -6.09043 q^{61} -4.11453 q^{62} +6.35126 q^{63} +1.00000 q^{64} +10.2231 q^{65} -2.15333 q^{66} +16.0033 q^{67} +0.563498 q^{68} -13.6498 q^{69} -14.7532 q^{70} +9.48287 q^{71} +1.63685 q^{72} -4.56247 q^{73} -0.728015 q^{74} -20.3635 q^{75} -5.44288 q^{76} +3.88018 q^{77} +5.78972 q^{78} +7.83601 q^{79} -3.80220 q^{80} -11.2313 q^{81} +9.65393 q^{82} +1.69925 q^{83} -8.35532 q^{84} -2.14253 q^{85} -0.443676 q^{86} +13.7724 q^{87} +1.00000 q^{88} -17.3736 q^{89} -6.22363 q^{90} -10.4327 q^{91} +6.33890 q^{92} +8.85995 q^{93} -0.446833 q^{94} +20.6949 q^{95} -2.15333 q^{96} -12.8995 q^{97} +8.05577 q^{98} +1.63685 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 15 q^{2} - 9 q^{3} + 15 q^{4} - 11 q^{5} - 9 q^{6} - 11 q^{7} + 15 q^{8} + 10 q^{9}+O(q^{10}) \) 15 * q + 15 * q^2 - 9 * q^3 + 15 * q^4 - 11 * q^5 - 9 * q^6 - 11 * q^7 + 15 * q^8 + 10 * q^9	\( 15 q + 15 q^{2} - 9 q^{3} + 15 q^{4} - 11 q^{5} - 9 q^{6} - 11 q^{7} + 15 q^{8} + 10 q^{9} - 11 q^{10} + 15 q^{11} - 9 q^{12} - 21 q^{13} - 11 q^{14} - 2 q^{15} + 15 q^{16} - 4 q^{17} + 10 q^{18} - 22 q^{19} - 11 q^{20} - 13 q^{21} + 15 q^{22} - 16 q^{23} - 9 q^{24} + 6 q^{25} - 21 q^{26} - 21 q^{27} - 11 q^{28} - 8 q^{29} - 2 q^{30} - 33 q^{31} + 15 q^{32} - 9 q^{33} - 4 q^{34} - 2 q^{35} + 10 q^{36} - q^{37} - 22 q^{38} + q^{39} - 11 q^{40} - 10 q^{41} - 13 q^{42} - 8 q^{43} + 15 q^{44} - 10 q^{45} - 16 q^{46} - 31 q^{47} - 9 q^{48} + 2 q^{49} + 6 q^{50} + 2 q^{51} - 21 q^{52} - 18 q^{53} - 21 q^{54} - 11 q^{55} - 11 q^{56} + 16 q^{57} - 8 q^{58} - 37 q^{59} - 2 q^{60} - 31 q^{61} - 33 q^{62} - 20 q^{63} + 15 q^{64} - 13 q^{65} - 9 q^{66} + q^{67} - 4 q^{68} - 25 q^{69} - 2 q^{70} - 28 q^{71} + 10 q^{72} - 20 q^{73} - q^{74} - 9 q^{75} - 22 q^{76} - 11 q^{77} + q^{78} - 6 q^{79} - 11 q^{80} + 3 q^{81} - 10 q^{82} - 15 q^{83} - 13 q^{84} - 31 q^{85} - 8 q^{86} - 16 q^{87} + 15 q^{88} - 17 q^{89} - 10 q^{90} - 21 q^{91} - 16 q^{92} + 10 q^{93} - 31 q^{94} - 3 q^{95} - 9 q^{96} - 9 q^{97} + 2 q^{98} + 10 q^{99}+O(q^{100}) \) 15 * q + 15 * q^2 - 9 * q^3 + 15 * q^4 - 11 * q^5 - 9 * q^6 - 11 * q^7 + 15 * q^8 + 10 * q^9 - 11 * q^10 + 15 * q^11 - 9 * q^12 - 21 * q^13 - 11 * q^14 - 2 * q^15 + 15 * q^16 - 4 * q^17 + 10 * q^18 - 22 * q^19 - 11 * q^20 - 13 * q^21 + 15 * q^22 - 16 * q^23 - 9 * q^24 + 6 * q^25 - 21 * q^26 - 21 * q^27 - 11 * q^28 - 8 * q^29 - 2 * q^30 - 33 * q^31 + 15 * q^32 - 9 * q^33 - 4 * q^34 - 2 * q^35 + 10 * q^36 - q^37 - 22 * q^38 + q^39 - 11 * q^40 - 10 * q^41 - 13 * q^42 - 8 * q^43 + 15 * q^44 - 10 * q^45 - 16 * q^46 - 31 * q^47 - 9 * q^48 + 2 * q^49 + 6 * q^50 + 2 * q^51 - 21 * q^52 - 18 * q^53 - 21 * q^54 - 11 * q^55 - 11 * q^56 + 16 * q^57 - 8 * q^58 - 37 * q^59 - 2 * q^60 - 31 * q^61 - 33 * q^62 - 20 * q^63 + 15 * q^64 - 13 * q^65 - 9 * q^66 + q^67 - 4 * q^68 - 25 * q^69 - 2 * q^70 - 28 * q^71 + 10 * q^72 - 20 * q^73 - q^74 - 9 * q^75 - 22 * q^76 - 11 * q^77 + q^78 - 6 * q^79 - 11 * q^80 + 3 * q^81 - 10 * q^82 - 15 * q^83 - 13 * q^84 - 31 * q^85 - 8 * q^86 - 16 * q^87 + 15 * q^88 - 17 * q^89 - 10 * q^90 - 21 * q^91 - 16 * q^92 + 10 * q^93 - 31 * q^94 - 3 * q^95 - 9 * q^96 - 9 * q^97 + 2 * q^98 + 10 * q^99

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