LMFDB - Embedded newform 6045.2.a.bg.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6045,2,Mod(1,6045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6045, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("6045.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6045 = 3 \cdot 5 \cdot 13 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6045.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:	$48.2695680219$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2 x^{15} - 27 x^{14} + 51 x^{13} + 294 x^{12} - 517 x^{11} - 1657 x^{10} + 2678 x^{9} + \cdots - 428 $ x^16 - 2x^15 - 27x^14 + 51x^13 + 294x^12 - 517x^11 - 1657x^10 + 2678x^9 + 5124x^8 - 7583x^7 - 8358x^6 + 11579x^5 + 6027x^4 - 8572x^3 - 780x^2 + 2178*x - 428
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.46449$ of defining polynomial
Character	$\chi$	$=$	6045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.46449 q^{2} -1.00000 q^{3} +4.07370 q^{4} -1.00000 q^{5} +2.46449 q^{6} -4.85009 q^{7} -5.11060 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.46449 q^{2} -1.00000 q^{3} +4.07370 q^{4} -1.00000 q^{5} +2.46449 q^{6} -4.85009 q^{7} -5.11060 q^{8} +1.00000 q^{9} +2.46449 q^{10} +3.32880 q^{11} -4.07370 q^{12} -1.00000 q^{13} +11.9530 q^{14} +1.00000 q^{15} +4.44762 q^{16} -6.96874 q^{17} -2.46449 q^{18} -1.90343 q^{19} -4.07370 q^{20} +4.85009 q^{21} -8.20379 q^{22} -0.550856 q^{23} +5.11060 q^{24} +1.00000 q^{25} +2.46449 q^{26} -1.00000 q^{27} -19.7578 q^{28} -7.80925 q^{29} -2.46449 q^{30} +1.00000 q^{31} -0.739899 q^{32} -3.32880 q^{33} +17.1744 q^{34} +4.85009 q^{35} +4.07370 q^{36} -3.69683 q^{37} +4.69099 q^{38} +1.00000 q^{39} +5.11060 q^{40} -12.0556 q^{41} -11.9530 q^{42} -10.9169 q^{43} +13.5605 q^{44} -1.00000 q^{45} +1.35758 q^{46} -1.42333 q^{47} -4.44762 q^{48} +16.5234 q^{49} -2.46449 q^{50} +6.96874 q^{51} -4.07370 q^{52} +11.1927 q^{53} +2.46449 q^{54} -3.32880 q^{55} +24.7869 q^{56} +1.90343 q^{57} +19.2458 q^{58} +0.524714 q^{59} +4.07370 q^{60} +8.37257 q^{61} -2.46449 q^{62} -4.85009 q^{63} -7.07177 q^{64} +1.00000 q^{65} +8.20379 q^{66} -2.34036 q^{67} -28.3885 q^{68} +0.550856 q^{69} -11.9530 q^{70} -11.6231 q^{71} -5.11060 q^{72} +10.4332 q^{73} +9.11078 q^{74} -1.00000 q^{75} -7.75401 q^{76} -16.1450 q^{77} -2.46449 q^{78} -3.51181 q^{79} -4.44762 q^{80} +1.00000 q^{81} +29.7108 q^{82} -17.7269 q^{83} +19.7578 q^{84} +6.96874 q^{85} +26.9045 q^{86} +7.80925 q^{87} -17.0122 q^{88} -8.64408 q^{89} +2.46449 q^{90} +4.85009 q^{91} -2.24402 q^{92} -1.00000 q^{93} +3.50778 q^{94} +1.90343 q^{95} +0.739899 q^{96} -13.9916 q^{97} -40.7216 q^{98} +3.32880 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 2 q^{2} - 16 q^{3} + 26 q^{4} - 16 q^{5} + 2 q^{6} - 2 q^{7} - 9 q^{8} + 16 q^{9}+O(q^{10}) $ 16 * q - 2 * q^2 - 16 * q^3 + 26 * q^4 - 16 * q^5 + 2 * q^6 - 2 * q^7 - 9 * q^8 + 16 * q^9	\( 16 q - 2 q^{2} - 16 q^{3} + 26 q^{4} - 16 q^{5} + 2 q^{6} - 2 q^{7} - 9 q^{8} + 16 q^{9} + 2 q^{10} + 3 q^{11} - 26 q^{12} - 16 q^{13} - 5 q^{14} + 16 q^{15} + 38 q^{16} - 13 q^{17} - 2 q^{18} - 26 q^{20} + 2 q^{21} + q^{22} - 15 q^{23} + 9 q^{24} + 16 q^{25} + 2 q^{26} - 16 q^{27} + 8 q^{28} - 4 q^{29} - 2 q^{30} + 16 q^{31} - 30 q^{32} - 3 q^{33} + 29 q^{34} + 2 q^{35} + 26 q^{36} + 12 q^{37} + 16 q^{39} + 9 q^{40} - 12 q^{41} + 5 q^{42} - 7 q^{43} - 13 q^{44} - 16 q^{45} + 14 q^{46} + 17 q^{47} - 38 q^{48} + 16 q^{49} - 2 q^{50} + 13 q^{51} - 26 q^{52} - 36 q^{53} + 2 q^{54} - 3 q^{55} + 41 q^{56} + 16 q^{58} + 53 q^{59} + 26 q^{60} + 34 q^{61} - 2 q^{62} - 2 q^{63} + 79 q^{64} + 16 q^{65} - q^{66} - 13 q^{67} - 39 q^{68} + 15 q^{69} + 5 q^{70} - 11 q^{71} - 9 q^{72} + 34 q^{73} - 12 q^{74} - 16 q^{75} + 86 q^{76} - 32 q^{77} - 2 q^{78} - 7 q^{79} - 38 q^{80} + 16 q^{81} + 27 q^{82} - 28 q^{83} - 8 q^{84} + 13 q^{85} + 38 q^{86} + 4 q^{87} + 23 q^{88} - 8 q^{89} + 2 q^{90} + 2 q^{91} - 71 q^{92} - 16 q^{93} + 66 q^{94} + 30 q^{96} + 4 q^{97} + 22 q^{98} + 3 q^{99}+O(q^{100}) \) 16 * q - 2 * q^2 - 16 * q^3 + 26 * q^4 - 16 * q^5 + 2 * q^6 - 2 * q^7 - 9 * q^8 + 16 * q^9 + 2 * q^10 + 3 * q^11 - 26 * q^12 - 16 * q^13 - 5 * q^14 + 16 * q^15 + 38 * q^16 - 13 * q^17 - 2 * q^18 - 26 * q^20 + 2 * q^21 + q^22 - 15 * q^23 + 9 * q^24 + 16 * q^25 + 2 * q^26 - 16 * q^27 + 8 * q^28 - 4 * q^29 - 2 * q^30 + 16 * q^31 - 30 * q^32 - 3 * q^33 + 29 * q^34 + 2 * q^35 + 26 * q^36 + 12 * q^37 + 16 * q^39 + 9 * q^40 - 12 * q^41 + 5 * q^42 - 7 * q^43 - 13 * q^44 - 16 * q^45 + 14 * q^46 + 17 * q^47 - 38 * q^48 + 16 * q^49 - 2 * q^50 + 13 * q^51 - 26 * q^52 - 36 * q^53 + 2 * q^54 - 3 * q^55 + 41 * q^56 + 16 * q^58 + 53 * q^59 + 26 * q^60 + 34 * q^61 - 2 * q^62 - 2 * q^63 + 79 * q^64 + 16 * q^65 - q^66 - 13 * q^67 - 39 * q^68 + 15 * q^69 + 5 * q^70 - 11 * q^71 - 9 * q^72 + 34 * q^73 - 12 * q^74 - 16 * q^75 + 86 * q^76 - 32 * q^77 - 2 * q^78 - 7 * q^79 - 38 * q^80 + 16 * q^81 + 27 * q^82 - 28 * q^83 - 8 * q^84 + 13 * q^85 + 38 * q^86 + 4 * q^87 + 23 * q^88 - 8 * q^89 + 2 * q^90 + 2 * q^91 - 71 * q^92 - 16 * q^93 + 66 * q^94 + 30 * q^96 + 4 * q^97 + 22 * q^98 + 3 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.46449	−1.74266	−0.871328	−	0.490701i	$-0.836741\pi$
$2$	−2.46449	−1.74266	−0.871328	+	0.490701i	$0.836741\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	4.07370	2.03685
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	2.46449	1.00612
$7$	−4.85009	−1.83316	−0.916581	−	0.399850i	$-0.869062\pi$
$7$	−4.85009	−1.83316	−0.916581	+	0.399850i	$0.869062\pi$
$8$	−5.11060	−1.80687
$9$	1.00000	0.333333
$10$	2.46449	0.779339
$11$	3.32880	1.00367	0.501835	−	0.864963i	$-0.332658\pi$
$11$	3.32880	1.00367	0.501835	+	0.864963i	$0.332658\pi$
$12$	−4.07370	−1.17598
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	11.9530	3.19457
$15$	1.00000	0.258199
$16$	4.44762	1.11191
$17$	−6.96874	−1.69017	−0.845084	−	0.534634i	$-0.820450\pi$
$17$	−6.96874	−1.69017	−0.845084	+	0.534634i	$0.820450\pi$
$18$	−2.46449	−0.580885
$19$	−1.90343	−0.436677	−0.218339	−	0.975873i	$-0.570064\pi$
$19$	−1.90343	−0.436677	−0.218339	+	0.975873i	$0.570064\pi$
$20$	−4.07370	−0.910907
$21$	4.85009	1.05838
$22$	−8.20379	−1.74905
$23$	−0.550856	−0.114861	−0.0574307	−	0.998349i	$-0.518291\pi$
$23$	−0.550856	−0.114861	−0.0574307	+	0.998349i	$0.518291\pi$
$24$	5.11060	1.04320
$25$	1.00000	0.200000
$26$	2.46449	0.483326
$27$	−1.00000	−0.192450
$28$	−19.7578	−3.73387
$29$	−7.80925	−1.45014	−0.725071	−	0.688674i	$-0.758193\pi$
$29$	−7.80925	−1.45014	−0.725071	+	0.688674i	$0.758193\pi$
$30$	−2.46449	−0.449952
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	−0.739899	−0.130797
$33$	−3.32880	−0.579470
$34$	17.1744	2.94538
$35$	4.85009	0.819815
$36$	4.07370	0.678950
$37$	−3.69683	−0.607754	−0.303877	−	0.952711i	$-0.598281\pi$
$37$	−3.69683	−0.607754	−0.303877	+	0.952711i	$0.598281\pi$
$38$	4.69099	0.760978
$39$	1.00000	0.160128
$40$	5.11060	0.808057
$41$	−12.0556	−1.88276	−0.941381	−	0.337347i	$-0.890471\pi$
$41$	−12.0556	−1.88276	−0.941381	+	0.337347i	$0.890471\pi$
$42$	−11.9530	−1.84439
$43$	−10.9169	−1.66481	−0.832404	−	0.554169i	$-0.813036\pi$
$43$	−10.9169	−1.66481	−0.832404	+	0.554169i	$0.813036\pi$
$44$	13.5605	2.04433
$45$	−1.00000	−0.149071
$46$	1.35758	0.200164
$47$	−1.42333	−0.207614	−0.103807	−	0.994597i	$-0.533102\pi$
$47$	−1.42333	−0.207614	−0.103807	+	0.994597i	$0.533102\pi$
$48$	−4.44762	−0.641959
$49$	16.5234	2.36048
$50$	−2.46449	−0.348531
$51$	6.96874	0.975819
$52$	−4.07370	−0.564920
$53$	11.1927	1.53744	0.768721	−	0.639584i	$-0.220893\pi$
$53$	11.1927	1.53744	0.768721	+	0.639584i	$0.220893\pi$
$54$	2.46449	0.335374
$55$	−3.32880	−0.448855
$56$	24.7869	3.31229
$57$	1.90343	0.252116
$58$	19.2458	2.52710
$59$	0.524714	0.0683119	0.0341559	−	0.999417i	$-0.489126\pi$
$59$	0.524714	0.0683119	0.0341559	+	0.999417i	$0.489126\pi$
$60$	4.07370	0.525912
$61$	8.37257	1.07200	0.535999	−	0.844219i	$-0.319935\pi$
$61$	8.37257	1.07200	0.535999	+	0.844219i	$0.319935\pi$
$62$	−2.46449	−0.312990
$63$	−4.85009	−0.611054
$64$	−7.07177	−0.883971
$65$	1.00000	0.124035
$66$	8.20379	1.00982
$67$	−2.34036	−0.285921	−0.142961	−	0.989728i	$-0.545662\pi$
$67$	−2.34036	−0.285921	−0.142961	+	0.989728i	$0.545662\pi$
$68$	−28.3885	−3.44262
$69$	0.550856	0.0663152
$70$	−11.9530	−1.42865
$71$	−11.6231	−1.37941	−0.689706	−	0.724089i	$-0.742260\pi$
$71$	−11.6231	−1.37941	−0.689706	+	0.724089i	$0.742260\pi$
$72$	−5.11060	−0.602290
$73$	10.4332	1.22112	0.610559	−	0.791971i	$-0.290945\pi$
$73$	10.4332	1.22112	0.610559	+	0.791971i	$0.290945\pi$
$74$	9.11078	1.05911
$75$	−1.00000	−0.115470
$76$	−7.75401	−0.889446
$77$	−16.1450	−1.83989
$78$	−2.46449	−0.279048
$79$	−3.51181	−0.395110	−0.197555	−	0.980292i	$-0.563300\pi$
$79$	−3.51181	−0.395110	−0.197555	+	0.980292i	$0.563300\pi$
$80$	−4.44762	−0.497259
$81$	1.00000	0.111111
$82$	29.7108	3.28100
$83$	−17.7269	−1.94578	−0.972888	−	0.231277i	$-0.925710\pi$
$83$	−17.7269	−1.94578	−0.972888	+	0.231277i	$0.925710\pi$
$84$	19.7578	2.15575
$85$	6.96874	0.755866
$86$	26.9045	2.90119
$87$	7.80925	0.837239
$88$	−17.0122	−1.81350
$89$	−8.64408	−0.916271	−0.458135	−	0.888882i	$-0.651482\pi$
$89$	−8.64408	−0.916271	−0.458135	+	0.888882i	$0.651482\pi$
$90$	2.46449	0.259780
$91$	4.85009	0.508427
$92$	−2.24402	−0.233955
$93$	−1.00000	−0.103695
$94$	3.50778	0.361800
$95$	1.90343	0.195288
$96$	0.739899	0.0755156
$97$	−13.9916	−1.42063	−0.710317	−	0.703882i	$-0.751448\pi$
$97$	−13.9916	−1.42063	−0.710317	+	0.703882i	$0.751448\pi$
$98$	−40.7216	−4.11350
$99$	3.32880	0.334557
$100$	4.07370	0.407370
$101$	−17.8911	−1.78023	−0.890114	−	0.455737i	$-0.849376\pi$
$101$	−17.8911	−1.78023	−0.890114	+	0.455737i	$0.849376\pi$
$102$	−17.1744	−1.70052
$103$	1.10847	0.109221	0.0546104	−	0.998508i	$-0.482608\pi$
$103$	1.10847	0.109221	0.0546104	+	0.998508i	$0.482608\pi$
$104$	5.11060	0.501136
$105$	−4.85009	−0.473320
$106$	−27.5844	−2.67923
$107$	−1.87626	−0.181385	−0.0906927	−	0.995879i	$-0.528908\pi$
$107$	−1.87626	−0.181385	−0.0906927	+	0.995879i	$0.528908\pi$
$108$	−4.07370	−0.391992
$109$	−10.5745	−1.01285	−0.506425	−	0.862284i	$-0.669033\pi$
$109$	−10.5745	−1.01285	−0.506425	+	0.862284i	$0.669033\pi$
$110$	8.20379	0.782200
$111$	3.69683	0.350887
$112$	−21.5714	−2.03830
$113$	−7.83019	−0.736603	−0.368301	−	0.929706i	$-0.620061\pi$
$113$	−7.83019	−0.736603	−0.368301	+	0.929706i	$0.620061\pi$
$114$	−4.69099	−0.439351
$115$	0.550856	0.0513676
$116$	−31.8125	−2.95372
$117$	−1.00000	−0.0924500
$118$	−1.29315	−0.119044
$119$	33.7990	3.09835
$120$	−5.11060	−0.466532
$121$	0.0809093	0.00735539
$122$	−20.6341	−1.86812
$123$	12.0556	1.08701
$124$	4.07370	0.365829
$125$	−1.00000	−0.0894427
$126$	11.9530	1.06486
$127$	10.0703	0.893590	0.446795	−	0.894636i	$-0.352565\pi$
$127$	10.0703	0.893590	0.446795	+	0.894636i	$0.352565\pi$
$128$	18.9081	1.67125
$129$	10.9169	0.961178
$130$	−2.46449	−0.216150
$131$	1.73644	0.151713	0.0758566	−	0.997119i	$-0.475831\pi$
$131$	1.73644	0.151713	0.0758566	+	0.997119i	$0.475831\pi$
$132$	−13.5605	−1.18029
$133$	9.23182	0.800500
$134$	5.76780	0.498262
$135$	1.00000	0.0860663
$136$	35.6145	3.05392
$137$	3.07547	0.262755	0.131378	−	0.991332i	$-0.458060\pi$
$137$	3.07547	0.262755	0.131378	+	0.991332i	$0.458060\pi$
$138$	−1.35758	−0.115565
$139$	12.5758	1.06667	0.533333	−	0.845905i	$-0.320939\pi$
$139$	12.5758	1.06667	0.533333	+	0.845905i	$0.320939\pi$
$140$	19.7578	1.66984
$141$	1.42333	0.119866
$142$	28.6451	2.40384
$143$	−3.32880	−0.278368
$144$	4.44762	0.370635
$145$	7.80925	0.648523
$146$	−25.7126	−2.12799
$147$	−16.5234	−1.36282
$148$	−15.0598	−1.23790
$149$	−11.1993	−0.917481	−0.458740	−	0.888570i	$-0.651699\pi$
$149$	−11.1993	−0.917481	−0.458740	+	0.888570i	$0.651699\pi$
$150$	2.46449	0.201225
$151$	−18.9909	−1.54546	−0.772729	−	0.634737i	$-0.781109\pi$
$151$	−18.9909	−1.54546	−0.772729	+	0.634737i	$0.781109\pi$
$152$	9.72769	0.789020
$153$	−6.96874	−0.563389
$154$	39.7891	3.20630
$155$	−1.00000	−0.0803219
$156$	4.07370	0.326157
$157$	5.79391	0.462404	0.231202	−	0.972906i	$-0.425734\pi$
$157$	5.79391	0.462404	0.231202	+	0.972906i	$0.425734\pi$
$158$	8.65482	0.688541
$159$	−11.1927	−0.887643
$160$	0.739899	0.0584941
$161$	2.67170	0.210559
$162$	−2.46449	−0.193628
$163$	12.2236	0.957429	0.478715	−	0.877971i	$-0.341103\pi$
$163$	12.2236	0.957429	0.478715	+	0.877971i	$0.341103\pi$
$164$	−49.1107	−3.83490
$165$	3.32880	0.259147
$166$	43.6876	3.39082
$167$	−12.1871	−0.943064	−0.471532	−	0.881849i	$-0.656299\pi$
$167$	−12.1871	−0.943064	−0.471532	+	0.881849i	$0.656299\pi$
$168$	−24.7869	−1.91235
$169$	1.00000	0.0769231
$170$	−17.1744	−1.31721
$171$	−1.90343	−0.145559
$172$	−44.4721	−3.39096
$173$	3.49998	0.266098	0.133049	−	0.991109i	$-0.457523\pi$
$173$	3.49998	0.266098	0.133049	+	0.991109i	$0.457523\pi$
$174$	−19.2458	−1.45902
$175$	−4.85009	−0.366632
$176$	14.8052	1.11599
$177$	−0.524714	−0.0394399
$178$	21.3032	1.59674
$179$	−19.1192	−1.42904	−0.714519	−	0.699616i	$-0.753354\pi$
$179$	−19.1192	−1.42904	−0.714519	+	0.699616i	$0.753354\pi$
$180$	−4.07370	−0.303636
$181$	5.25064	0.390277	0.195138	−	0.980776i	$-0.437484\pi$
$181$	5.25064	0.390277	0.195138	+	0.980776i	$0.437484\pi$
$182$	−11.9530	−0.886014
$183$	−8.37257	−0.618918
$184$	2.81520	0.207540
$185$	3.69683	0.271796
$186$	2.46449	0.180705
$187$	−23.1975	−1.69637
$188$	−5.79822	−0.422879
$189$	4.85009	0.352792
$190$	−4.69099	−0.340320
$191$	−17.0100	−1.23080	−0.615399	−	0.788216i	$-0.711005\pi$
$191$	−17.0100	−1.23080	−0.615399	+	0.788216i	$0.711005\pi$
$192$	7.07177	0.510361
$193$	−16.1066	−1.15938	−0.579689	−	0.814838i	$-0.696826\pi$
$193$	−16.1066	−1.15938	−0.579689	+	0.814838i	$0.696826\pi$
$194$	34.4822	2.47568
$195$	−1.00000	−0.0716115
$196$	67.3112	4.80794
$197$	−4.29932	−0.306314	−0.153157	−	0.988202i	$-0.548944\pi$
$197$	−4.29932	−0.306314	−0.153157	+	0.988202i	$0.548944\pi$
$198$	−8.20379	−0.583018
$199$	20.3250	1.44080	0.720400	−	0.693558i	$-0.243958\pi$
$199$	20.3250	1.44080	0.720400	+	0.693558i	$0.243958\pi$
$200$	−5.11060	−0.361374
$201$	2.34036	0.165077
$202$	44.0923	3.10233
$203$	37.8756	2.65834
$204$	28.3885	1.98760
$205$	12.0556	0.841996
$206$	−2.73181	−0.190334
$207$	−0.550856	−0.0382871
$208$	−4.44762	−0.308387
$209$	−6.33615	−0.438280
$210$	11.9530	0.824834
$211$	−5.60386	−0.385786	−0.192893	−	0.981220i	$-0.561787\pi$
$211$	−5.60386	−0.385786	−0.192893	+	0.981220i	$0.561787\pi$
$212$	45.5959	3.13154
$213$	11.6231	0.796404
$214$	4.62403	0.316092
$215$	10.9169	0.744525
$216$	5.11060	0.347732
$217$	−4.85009	−0.329245
$218$	26.0606	1.76505
$219$	−10.4332	−0.705013
$220$	−13.5605	−0.914251
$221$	6.96874	0.468768
$222$	−9.11078	−0.611475
$223$	−18.0398	−1.20804	−0.604018	−	0.796971i	$-0.706435\pi$
$223$	−18.0398	−1.20804	−0.604018	+	0.796971i	$0.706435\pi$
$224$	3.58857	0.239772
$225$	1.00000	0.0666667
$226$	19.2974	1.28364
$227$	−6.89095	−0.457368	−0.228684	−	0.973501i	$-0.573442\pi$
$227$	−6.89095	−0.457368	−0.228684	+	0.973501i	$0.573442\pi$
$228$	7.75401	0.513522
$229$	−7.08917	−0.468466	−0.234233	−	0.972181i	$-0.575258\pi$
$229$	−7.08917	−0.468466	−0.234233	+	0.972181i	$0.575258\pi$
$230$	−1.35758	−0.0895160
$231$	16.1450	1.06226
$232$	39.9100	2.62022
$233$	−3.78681	−0.248082	−0.124041	−	0.992277i	$-0.539585\pi$
$233$	−3.78681	−0.248082	−0.124041	+	0.992277i	$0.539585\pi$
$234$	2.46449	0.161109
$235$	1.42333	0.0928479
$236$	2.13753	0.139141
$237$	3.51181	0.228117
$238$	−83.2972	−5.39936
$239$	12.1239	0.784228	0.392114	−	0.919917i	$-0.371744\pi$
$239$	12.1239	0.784228	0.392114	+	0.919917i	$0.371744\pi$
$240$	4.44762	0.287093
$241$	9.78184	0.630104	0.315052	−	0.949074i	$-0.397978\pi$
$241$	9.78184	0.630104	0.315052	+	0.949074i	$0.397978\pi$
$242$	−0.199400	−0.0128179
$243$	−1.00000	−0.0641500
$244$	34.1073	2.18350
$245$	−16.5234	−1.05564
$246$	−29.7108	−1.89429
$247$	1.90343	0.121113
$248$	−5.11060	−0.324524
$249$	17.7269	1.12339
$250$	2.46449	0.155868
$251$	−13.3526	−0.842808	−0.421404	−	0.906873i	$-0.638463\pi$
$251$	−13.3526	−0.842808	−0.421404	+	0.906873i	$0.638463\pi$
$252$	−19.7578	−1.24462
$253$	−1.83369	−0.115283
$254$	−24.8180	−1.55722
$255$	−6.96874	−0.436399
$256$	−32.4552	−2.02845
$257$	1.76780	0.110272	0.0551362	−	0.998479i	$-0.482441\pi$
$257$	1.76780	0.110272	0.0551362	+	0.998479i	$0.482441\pi$
$258$	−26.9045	−1.67500
$259$	17.9299	1.11411
$260$	4.07370	0.252640
$261$	−7.80925	−0.483380
$262$	−4.27943	−0.264384
$263$	4.65218	0.286866	0.143433	−	0.989660i	$-0.454186\pi$
$263$	4.65218	0.286866	0.143433	+	0.989660i	$0.454186\pi$
$264$	17.0122	1.04703
$265$	−11.1927	−0.687565
$266$	−22.7517	−1.39500
$267$	8.64408	0.529009
$268$	−9.53394	−0.582378
$269$	−8.16445	−0.497795	−0.248898	−	0.968530i	$-0.580068\pi$
$269$	−8.16445	−0.497795	−0.248898	+	0.968530i	$0.580068\pi$
$270$	−2.46449	−0.149984
$271$	−27.2612	−1.65600	−0.827999	−	0.560730i	$-0.810521\pi$
$271$	−27.2612	−1.65600	−0.827999	+	0.560730i	$0.810521\pi$
$272$	−30.9943	−1.87931
$273$	−4.85009	−0.293541
$274$	−7.57947	−0.457892
$275$	3.32880	0.200734
$276$	2.24402	0.135074
$277$	−28.7214	−1.72570	−0.862850	−	0.505460i	$-0.831323\pi$
$277$	−28.7214	−1.72570	−0.862850	+	0.505460i	$0.831323\pi$
$278$	−30.9929	−1.85883
$279$	1.00000	0.0598684
$280$	−24.7869	−1.48130
$281$	−9.05790	−0.540349	−0.270174	−	0.962811i	$-0.587081\pi$
$281$	−9.05790	−0.540349	−0.270174	+	0.962811i	$0.587081\pi$
$282$	−3.50778	−0.208885
$283$	27.0652	1.60886	0.804429	−	0.594049i	$-0.202472\pi$
$283$	27.0652	1.60886	0.804429	+	0.594049i	$0.202472\pi$
$284$	−47.3491	−2.80965
$285$	−1.90343	−0.112750
$286$	8.20379	0.485100
$287$	58.4705	3.45140
$288$	−0.739899	−0.0435989
$289$	31.5633	1.85667
$290$	−19.2458	−1.13015
$291$	13.9916	0.820203
$292$	42.5018	2.48723
$293$	−15.2064	−0.888370	−0.444185	−	0.895935i	$-0.646507\pi$
$293$	−15.2064	−0.888370	−0.444185	+	0.895935i	$0.646507\pi$
$294$	40.7216	2.37493
$295$	−0.524714	−0.0305500
$296$	18.8930	1.09813
$297$	−3.32880	−0.193157
$298$	27.6005	1.59885
$299$	0.550856	0.0318568
$300$	−4.07370	−0.235195
$301$	52.9478	3.05186
$302$	46.8028	2.69320
$303$	17.8911	1.02782
$304$	−8.46575	−0.485544
$305$	−8.37257	−0.479412
$306$	17.1744	0.981793
$307$	16.1701	0.922874	0.461437	−	0.887173i	$-0.347334\pi$
$307$	16.1701	0.922874	0.461437	+	0.887173i	$0.347334\pi$
$308$	−65.7698	−3.74758
$309$	−1.10847	−0.0630586
$310$	2.46449	0.139973
$311$	−8.39622	−0.476106	−0.238053	−	0.971252i	$-0.576509\pi$
$311$	−8.39622	−0.476106	−0.238053	+	0.971252i	$0.576509\pi$
$312$	−5.11060	−0.289331
$313$	7.64324	0.432021	0.216011	−	0.976391i	$-0.430695\pi$
$313$	7.64324	0.432021	0.216011	+	0.976391i	$0.430695\pi$
$314$	−14.2790	−0.805811
$315$	4.85009	0.273272
$316$	−14.3061	−0.804779
$317$	−5.68994	−0.319579	−0.159789	−	0.987151i	$-0.551082\pi$
$317$	−5.68994	−0.319579	−0.159789	+	0.987151i	$0.551082\pi$
$318$	27.5844	1.54686
$319$	−25.9954	−1.45546
$320$	7.07177	0.395324
$321$	1.87626	0.104723
$322$	−6.58437	−0.366932
$323$	13.2645	0.738058
$324$	4.07370	0.226317
$325$	−1.00000	−0.0554700
$326$	−30.1250	−1.66847
$327$	10.5745	0.584769
$328$	61.6111	3.40191
$329$	6.90328	0.380590
$330$	−8.20379	−0.451604
$331$	−23.4047	−1.28644	−0.643220	−	0.765682i	$-0.722402\pi$
$331$	−23.4047	−1.28644	−0.643220	+	0.765682i	$0.722402\pi$
$332$	−72.2139	−3.96325
$333$	−3.69683	−0.202585
$334$	30.0349	1.64344
$335$	2.34036	0.127868
$336$	21.5714	1.17681
$337$	−4.94709	−0.269485	−0.134743	−	0.990881i	$-0.543021\pi$
$337$	−4.94709	−0.269485	−0.134743	+	0.990881i	$0.543021\pi$
$338$	−2.46449	−0.134050
$339$	7.83019	0.425278
$340$	28.3885	1.53958
$341$	3.32880	0.180265
$342$	4.69099	0.253659
$343$	−46.1891	−2.49398
$344$	55.7919	3.00810
$345$	−0.550856	−0.0296571
$346$	−8.62565	−0.463718
$347$	−20.9789	−1.12621	−0.563104	−	0.826386i	$-0.690393\pi$
$347$	−20.9789	−1.12621	−0.563104	+	0.826386i	$0.690393\pi$
$348$	31.8125	1.70533
$349$	28.4428	1.52251	0.761255	−	0.648453i	$-0.224584\pi$
$349$	28.4428	1.52251	0.761255	+	0.648453i	$0.224584\pi$
$350$	11.9530	0.638914
$351$	1.00000	0.0533761
$352$	−2.46297	−0.131277
$353$	25.5970	1.36239	0.681194	−	0.732103i	$-0.261461\pi$
$353$	25.5970	1.36239	0.681194	+	0.732103i	$0.261461\pi$
$354$	1.29315	0.0687302
$355$	11.6231	0.616892
$356$	−35.2134	−1.86630
$357$	−33.7990	−1.78883
$358$	47.1191	2.49032
$359$	−19.2665	−1.01685	−0.508424	−	0.861107i	$-0.669772\pi$
$359$	−19.2665	−1.01685	−0.508424	+	0.861107i	$0.669772\pi$
$360$	5.11060	0.269352
$361$	−15.3769	−0.809313
$362$	−12.9401	−0.680118
$363$	−0.0809093	−0.00424664
$364$	19.7578	1.03559
$365$	−10.4332	−0.546100
$366$	20.6341	1.07856
$367$	−13.7485	−0.717663	−0.358832	−	0.933402i	$-0.616825\pi$
$367$	−13.7485	−0.717663	−0.358832	+	0.933402i	$0.616825\pi$
$368$	−2.45000	−0.127715
$369$	−12.0556	−0.627587
$370$	−9.11078	−0.473647
$371$	−54.2858	−2.81838
$372$	−4.07370	−0.211211
$373$	1.05699	0.0547288	0.0273644	−	0.999626i	$-0.491289\pi$
$373$	1.05699	0.0547288	0.0273644	+	0.999626i	$0.491289\pi$
$374$	57.1700	2.95619
$375$	1.00000	0.0516398
$376$	7.27408	0.375132
$377$	7.80925	0.402197
$378$	−11.9530	−0.614795
$379$	−32.0430	−1.64594	−0.822969	−	0.568086i	$-0.807684\pi$
$379$	−32.0430	−1.64594	−0.822969	+	0.568086i	$0.807684\pi$
$380$	7.75401	0.397772
$381$	−10.0703	−0.515915
$382$	41.9208	2.14486
$383$	21.4696	1.09705	0.548524	−	0.836135i	$-0.315190\pi$
$383$	21.4696	1.09705	0.548524	+	0.836135i	$0.315190\pi$
$384$	−18.9081	−0.964899
$385$	16.1450	0.822824
$386$	39.6945	2.02040
$387$	−10.9169	−0.554936
$388$	−56.9976	−2.89362
$389$	1.63973	0.0831374	0.0415687	−	0.999136i	$-0.486764\pi$
$389$	1.63973	0.0831374	0.0415687	+	0.999136i	$0.486764\pi$
$390$	2.46449	0.124794
$391$	3.83877	0.194135
$392$	−84.4443	−4.26508
$393$	−1.73644	−0.0875917
$394$	10.5956	0.533800
$395$	3.51181	0.176699
$396$	13.5605	0.681442
$397$	−17.6040	−0.883521	−0.441760	−	0.897133i	$-0.645646\pi$
$397$	−17.6040	−0.883521	−0.441760	+	0.897133i	$0.645646\pi$
$398$	−50.0907	−2.51082
$399$	−9.23182	−0.462169
$400$	4.44762	0.222381
$401$	−22.2198	−1.10960	−0.554802	−	0.831982i	$-0.687206\pi$
$401$	−22.2198	−1.10960	−0.554802	+	0.831982i	$0.687206\pi$
$402$	−5.76780	−0.287672
$403$	−1.00000	−0.0498135
$404$	−72.8829	−3.62606
$405$	−1.00000	−0.0496904
$406$	−93.3438	−4.63258
$407$	−12.3060	−0.609985
$408$	−35.6145	−1.76318
$409$	27.8281	1.37601	0.688006	−	0.725705i	$-0.258486\pi$
$409$	27.8281	1.37601	0.688006	+	0.725705i	$0.258486\pi$
$410$	−29.7108	−1.46731
$411$	−3.07547	−0.151702
$412$	4.51557	0.222466
$413$	−2.54491	−0.125227
$414$	1.35758	0.0667213
$415$	17.7269	0.870177
$416$	0.739899	0.0362765
$417$	−12.5758	−0.615840
$418$	15.6154	0.763772
$419$	32.2062	1.57337	0.786687	−	0.617352i	$-0.211795\pi$
$419$	32.2062	1.57337	0.786687	+	0.617352i	$0.211795\pi$
$420$	−19.7578	−0.964082
$421$	19.1247	0.932083	0.466042	−	0.884763i	$-0.345680\pi$
$421$	19.1247	0.932083	0.466042	+	0.884763i	$0.345680\pi$
$422$	13.8107	0.672292
$423$	−1.42333	−0.0692047
$424$	−57.2017	−2.77796
$425$	−6.96874	−0.338034
$426$	−28.6451	−1.38786
$427$	−40.6077	−1.96514
$428$	−7.64334	−0.369455
$429$	3.32880	0.160716
$430$	−26.9045	−1.29745
$431$	32.1731	1.54972	0.774861	−	0.632131i	$-0.217820\pi$
$431$	32.1731	1.54972	0.774861	+	0.632131i	$0.217820\pi$
$432$	−4.44762	−0.213986
$433$	12.8666	0.618328	0.309164	−	0.951009i	$-0.399951\pi$
$433$	12.8666	0.618328	0.309164	+	0.951009i	$0.399951\pi$
$434$	11.9530	0.573761
$435$	−7.80925	−0.374425
$436$	−43.0771	−2.06302
$437$	1.04852	0.0501574
$438$	25.7126	1.22859
$439$	10.8398	0.517357	0.258679	−	0.965963i	$-0.416713\pi$
$439$	10.8398	0.517357	0.258679	+	0.965963i	$0.416713\pi$
$440$	17.0122	0.811024
$441$	16.5234	0.786826
$442$	−17.1744	−0.816902
$443$	1.65784	0.0787664	0.0393832	−	0.999224i	$-0.487461\pi$
$443$	1.65784	0.0787664	0.0393832	+	0.999224i	$0.487461\pi$
$444$	15.0598	0.714704
$445$	8.64408	0.409769
$446$	44.4589	2.10519
$447$	11.1993	0.529708
$448$	34.2987	1.62046
$449$	−33.5924	−1.58532	−0.792662	−	0.609661i	$-0.791305\pi$
$449$	−33.5924	−1.58532	−0.792662	+	0.609661i	$0.791305\pi$
$450$	−2.46449	−0.116177
$451$	−40.1305	−1.88967
$452$	−31.8978	−1.50035
$453$	18.9909	0.892270
$454$	16.9827	0.797035
$455$	−4.85009	−0.227376
$456$	−9.72769	−0.455541
$457$	−13.0403	−0.609999	−0.304999	−	0.952353i	$-0.598656\pi$
$457$	−13.0403	−0.609999	−0.304999	+	0.952353i	$0.598656\pi$
$458$	17.4712	0.816374
$459$	6.96874	0.325273
$460$	2.24402	0.104628
$461$	−14.9086	−0.694363	−0.347182	−	0.937798i	$-0.612861\pi$
$461$	−14.9086	−0.694363	−0.347182	+	0.937798i	$0.612861\pi$
$462$	−39.7891	−1.85116
$463$	11.2169	0.521292	0.260646	−	0.965434i	$-0.416064\pi$
$463$	11.2169	0.521292	0.260646	+	0.965434i	$0.416064\pi$
$464$	−34.7326	−1.61242
$465$	1.00000	0.0463739
$466$	9.33255	0.432322
$467$	−1.89576	−0.0877251	−0.0438626	−	0.999038i	$-0.513966\pi$
$467$	−1.89576	−0.0877251	−0.0438626	+	0.999038i	$0.513966\pi$
$468$	−4.07370	−0.188307
$469$	11.3510	0.524139
$470$	−3.50778	−0.161802
$471$	−5.79391	−0.266969
$472$	−2.68160	−0.123431
$473$	−36.3401	−1.67092
$474$	−8.65482	−0.397529
$475$	−1.90343	−0.0873355
$476$	137.687	6.31087
$477$	11.1927	0.512481
$478$	−29.8791	−1.36664
$479$	8.09386	0.369818	0.184909	−	0.982756i	$-0.440801\pi$
$479$	8.09386	0.369818	0.184909	+	0.982756i	$0.440801\pi$
$480$	−0.739899	−0.0337716
$481$	3.69683	0.168561
$482$	−24.1072	−1.09805
$483$	−2.67170	−0.121566
$484$	0.329600	0.0149818
$485$	13.9916	0.635327
$486$	2.46449	0.111791
$487$	3.21150	0.145527	0.0727634	−	0.997349i	$-0.476818\pi$
$487$	3.21150	0.145527	0.0727634	+	0.997349i	$0.476818\pi$
$488$	−42.7889	−1.93696
$489$	−12.2236	−0.552772
$490$	40.7216	1.83961
$491$	31.0728	1.40230	0.701149	−	0.713015i	$-0.252671\pi$
$491$	31.0728	1.40230	0.701149	+	0.713015i	$0.252671\pi$
$492$	49.1107	2.21408
$493$	54.4206	2.45098
$494$	−4.69099	−0.211057
$495$	−3.32880	−0.149618
$496$	4.44762	0.199704
$497$	56.3732	2.52869
$498$	−43.6876	−1.95769
$499$	16.7096	0.748024	0.374012	−	0.927424i	$-0.377982\pi$
$499$	16.7096	0.748024	0.374012	+	0.927424i	$0.377982\pi$
$500$	−4.07370	−0.182181
$501$	12.1871	0.544478
$502$	32.9073	1.46872
$503$	−34.7410	−1.54902	−0.774512	−	0.632559i	$-0.782005\pi$
$503$	−34.7410	−1.54902	−0.774512	+	0.632559i	$0.782005\pi$
$504$	24.7869	1.10410
$505$	17.8911	0.796143
$506$	4.51910	0.200899
$507$	−1.00000	−0.0444116
$508$	41.0232	1.82011
$509$	−27.4542	−1.21689	−0.608443	−	0.793598i	$-0.708206\pi$
$509$	−27.4542	−1.21689	−0.608443	+	0.793598i	$0.708206\pi$
$510$	17.1744	0.760494
$511$	−50.6021	−2.23851
$512$	42.1693	1.86364
$513$	1.90343	0.0840386
$514$	−4.35673	−0.192167
$515$	−1.10847	−0.0488450
$516$	44.4721	1.95777
$517$	−4.73799	−0.208376
$518$	−44.1881	−1.94151
$519$	−3.49998	−0.153632
$520$	−5.11060	−0.224115
$521$	31.0655	1.36100	0.680501	−	0.732747i	$-0.261762\pi$
$521$	31.0655	1.36100	0.680501	+	0.732747i	$0.261762\pi$
$522$	19.2458	0.842366
$523$	34.2392	1.49717	0.748587	−	0.663036i	$-0.230732\pi$
$523$	34.2392	1.49717	0.748587	+	0.663036i	$0.230732\pi$
$524$	7.07372	0.309017
$525$	4.85009	0.211675
$526$	−11.4653	−0.499909
$527$	−6.96874	−0.303563
$528$	−14.8052	−0.644315
$529$	−22.6966	−0.986807
$530$	27.5844	1.19819
$531$	0.524714	0.0227706
$532$	37.6076	1.63050
$533$	12.0556	0.522184
$534$	−21.3032	−0.921881
$535$	1.87626	0.0811180
$536$	11.9607	0.516622
$537$	19.1192	0.825056
$538$	20.1212	0.867486
$539$	55.0029	2.36914
$540$	4.07370	0.175304
$541$	8.86482	0.381128	0.190564	−	0.981675i	$-0.438968\pi$
$541$	8.86482	0.381128	0.190564	+	0.981675i	$0.438968\pi$
$542$	67.1848	2.88583
$543$	−5.25064	−0.225326
$544$	5.15616	0.221069
$545$	10.5745	0.452960
$546$	11.9530	0.511540
$547$	16.8910	0.722206	0.361103	−	0.932526i	$-0.382400\pi$
$547$	16.8910	0.722206	0.361103	+	0.932526i	$0.382400\pi$
$548$	12.5286	0.535193
$549$	8.37257	0.357332
$550$	−8.20379	−0.349811
$551$	14.8644	0.633244
$552$	−2.81520	−0.119823
$553$	17.0326	0.724300
$554$	70.7835	3.00730
$555$	−3.69683	−0.156921
$556$	51.2301	2.17264
$557$	−34.4689	−1.46049	−0.730247	−	0.683183i	$-0.760595\pi$
$557$	−34.4689	−1.46049	−0.730247	+	0.683183i	$0.760595\pi$
$558$	−2.46449	−0.104330
$559$	10.9169	0.461735
$560$	21.5714	0.911556
$561$	23.1975	0.979401
$562$	22.3231	0.941642
$563$	17.5114	0.738017	0.369008	−	0.929426i	$-0.379697\pi$
$563$	17.5114	0.738017	0.369008	+	0.929426i	$0.379697\pi$
$564$	5.79822	0.244149
$565$	7.83019	0.329419
$566$	−66.7018	−2.80368
$567$	−4.85009	−0.203685
$568$	59.4012	2.49242
$569$	29.7887	1.24881	0.624404	−	0.781102i	$-0.285342\pi$
$569$	29.7887	1.24881	0.624404	+	0.781102i	$0.285342\pi$
$570$	4.69099	0.196484
$571$	8.75971	0.366582	0.183291	−	0.983059i	$-0.441325\pi$
$571$	8.75971	0.366582	0.183291	+	0.983059i	$0.441325\pi$
$572$	−13.5605	−0.566994
$573$	17.0100	0.710601
$574$	−144.100	−6.01461
$575$	−0.550856	−0.0229723
$576$	−7.07177	−0.294657
$577$	5.38443	0.224157	0.112078	−	0.993699i	$-0.464249\pi$
$577$	5.38443	0.224157	0.112078	+	0.993699i	$0.464249\pi$
$578$	−77.7874	−3.23553
$579$	16.1066	0.669367
$580$	31.8125	1.32094
$581$	85.9768	3.56692
$582$	−34.4822	−1.42933
$583$	37.2584	1.54309
$584$	−53.3201	−2.20640
$585$	1.00000	0.0413449
$586$	37.4761	1.54812
$587$	24.3239	1.00396	0.501978	−	0.864880i	$-0.332606\pi$
$587$	24.3239	1.00396	0.501978	+	0.864880i	$0.332606\pi$
$588$	−67.3112	−2.77587
$589$	−1.90343	−0.0784296
$590$	1.29315	0.0532382
$591$	4.29932	0.176850
$592$	−16.4421	−0.675765
$593$	18.0431	0.740943	0.370471	−	0.928844i	$-0.379196\pi$
$593$	18.0431	0.740943	0.370471	+	0.928844i	$0.379196\pi$
$594$	8.20379	0.336605
$595$	−33.7990	−1.38562
$596$	−45.6225	−1.86877
$597$	−20.3250	−0.831847
$598$	−1.35758	−0.0555154
$599$	46.8262	1.91327	0.956634	−	0.291294i	$-0.0940857\pi$
$599$	46.8262	1.91327	0.956634	+	0.291294i	$0.0940857\pi$
$600$	5.11060	0.208639
$601$	20.1074	0.820198	0.410099	−	0.912041i	$-0.365494\pi$
$601$	20.1074	0.820198	0.410099	+	0.912041i	$0.365494\pi$
$602$	−130.489	−5.31835
$603$	−2.34036	−0.0953070
$604$	−77.3632	−3.14786
$605$	−0.0809093	−0.00328943
$606$	−44.0923	−1.79113
$607$	−9.28144	−0.376722	−0.188361	−	0.982100i	$-0.560318\pi$
$607$	−9.28144	−0.376722	−0.188361	+	0.982100i	$0.560318\pi$
$608$	1.40835	0.0571160
$609$	−37.8756	−1.53479
$610$	20.6341	0.835450
$611$	1.42333	0.0575818
$612$	−28.3885	−1.14754
$613$	−43.1297	−1.74199	−0.870996	−	0.491289i	$-0.836526\pi$
$613$	−43.1297	−1.74199	−0.870996	+	0.491289i	$0.836526\pi$
$614$	−39.8509	−1.60825
$615$	−12.0556	−0.486127
$616$	82.5106	3.32444
$617$	12.5159	0.503869	0.251935	−	0.967744i	$-0.418933\pi$
$617$	12.5159	0.503869	0.251935	+	0.967744i	$0.418933\pi$
$618$	2.73181	0.109889
$619$	−7.69742	−0.309386	−0.154693	−	0.987963i	$-0.549439\pi$
$619$	−7.69742	−0.309386	−0.154693	+	0.987963i	$0.549439\pi$
$620$	−4.07370	−0.163604
$621$	0.550856	0.0221051
$622$	20.6924	0.829689
$623$	41.9245	1.67967
$624$	4.44762	0.178047
$625$	1.00000	0.0400000
$626$	−18.8367	−0.752865
$627$	6.33615	0.253041
$628$	23.6026	0.941847
$629$	25.7622	1.02721
$630$	−11.9530	−0.476218
$631$	46.4219	1.84803	0.924014	−	0.382358i	$-0.124888\pi$
$631$	46.4219	1.84803	0.924014	+	0.382358i	$0.124888\pi$
$632$	17.9475	0.713913
$633$	5.60386	0.222734
$634$	14.0228	0.556916
$635$	−10.0703	−0.399626
$636$	−45.5959	−1.80799
$637$	−16.5234	−0.654679
$638$	64.0654	2.53637
$639$	−11.6231	−0.459804
$640$	−18.9081	−0.747408
$641$	−21.4622	−0.847707	−0.423853	−	0.905731i	$-0.639323\pi$
$641$	−21.4622	−0.847707	−0.423853	+	0.905731i	$0.639323\pi$
$642$	−4.62403	−0.182496
$643$	32.2560	1.27205	0.636026	−	0.771667i	$-0.280577\pi$
$643$	32.2560	1.27205	0.636026	+	0.771667i	$0.280577\pi$
$644$	10.8837	0.428878
$645$	−10.9169	−0.429852
$646$	−32.6903	−1.28618
$647$	39.9653	1.57120	0.785598	−	0.618737i	$-0.212355\pi$
$647$	39.9653	1.57120	0.785598	+	0.618737i	$0.212355\pi$
$648$	−5.11060	−0.200763
$649$	1.74667	0.0685627
$650$	2.46449	0.0966651
$651$	4.85009	0.190090
$652$	49.7954	1.95014
$653$	−15.9133	−0.622735	−0.311368	−	0.950290i	$-0.600787\pi$
$653$	−15.9133	−0.622735	−0.311368	+	0.950290i	$0.600787\pi$
$654$	−26.0606	−1.01905
$655$	−1.73644	−0.0678482
$656$	−53.6185	−2.09345
$657$	10.4332	0.407039
$658$	−17.0131	−0.663238
$659$	−19.7425	−0.769058	−0.384529	−	0.923113i	$-0.625636\pi$
$659$	−19.7425	−0.769058	−0.384529	+	0.923113i	$0.625636\pi$
$660$	13.5605	0.527843
$661$	4.86432	0.189200	0.0946001	−	0.995515i	$-0.469843\pi$
$661$	4.86432	0.189200	0.0946001	+	0.995515i	$0.469843\pi$
$662$	57.6806	2.24182
$663$	−6.96874	−0.270643
$664$	90.5950	3.51577
$665$	−9.23182	−0.357995
$666$	9.11078	0.353036
$667$	4.30177	0.166565
$668$	−49.6465	−1.92088
$669$	18.0398	0.697460
$670$	−5.76780	−0.222829
$671$	27.8706	1.07593
$672$	−3.58857	−0.138432
$673$	9.54488	0.367928	0.183964	−	0.982933i	$-0.441107\pi$
$673$	9.54488	0.367928	0.183964	+	0.982933i	$0.441107\pi$
$674$	12.1920	0.469620
$675$	−1.00000	−0.0384900
$676$	4.07370	0.156681
$677$	6.06722	0.233182	0.116591	−	0.993180i	$-0.462803\pi$
$677$	6.06722	0.233182	0.116591	+	0.993180i	$0.462803\pi$
$678$	−19.2974	−0.741113
$679$	67.8606	2.60425
$680$	−35.6145	−1.36575
$681$	6.89095	0.264062
$682$	−8.20379	−0.314139
$683$	23.6043	0.903194	0.451597	−	0.892222i	$-0.350855\pi$
$683$	23.6043	0.903194	0.451597	+	0.892222i	$0.350855\pi$
$684$	−7.75401	−0.296482
$685$	−3.07547	−0.117508
$686$	113.833	4.34615
$687$	7.08917	0.270469
$688$	−48.5542	−1.85111
$689$	−11.1927	−0.426410
$690$	1.35758	0.0516821
$691$	−36.1628	−1.37570	−0.687849	−	0.725854i	$-0.741445\pi$
$691$	−36.1628	−1.37570	−0.687849	+	0.725854i	$0.741445\pi$
$692$	14.2579	0.542002
$693$	−16.1450	−0.613297
$694$	51.7023	1.96259
$695$	−12.5758	−0.477028
$696$	−39.9100	−1.51278
$697$	84.0120	3.18218
$698$	−70.0970	−2.65321
$699$	3.78681	0.143230
$700$	−19.7578	−0.746774
$701$	1.27222	0.0480510	0.0240255	−	0.999711i	$-0.492352\pi$
$701$	1.27222	0.0480510	0.0240255	+	0.999711i	$0.492352\pi$
$702$	−2.46449	−0.0930161
$703$	7.03666	0.265393
$704$	−23.5405	−0.887216
$705$	−1.42333	−0.0536058
$706$	−63.0834	−2.37417
$707$	86.7733	3.26345
$708$	−2.13753	−0.0803331
$709$	−44.3417	−1.66529	−0.832644	−	0.553808i	$-0.813174\pi$
$709$	−44.3417	−1.66529	−0.832644	+	0.553808i	$0.813174\pi$
$710$	−28.6451	−1.07503
$711$	−3.51181	−0.131703
$712$	44.1765	1.65558
$713$	−0.550856	−0.0206297
$714$	83.2972	3.11732
$715$	3.32880	0.124490
$716$	−77.8860	−2.91074
$717$	−12.1239	−0.452774
$718$	47.4821	1.77202
$719$	24.9791	0.931565	0.465783	−	0.884899i	$-0.345773\pi$
$719$	24.9791	0.931565	0.465783	+	0.884899i	$0.345773\pi$
$720$	−4.44762	−0.165753
$721$	−5.37617	−0.200219
$722$	37.8963	1.41035
$723$	−9.78184	−0.363791
$724$	21.3895	0.794935
$725$	−7.80925	−0.290028
$726$	0.199400	0.00740043
$727$	−0.597497	−0.0221599	−0.0110800	−	0.999939i	$-0.503527\pi$
$727$	−0.597497	−0.0221599	−0.0110800	+	0.999939i	$0.503527\pi$
$728$	−24.7869	−0.918663
$729$	1.00000	0.0370370
$730$	25.7126	0.951665
$731$	76.0769	2.81381
$732$	−34.1073	−1.26064
$733$	−46.2134	−1.70693	−0.853466	−	0.521148i	$-0.825504\pi$
$733$	−46.2134	−1.70693	−0.853466	+	0.521148i	$0.825504\pi$
$734$	33.8829	1.25064
$735$	16.5234	0.609473
$736$	0.407577	0.0150235
$737$	−7.79060	−0.286971
$738$	29.7108	1.09367
$739$	−2.85355	−0.104970	−0.0524848	−	0.998622i	$-0.516714\pi$
$739$	−2.85355	−0.104970	−0.0524848	+	0.998622i	$0.516714\pi$
$740$	15.0598	0.553607
$741$	−1.90343	−0.0699244
$742$	133.787	4.91147
$743$	−0.634672	−0.0232839	−0.0116419	−	0.999932i	$-0.503706\pi$
$743$	−0.634672	−0.0232839	−0.0116419	+	0.999932i	$0.503706\pi$
$744$	5.11060	0.187364
$745$	11.1993	0.410310
$746$	−2.60494	−0.0953735
$747$	−17.7269	−0.648592
$748$	−94.4998	−3.45525
$749$	9.10005	0.332509
$750$	−2.46449	−0.0899904
$751$	39.4436	1.43932	0.719659	−	0.694328i	$-0.244298\pi$
$751$	39.4436	1.43932	0.719659	+	0.694328i	$0.244298\pi$
$752$	−6.33044	−0.230847
$753$	13.3526	0.486596
$754$	−19.2458	−0.700891
$755$	18.9909	0.691149
$756$	19.7578	0.718584
$757$	−42.6925	−1.55169	−0.775843	−	0.630926i	$-0.782675\pi$
$757$	−42.6925	−1.55169	−0.775843	+	0.630926i	$0.782675\pi$
$758$	78.9696	2.86830
$759$	1.83369	0.0665587
$760$	−9.72769	−0.352860
$761$	−28.6590	−1.03889	−0.519444	−	0.854504i	$-0.673861\pi$
$761$	−28.6590	−1.03889	−0.519444	+	0.854504i	$0.673861\pi$
$762$	24.8180	0.899062
$763$	51.2870	1.85672
$764$	−69.2934	−2.50695
$765$	6.96874	0.251955
$766$	−52.9117	−1.91178
$767$	−0.524714	−0.0189463
$768$	32.4552	1.17113
$769$	−2.73195	−0.0985167	−0.0492583	−	0.998786i	$-0.515686\pi$
$769$	−2.73195	−0.0985167	−0.0492583	+	0.998786i	$0.515686\pi$
$770$	−39.7891	−1.43390
$771$	−1.76780	−0.0636658
$772$	−65.6134	−2.36148
$773$	43.4731	1.56362	0.781809	−	0.623518i	$-0.214297\pi$
$773$	43.4731	1.56362	0.781809	+	0.623518i	$0.214297\pi$
$774$	26.9045	0.967063
$775$	1.00000	0.0359211
$776$	71.5056	2.56690
$777$	−17.9299	−0.643233
$778$	−4.04108	−0.144880
$779$	22.9469	0.822159
$780$	−4.07370	−0.145862
$781$	−38.6911	−1.38448
$782$	−9.46060	−0.338310
$783$	7.80925	0.279080
$784$	73.4896	2.62463
$785$	−5.79391	−0.206793
$786$	4.27943	0.152642
$787$	40.4204	1.44083	0.720416	−	0.693543i	$-0.243951\pi$
$787$	40.4204	1.44083	0.720416	+	0.693543i	$0.243951\pi$
$788$	−17.5141	−0.623915
$789$	−4.65218	−0.165622
$790$	−8.65482	−0.307925
$791$	37.9771	1.35031
$792$	−17.0122	−0.604501
$793$	−8.37257	−0.297319
$794$	43.3849	1.53967
$795$	11.1927	0.396966
$796$	82.7979	2.93469
$797$	8.53927	0.302476	0.151238	−	0.988497i	$-0.451674\pi$
$797$	8.53927	0.302476	0.151238	+	0.988497i	$0.451674\pi$
$798$	22.7517	0.805401
$799$	9.91883	0.350903
$800$	−0.739899	−0.0261594
$801$	−8.64408	−0.305424
$802$	54.7604	1.93366
$803$	34.7301	1.22560
$804$	9.53394	0.336236
$805$	−2.67170	−0.0941650
$806$	2.46449	0.0868079
$807$	8.16445	0.287402
$808$	91.4342	3.21664
$809$	2.31493	0.0813887	0.0406943	−	0.999172i	$-0.487043\pi$
$809$	2.31493	0.0813887	0.0406943	+	0.999172i	$0.487043\pi$
$810$	2.46449	0.0865933
$811$	−23.7511	−0.834013	−0.417006	−	0.908904i	$-0.636921\pi$
$811$	−23.7511	−0.834013	−0.417006	+	0.908904i	$0.636921\pi$
$812$	154.294	5.41464
$813$	27.2612	0.956091
$814$	30.3280	1.06299
$815$	−12.2236	−0.428175
$816$	30.9943	1.08502
$817$	20.7796	0.726985
$818$	−68.5820	−2.39791
$819$	4.85009	0.169476
$820$	49.1107	1.71502
$821$	−0.562096	−0.0196173	−0.00980865	−	0.999952i	$-0.503122\pi$
$821$	−0.562096	−0.0196173	−0.00980865	+	0.999952i	$0.503122\pi$
$822$	7.57947	0.264364
$823$	−11.5838	−0.403784	−0.201892	−	0.979408i	$-0.564709\pi$
$823$	−11.5838	−0.403784	−0.201892	+	0.979408i	$0.564709\pi$
$824$	−5.66495	−0.197348
$825$	−3.32880	−0.115894
$826$	6.27189	0.218227
$827$	18.5084	0.643598	0.321799	−	0.946808i	$-0.395712\pi$
$827$	18.5084	0.643598	0.321799	+	0.946808i	$0.395712\pi$
$828$	−2.24402	−0.0779851
$829$	−33.3094	−1.15688	−0.578442	−	0.815724i	$-0.696339\pi$
$829$	−33.3094	−1.15688	−0.578442	+	0.815724i	$0.696339\pi$
$830$	−43.6876	−1.51642
$831$	28.7214	0.996334
$832$	7.07177	0.245170
$833$	−115.147	−3.98961
$834$	30.9929	1.07320
$835$	12.1871	0.421751
$836$	−25.8116	−0.892711
$837$	−1.00000	−0.0345651
$838$	−79.3717	−2.74185
$839$	44.5266	1.53723	0.768614	−	0.639713i	$-0.220947\pi$
$839$	44.5266	1.53723	0.768614	+	0.639713i	$0.220947\pi$
$840$	24.7869	0.855228
$841$	31.9844	1.10291
$842$	−47.1327	−1.62430
$843$	9.05790	0.311971
$844$	−22.8285	−0.785788
$845$	−1.00000	−0.0344010
$846$	3.50778	0.120600
$847$	−0.392417	−0.0134836
$848$	49.7811	1.70949
$849$	−27.0652	−0.928874
$850$	17.1744	0.589076
$851$	2.03642	0.0698075
$852$	47.3491	1.62215
$853$	−18.0371	−0.617580	−0.308790	−	0.951130i	$-0.599924\pi$
$853$	−18.0371	−0.617580	−0.308790	+	0.951130i	$0.599924\pi$
$854$	100.077	3.42457
$855$	1.90343	0.0650960
$856$	9.58884	0.327740
$857$	−16.8279	−0.574830	−0.287415	−	0.957806i	$-0.592796\pi$
$857$	−16.8279	−0.574830	−0.287415	+	0.957806i	$0.592796\pi$
$858$	−8.20379	−0.280073
$859$	48.6391	1.65954	0.829772	−	0.558102i	$-0.188470\pi$
$859$	48.6391	1.65954	0.829772	+	0.558102i	$0.188470\pi$
$860$	44.4721	1.51649
$861$	−58.4705	−1.99267
$862$	−79.2902	−2.70063
$863$	−49.6292	−1.68940	−0.844699	−	0.535242i	$-0.820220\pi$
$863$	−49.6292	−1.68940	−0.844699	+	0.535242i	$0.820220\pi$
$864$	0.739899	0.0251719
$865$	−3.49998	−0.119003
$866$	−31.7095	−1.07753
$867$	−31.5633	−1.07195
$868$	−19.7578	−0.670623
$869$	−11.6901	−0.396560
$870$	19.2458	0.652494
$871$	2.34036	0.0793002
$872$	54.0418	1.83009
$873$	−13.9916	−0.473545
$874$	−2.58406	−0.0874070
$875$	4.85009	0.163963
$876$	−42.5018	−1.43600
$877$	8.18053	0.276237	0.138119	−	0.990416i	$-0.455895\pi$
$877$	8.18053	0.276237	0.138119	+	0.990416i	$0.455895\pi$
$878$	−26.7146	−0.901576
$879$	15.2064	0.512901
$880$	−14.8052	−0.499085
$881$	8.76725	0.295376	0.147688	−	0.989034i	$-0.452817\pi$
$881$	8.76725	0.295376	0.147688	+	0.989034i	$0.452817\pi$
$882$	−40.7216	−1.37117
$883$	−16.0872	−0.541377	−0.270689	−	0.962667i	$-0.587251\pi$
$883$	−16.0872	−0.541377	−0.270689	+	0.962667i	$0.587251\pi$
$884$	28.3885	0.954810
$885$	0.524714	0.0176381
$886$	−4.08573	−0.137263
$887$	−22.3013	−0.748804	−0.374402	−	0.927266i	$-0.622152\pi$
$887$	−22.3013	−0.748804	−0.374402	+	0.927266i	$0.622152\pi$
$888$	−18.8930	−0.634008
$889$	−48.8416	−1.63810
$890$	−21.3032	−0.714086
$891$	3.32880	0.111519
$892$	−73.4888	−2.46059
$893$	2.70922	0.0906604
$894$	−27.6005	−0.923098
$895$	19.1192	0.639085
$896$	−91.7059	−3.06368
$897$	−0.550856	−0.0183925
$898$	82.7881	2.76267
$899$	−7.80925	−0.260453
$900$	4.07370	0.135790
$901$	−77.9994	−2.59854
$902$	98.9012	3.29305
$903$	−52.9478	−1.76199
$904$	40.0170	1.33095
$905$	−5.25064	−0.174537
$906$	−46.8028	−1.55492
$907$	−50.1066	−1.66376	−0.831881	−	0.554954i	$-0.812736\pi$
$907$	−50.1066	−1.66376	−0.831881	+	0.554954i	$0.812736\pi$
$908$	−28.0716	−0.931590
$909$	−17.8911	−0.593410
$910$	11.9530	0.396237
$911$	49.3252	1.63422	0.817108	−	0.576485i	$-0.195576\pi$
$911$	49.3252	1.63422	0.817108	+	0.576485i	$0.195576\pi$
$912$	8.46575	0.280329
$913$	−59.0092	−1.95292
$914$	32.1376	1.06302
$915$	8.37257	0.276788
$916$	−28.8791	−0.954194
$917$	−8.42187	−0.278115
$918$	−17.1744	−0.566839
$919$	45.1671	1.48993	0.744963	−	0.667106i	$-0.232467\pi$
$919$	45.1671	1.48993	0.744963	+	0.667106i	$0.232467\pi$
$920$	−2.81520	−0.0928146
$921$	−16.1701	−0.532822
$922$	36.7421	1.21004
$923$	11.6231	0.382580
$924$	65.7698	2.16367
$925$	−3.69683	−0.121551
$926$	−27.6438	−0.908433
$927$	1.10847	0.0364069
$928$	5.77805	0.189674
$929$	11.6150	0.381075	0.190538	−	0.981680i	$-0.438977\pi$
$929$	11.6150	0.381075	0.190538	+	0.981680i	$0.438977\pi$
$930$	−2.46449	−0.0808137
$931$	−31.4511	−1.03077
$932$	−15.4263	−0.505306
$933$	8.39622	0.274880
$934$	4.67207	0.152875
$935$	23.1975	0.758641
$936$	5.11060	0.167045
$937$	7.10150	0.231996	0.115998	−	0.993249i	$-0.462993\pi$
$937$	7.10150	0.231996	0.115998	+	0.993249i	$0.462993\pi$
$938$	−27.9743	−0.913394
$939$	−7.64324	−0.249428
$940$	5.79822	0.189117
$941$	6.11300	0.199278	0.0996391	−	0.995024i	$-0.468231\pi$
$941$	6.11300	0.199278	0.0996391	+	0.995024i	$0.468231\pi$
$942$	14.2790	0.465235
$943$	6.64087	0.216256
$944$	2.33373	0.0759564
$945$	−4.85009	−0.157773
$946$	89.5598	2.91184
$947$	58.1976	1.89117	0.945583	−	0.325381i	$-0.105492\pi$
$947$	58.1976	1.89117	0.945583	+	0.325381i	$0.105492\pi$
$948$	14.3061	0.464639
$949$	−10.4332	−0.338677
$950$	4.69099	0.152196
$951$	5.68994	0.184509
$952$	−172.733	−5.59832
$953$	3.69193	0.119593	0.0597966	−	0.998211i	$-0.480955\pi$
$953$	3.69193	0.119593	0.0597966	+	0.998211i	$0.480955\pi$
$954$	−27.5844	−0.893078
$955$	17.0100	0.550429
$956$	49.3890	1.59735
$957$	25.9954	0.840313
$958$	−19.9472	−0.644465
$959$	−14.9163	−0.481673
$960$	−7.07177	−0.228240
$961$	1.00000	0.0322581
$962$	−9.11078	−0.293743
$963$	−1.87626	−0.0604618
$964$	39.8483	1.28343
$965$	16.1066	0.518490
$966$	6.58437	0.211849
$967$	−13.6982	−0.440504	−0.220252	−	0.975443i	$-0.570688\pi$
$967$	−13.6982	−0.440504	−0.220252	+	0.975443i	$0.570688\pi$
$968$	−0.413495	−0.0132902
$969$	−13.2645	−0.426118
$970$	−34.4822	−1.10716
$971$	−43.6680	−1.40137	−0.700686	−	0.713470i	$-0.747122\pi$
$971$	−43.6680	−1.40137	−0.700686	+	0.713470i	$0.747122\pi$
$972$	−4.07370	−0.130664
$973$	−60.9938	−1.95537
$974$	−7.91469	−0.253603
$975$	1.00000	0.0320256
$976$	37.2380	1.19196
$977$	−7.32338	−0.234296	−0.117148	−	0.993114i	$-0.537375\pi$
$977$	−7.32338	−0.234296	−0.117148	+	0.993114i	$0.537375\pi$
$978$	30.1250	0.963292
$979$	−28.7744	−0.919634
$980$	−67.3112	−2.15018
$981$	−10.5745	−0.337616
$982$	−76.5786	−2.44372
$983$	37.4964	1.19595	0.597974	−	0.801515i	$-0.295972\pi$
$983$	37.4964	1.19595	0.597974	+	0.801515i	$0.295972\pi$
$984$	−61.6111	−1.96409
$985$	4.29932	0.136988
$986$	−134.119	−4.27122
$987$	−6.90328	−0.219734
$988$	7.75401	0.246688
$989$	6.01363	0.191222
$990$	8.20379	0.260733
$991$	−1.04177	−0.0330930	−0.0165465	−	0.999863i	$-0.505267\pi$
$991$	−1.04177	−0.0330930	−0.0165465	+	0.999863i	$0.505267\pi$
$992$	−0.739899	−0.0234918
$993$	23.4047	0.742726
$994$	−138.931	−4.40663
$995$	−20.3250	−0.644346
$996$	72.2139	2.28818
$997$	36.2950	1.14947	0.574737	−	0.818338i	$-0.305105\pi$
$997$	36.2950	1.14947	0.574737	+	0.818338i	$0.305105\pi$
$998$	−41.1806	−1.30355
$999$	3.69683	0.116962

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6045.2.a.bg.1.3	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6045.2.a.bg.1.3	✓	16	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:	\( N \)	\(=\)	\( 6045 = 3 \cdot 5 \cdot 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6045.a (trivial)

\(f(q)\)	\(=\)	\(q-2.46449 q^{2} -1.00000 q^{3} +4.07370 q^{4} -1.00000 q^{5} +2.46449 q^{6} -4.85009 q^{7} -5.11060 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.46449 q^{2} -1.00000 q^{3} +4.07370 q^{4} -1.00000 q^{5} +2.46449 q^{6} -4.85009 q^{7} -5.11060 q^{8} +1.00000 q^{9} +2.46449 q^{10} +3.32880 q^{11} -4.07370 q^{12} -1.00000 q^{13} +11.9530 q^{14} +1.00000 q^{15} +4.44762 q^{16} -6.96874 q^{17} -2.46449 q^{18} -1.90343 q^{19} -4.07370 q^{20} +4.85009 q^{21} -8.20379 q^{22} -0.550856 q^{23} +5.11060 q^{24} +1.00000 q^{25} +2.46449 q^{26} -1.00000 q^{27} -19.7578 q^{28} -7.80925 q^{29} -2.46449 q^{30} +1.00000 q^{31} -0.739899 q^{32} -3.32880 q^{33} +17.1744 q^{34} +4.85009 q^{35} +4.07370 q^{36} -3.69683 q^{37} +4.69099 q^{38} +1.00000 q^{39} +5.11060 q^{40} -12.0556 q^{41} -11.9530 q^{42} -10.9169 q^{43} +13.5605 q^{44} -1.00000 q^{45} +1.35758 q^{46} -1.42333 q^{47} -4.44762 q^{48} +16.5234 q^{49} -2.46449 q^{50} +6.96874 q^{51} -4.07370 q^{52} +11.1927 q^{53} +2.46449 q^{54} -3.32880 q^{55} +24.7869 q^{56} +1.90343 q^{57} +19.2458 q^{58} +0.524714 q^{59} +4.07370 q^{60} +8.37257 q^{61} -2.46449 q^{62} -4.85009 q^{63} -7.07177 q^{64} +1.00000 q^{65} +8.20379 q^{66} -2.34036 q^{67} -28.3885 q^{68} +0.550856 q^{69} -11.9530 q^{70} -11.6231 q^{71} -5.11060 q^{72} +10.4332 q^{73} +9.11078 q^{74} -1.00000 q^{75} -7.75401 q^{76} -16.1450 q^{77} -2.46449 q^{78} -3.51181 q^{79} -4.44762 q^{80} +1.00000 q^{81} +29.7108 q^{82} -17.7269 q^{83} +19.7578 q^{84} +6.96874 q^{85} +26.9045 q^{86} +7.80925 q^{87} -17.0122 q^{88} -8.64408 q^{89} +2.46449 q^{90} +4.85009 q^{91} -2.24402 q^{92} -1.00000 q^{93} +3.50778 q^{94} +1.90343 q^{95} +0.739899 q^{96} -13.9916 q^{97} -40.7216 q^{98} +3.32880 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 2 q^{2} - 16 q^{3} + 26 q^{4} - 16 q^{5} + 2 q^{6} - 2 q^{7} - 9 q^{8} + 16 q^{9}+O(q^{10}) \) 16 * q - 2 * q^2 - 16 * q^3 + 26 * q^4 - 16 * q^5 + 2 * q^6 - 2 * q^7 - 9 * q^8 + 16 * q^9	\( 16 q - 2 q^{2} - 16 q^{3} + 26 q^{4} - 16 q^{5} + 2 q^{6} - 2 q^{7} - 9 q^{8} + 16 q^{9} + 2 q^{10} + 3 q^{11} - 26 q^{12} - 16 q^{13} - 5 q^{14} + 16 q^{15} + 38 q^{16} - 13 q^{17} - 2 q^{18} - 26 q^{20} + 2 q^{21} + q^{22} - 15 q^{23} + 9 q^{24} + 16 q^{25} + 2 q^{26} - 16 q^{27} + 8 q^{28} - 4 q^{29} - 2 q^{30} + 16 q^{31} - 30 q^{32} - 3 q^{33} + 29 q^{34} + 2 q^{35} + 26 q^{36} + 12 q^{37} + 16 q^{39} + 9 q^{40} - 12 q^{41} + 5 q^{42} - 7 q^{43} - 13 q^{44} - 16 q^{45} + 14 q^{46} + 17 q^{47} - 38 q^{48} + 16 q^{49} - 2 q^{50} + 13 q^{51} - 26 q^{52} - 36 q^{53} + 2 q^{54} - 3 q^{55} + 41 q^{56} + 16 q^{58} + 53 q^{59} + 26 q^{60} + 34 q^{61} - 2 q^{62} - 2 q^{63} + 79 q^{64} + 16 q^{65} - q^{66} - 13 q^{67} - 39 q^{68} + 15 q^{69} + 5 q^{70} - 11 q^{71} - 9 q^{72} + 34 q^{73} - 12 q^{74} - 16 q^{75} + 86 q^{76} - 32 q^{77} - 2 q^{78} - 7 q^{79} - 38 q^{80} + 16 q^{81} + 27 q^{82} - 28 q^{83} - 8 q^{84} + 13 q^{85} + 38 q^{86} + 4 q^{87} + 23 q^{88} - 8 q^{89} + 2 q^{90} + 2 q^{91} - 71 q^{92} - 16 q^{93} + 66 q^{94} + 30 q^{96} + 4 q^{97} + 22 q^{98} + 3 q^{99}+O(q^{100}) \) 16 * q - 2 * q^2 - 16 * q^3 + 26 * q^4 - 16 * q^5 + 2 * q^6 - 2 * q^7 - 9 * q^8 + 16 * q^9 + 2 * q^10 + 3 * q^11 - 26 * q^12 - 16 * q^13 - 5 * q^14 + 16 * q^15 + 38 * q^16 - 13 * q^17 - 2 * q^18 - 26 * q^20 + 2 * q^21 + q^22 - 15 * q^23 + 9 * q^24 + 16 * q^25 + 2 * q^26 - 16 * q^27 + 8 * q^28 - 4 * q^29 - 2 * q^30 + 16 * q^31 - 30 * q^32 - 3 * q^33 + 29 * q^34 + 2 * q^35 + 26 * q^36 + 12 * q^37 + 16 * q^39 + 9 * q^40 - 12 * q^41 + 5 * q^42 - 7 * q^43 - 13 * q^44 - 16 * q^45 + 14 * q^46 + 17 * q^47 - 38 * q^48 + 16 * q^49 - 2 * q^50 + 13 * q^51 - 26 * q^52 - 36 * q^53 + 2 * q^54 - 3 * q^55 + 41 * q^56 + 16 * q^58 + 53 * q^59 + 26 * q^60 + 34 * q^61 - 2 * q^62 - 2 * q^63 + 79 * q^64 + 16 * q^65 - q^66 - 13 * q^67 - 39 * q^68 + 15 * q^69 + 5 * q^70 - 11 * q^71 - 9 * q^72 + 34 * q^73 - 12 * q^74 - 16 * q^75 + 86 * q^76 - 32 * q^77 - 2 * q^78 - 7 * q^79 - 38 * q^80 + 16 * q^81 + 27 * q^82 - 28 * q^83 - 8 * q^84 + 13 * q^85 + 38 * q^86 + 4 * q^87 + 23 * q^88 - 8 * q^89 + 2 * q^90 + 2 * q^91 - 71 * q^92 - 16 * q^93 + 66 * q^94 + 30 * q^96 + 4 * q^97 + 22 * q^98 + 3 * q^99

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