LMFDB - Embedded newform 6045.2.a.y.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:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} - 15 x^{10} + 46 x^{9} + 78 x^{8} - 252 x^{7} - 149 x^{6} + 583 x^{5} + 18 x^{4} + \cdots - 26 $ x^12 - 3x^11 - 15x^10 + 46x^9 + 78x^8 - 252x^7 - 149x^6 + 583x^5 + 18x^4 - 511x^3 + 134x^2 + 98*x - 26
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.32897$ 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)

$f(q)$	$=$	$q-2.32897 q^{2} +1.00000 q^{3} +3.42411 q^{4} -1.00000 q^{5} -2.32897 q^{6} -4.93080 q^{7} -3.31671 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.32897 q^{2} +1.00000 q^{3} +3.42411 q^{4} -1.00000 q^{5} -2.32897 q^{6} -4.93080 q^{7} -3.31671 q^{8} +1.00000 q^{9} +2.32897 q^{10} -6.00408 q^{11} +3.42411 q^{12} +1.00000 q^{13} +11.4837 q^{14} -1.00000 q^{15} +0.876314 q^{16} -0.940198 q^{17} -2.32897 q^{18} +7.87235 q^{19} -3.42411 q^{20} -4.93080 q^{21} +13.9833 q^{22} +7.30398 q^{23} -3.31671 q^{24} +1.00000 q^{25} -2.32897 q^{26} +1.00000 q^{27} -16.8836 q^{28} -4.21777 q^{29} +2.32897 q^{30} -1.00000 q^{31} +4.59252 q^{32} -6.00408 q^{33} +2.18969 q^{34} +4.93080 q^{35} +3.42411 q^{36} -4.83866 q^{37} -18.3345 q^{38} +1.00000 q^{39} +3.31671 q^{40} -3.06340 q^{41} +11.4837 q^{42} +1.58879 q^{43} -20.5586 q^{44} -1.00000 q^{45} -17.0108 q^{46} +6.21554 q^{47} +0.876314 q^{48} +17.3127 q^{49} -2.32897 q^{50} -0.940198 q^{51} +3.42411 q^{52} -0.284478 q^{53} -2.32897 q^{54} +6.00408 q^{55} +16.3540 q^{56} +7.87235 q^{57} +9.82307 q^{58} -9.05335 q^{59} -3.42411 q^{60} +1.36460 q^{61} +2.32897 q^{62} -4.93080 q^{63} -12.4485 q^{64} -1.00000 q^{65} +13.9833 q^{66} +9.52801 q^{67} -3.21934 q^{68} +7.30398 q^{69} -11.4837 q^{70} -5.95836 q^{71} -3.31671 q^{72} +8.26432 q^{73} +11.2691 q^{74} +1.00000 q^{75} +26.9558 q^{76} +29.6049 q^{77} -2.32897 q^{78} -9.05249 q^{79} -0.876314 q^{80} +1.00000 q^{81} +7.13458 q^{82} +4.39592 q^{83} -16.8836 q^{84} +0.940198 q^{85} -3.70025 q^{86} -4.21777 q^{87} +19.9138 q^{88} -1.70370 q^{89} +2.32897 q^{90} -4.93080 q^{91} +25.0096 q^{92} -1.00000 q^{93} -14.4758 q^{94} -7.87235 q^{95} +4.59252 q^{96} -0.180652 q^{97} -40.3209 q^{98} -6.00408 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 3 q^{2} + 12 q^{3} + 15 q^{4} - 12 q^{5} - 3 q^{6} + 2 q^{7} - 12 q^{8} + 12 q^{9}+O(q^{10}) $ 12 * q - 3 * q^2 + 12 * q^3 + 15 * q^4 - 12 * q^5 - 3 * q^6 + 2 * q^7 - 12 * q^8 + 12 * q^9	\( 12 q - 3 q^{2} + 12 q^{3} + 15 q^{4} - 12 q^{5} - 3 q^{6} + 2 q^{7} - 12 q^{8} + 12 q^{9} + 3 q^{10} - 24 q^{11} + 15 q^{12} + 12 q^{13} - 13 q^{14} - 12 q^{15} + 21 q^{16} - 9 q^{17} - 3 q^{18} - 2 q^{19} - 15 q^{20} + 2 q^{21} + 9 q^{22} + 7 q^{23} - 12 q^{24} + 12 q^{25} - 3 q^{26} + 12 q^{27} - 16 q^{28} - 24 q^{29} + 3 q^{30} - 12 q^{31} - 57 q^{32} - 24 q^{33} - 15 q^{34} - 2 q^{35} + 15 q^{36} - 13 q^{37} - 22 q^{38} + 12 q^{39} + 12 q^{40} - 46 q^{41} - 13 q^{42} - 5 q^{43} - 49 q^{44} - 12 q^{45} - 2 q^{46} - 39 q^{47} + 21 q^{48} - 3 q^{50} - 9 q^{51} + 15 q^{52} - 12 q^{53} - 3 q^{54} + 24 q^{55} + q^{56} - 2 q^{57} - 8 q^{58} - 31 q^{59} - 15 q^{60} + 2 q^{61} + 3 q^{62} + 2 q^{63} + 18 q^{64} - 12 q^{65} + 9 q^{66} + 5 q^{67} - 5 q^{68} + 7 q^{69} + 13 q^{70} - 31 q^{71} - 12 q^{72} + 3 q^{73} + 16 q^{74} + 12 q^{75} - 2 q^{76} + 2 q^{77} - 3 q^{78} + 5 q^{79} - 21 q^{80} + 12 q^{81} + 25 q^{82} + 2 q^{83} - 16 q^{84} + 9 q^{85} - 26 q^{86} - 24 q^{87} + 27 q^{88} - 13 q^{89} + 3 q^{90} + 2 q^{91} + 5 q^{92} - 12 q^{93} - 12 q^{94} + 2 q^{95} - 57 q^{96} - 28 q^{97} - 29 q^{98} - 24 q^{99}+O(q^{100}) \) 12 * q - 3 * q^2 + 12 * q^3 + 15 * q^4 - 12 * q^5 - 3 * q^6 + 2 * q^7 - 12 * q^8 + 12 * q^9 + 3 * q^10 - 24 * q^11 + 15 * q^12 + 12 * q^13 - 13 * q^14 - 12 * q^15 + 21 * q^16 - 9 * q^17 - 3 * q^18 - 2 * q^19 - 15 * q^20 + 2 * q^21 + 9 * q^22 + 7 * q^23 - 12 * q^24 + 12 * q^25 - 3 * q^26 + 12 * q^27 - 16 * q^28 - 24 * q^29 + 3 * q^30 - 12 * q^31 - 57 * q^32 - 24 * q^33 - 15 * q^34 - 2 * q^35 + 15 * q^36 - 13 * q^37 - 22 * q^38 + 12 * q^39 + 12 * q^40 - 46 * q^41 - 13 * q^42 - 5 * q^43 - 49 * q^44 - 12 * q^45 - 2 * q^46 - 39 * q^47 + 21 * q^48 - 3 * q^50 - 9 * q^51 + 15 * q^52 - 12 * q^53 - 3 * q^54 + 24 * q^55 + q^56 - 2 * q^57 - 8 * q^58 - 31 * q^59 - 15 * q^60 + 2 * q^61 + 3 * q^62 + 2 * q^63 + 18 * q^64 - 12 * q^65 + 9 * q^66 + 5 * q^67 - 5 * q^68 + 7 * q^69 + 13 * q^70 - 31 * q^71 - 12 * q^72 + 3 * q^73 + 16 * q^74 + 12 * q^75 - 2 * q^76 + 2 * q^77 - 3 * q^78 + 5 * q^79 - 21 * q^80 + 12 * q^81 + 25 * q^82 + 2 * q^83 - 16 * q^84 + 9 * q^85 - 26 * q^86 - 24 * q^87 + 27 * q^88 - 13 * q^89 + 3 * q^90 + 2 * q^91 + 5 * q^92 - 12 * q^93 - 12 * q^94 + 2 * q^95 - 57 * q^96 - 28 * q^97 - 29 * q^98 - 24 * 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 $

$2$	−2.32897	−1.64683	−0.823416	−	0.567438i	$-0.807935\pi$
$2$	−2.32897	−1.64683	−0.823416	+	0.567438i	$0.807935\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	3.42411	1.71206
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−2.32897	−0.950799
$7$	−4.93080	−1.86367	−0.931833	−	0.362888i	$-0.881791\pi$
$7$	−4.93080	−1.86367	−0.931833	+	0.362888i	$0.881791\pi$
$8$	−3.31671	−1.17264
$9$	1.00000	0.333333
$10$	2.32897	0.736486
$11$	−6.00408	−1.81030	−0.905149	−	0.425094i	$-0.860241\pi$
$11$	−6.00408	−1.81030	−0.905149	+	0.425094i	$0.860241\pi$
$12$	3.42411	0.988456
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	11.4837	3.06914
$15$	−1.00000	−0.258199
$16$	0.876314	0.219079
$17$	−0.940198	−0.228031	−0.114016	−	0.993479i	$-0.536371\pi$
$17$	−0.940198	−0.228031	−0.114016	+	0.993479i	$0.536371\pi$
$18$	−2.32897	−0.548944
$19$	7.87235	1.80604	0.903021	−	0.429597i	$-0.141344\pi$
$19$	7.87235	1.80604	0.903021	+	0.429597i	$0.141344\pi$
$20$	−3.42411	−0.765655
$21$	−4.93080	−1.07599
$22$	13.9833	2.98126
$23$	7.30398	1.52299	0.761493	−	0.648173i	$-0.224467\pi$
$23$	7.30398	1.52299	0.761493	+	0.648173i	$0.224467\pi$
$24$	−3.31671	−0.677022
$25$	1.00000	0.200000
$26$	−2.32897	−0.456749
$27$	1.00000	0.192450
$28$	−16.8836	−3.19070
$29$	−4.21777	−0.783220	−0.391610	−	0.920131i	$-0.628082\pi$
$29$	−4.21777	−0.783220	−0.391610	+	0.920131i	$0.628082\pi$
$30$	2.32897	0.425210
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	4.59252	0.811850
$33$	−6.00408	−1.04518
$34$	2.18969	0.375530
$35$	4.93080	0.833456
$36$	3.42411	0.570685
$37$	−4.83866	−0.795470	−0.397735	−	0.917500i	$-0.630204\pi$
$37$	−4.83866	−0.795470	−0.397735	+	0.917500i	$0.630204\pi$
$38$	−18.3345	−2.97425
$39$	1.00000	0.160128
$40$	3.31671	0.524419
$41$	−3.06340	−0.478423	−0.239212	−	0.970967i	$-0.576889\pi$
$41$	−3.06340	−0.478423	−0.239212	+	0.970967i	$0.576889\pi$
$42$	11.4837	1.77197
$43$	1.58879	0.242288	0.121144	−	0.992635i	$-0.461344\pi$
$43$	1.58879	0.242288	0.121144	+	0.992635i	$0.461344\pi$
$44$	−20.5586	−3.09933
$45$	−1.00000	−0.149071
$46$	−17.0108	−2.50810
$47$	6.21554	0.906630	0.453315	−	0.891350i	$-0.350241\pi$
$47$	6.21554	0.906630	0.453315	+	0.891350i	$0.350241\pi$
$48$	0.876314	0.126485
$49$	17.3127	2.47325
$50$	−2.32897	−0.329366
$51$	−0.940198	−0.131654
$52$	3.42411	0.474839
$53$	−0.284478	−0.0390761	−0.0195380	−	0.999809i	$-0.506220\pi$
$53$	−0.284478	−0.0390761	−0.0195380	+	0.999809i	$0.506220\pi$
$54$	−2.32897	−0.316933
$55$	6.00408	0.809590
$56$	16.3540	2.18540
$57$	7.87235	1.04272
$58$	9.82307	1.28983
$59$	−9.05335	−1.17865	−0.589323	−	0.807897i	$-0.700606\pi$
$59$	−9.05335	−1.17865	−0.589323	+	0.807897i	$0.700606\pi$
$60$	−3.42411	−0.442051
$61$	1.36460	0.174720	0.0873599	−	0.996177i	$-0.472157\pi$
$61$	1.36460	0.174720	0.0873599	+	0.996177i	$0.472157\pi$
$62$	2.32897	0.295780
$63$	−4.93080	−0.621222
$64$	−12.4485	−1.55606
$65$	−1.00000	−0.124035
$66$	13.9833	1.72123
$67$	9.52801	1.16403	0.582016	−	0.813177i	$-0.302264\pi$
$67$	9.52801	1.16403	0.582016	+	0.813177i	$0.302264\pi$
$68$	−3.21934	−0.390403
$69$	7.30398	0.879296
$70$	−11.4837	−1.37256
$71$	−5.95836	−0.707127	−0.353563	−	0.935411i	$-0.615030\pi$
$71$	−5.95836	−0.707127	−0.353563	+	0.935411i	$0.615030\pi$
$72$	−3.31671	−0.390879
$73$	8.26432	0.967266	0.483633	−	0.875271i	$-0.339317\pi$
$73$	8.26432	0.967266	0.483633	+	0.875271i	$0.339317\pi$
$74$	11.2691	1.31001
$75$	1.00000	0.115470
$76$	26.9558	3.09204
$77$	29.6049	3.37379
$78$	−2.32897	−0.263704
$79$	−9.05249	−1.01848	−0.509242	−	0.860623i	$-0.670074\pi$
$79$	−9.05249	−1.01848	−0.509242	+	0.860623i	$0.670074\pi$
$80$	−0.876314	−0.0979749
$81$	1.00000	0.111111
$82$	7.13458	0.787883
$83$	4.39592	0.482515	0.241258	−	0.970461i	$-0.422440\pi$
$83$	4.39592	0.482515	0.241258	+	0.970461i	$0.422440\pi$
$84$	−16.8836	−1.84215
$85$	0.940198	0.101979
$86$	−3.70025	−0.399008
$87$	−4.21777	−0.452192
$88$	19.9138	2.12282
$89$	−1.70370	−0.180592	−0.0902961	−	0.995915i	$-0.528781\pi$
$89$	−1.70370	−0.180592	−0.0902961	+	0.995915i	$0.528781\pi$
$90$	2.32897	0.245495
$91$	−4.93080	−0.516888
$92$	25.0096	2.60744
$93$	−1.00000	−0.103695
$94$	−14.4758	−1.49307
$95$	−7.87235	−0.807686
$96$	4.59252	0.468722
$97$	−0.180652	−0.0183424	−0.00917120	−	0.999958i	$-0.502919\pi$
$97$	−0.180652	−0.0183424	−0.00917120	+	0.999958i	$0.502919\pi$
$98$	−40.3209	−4.07302
$99$	−6.00408	−0.603433
$100$	3.42411	0.342411
$101$	11.5574	1.15000	0.575002	−	0.818152i	$-0.305001\pi$
$101$	11.5574	1.15000	0.575002	+	0.818152i	$0.305001\pi$
$102$	2.18969	0.216812
$103$	11.1097	1.09467	0.547335	−	0.836913i	$-0.315642\pi$
$103$	11.1097	1.09467	0.547335	+	0.836913i	$0.315642\pi$
$104$	−3.31671	−0.325231
$105$	4.93080	0.481196
$106$	0.662541	0.0643517
$107$	17.1849	1.66133	0.830666	−	0.556772i	$-0.187960\pi$
$107$	17.1849	1.66133	0.830666	+	0.556772i	$0.187960\pi$
$108$	3.42411	0.329485
$109$	16.3676	1.56773	0.783866	−	0.620930i	$-0.213245\pi$
$109$	16.3676	1.56773	0.783866	+	0.620930i	$0.213245\pi$
$110$	−13.9833	−1.33326
$111$	−4.83866	−0.459265
$112$	−4.32093	−0.408289
$113$	−19.6432	−1.84788	−0.923938	−	0.382541i	$-0.875049\pi$
$113$	−19.6432	−1.84788	−0.923938	+	0.382541i	$0.875049\pi$
$114$	−18.3345	−1.71718
$115$	−7.30398	−0.681100
$116$	−14.4421	−1.34092
$117$	1.00000	0.0924500
$118$	21.0850	1.94103
$119$	4.63592	0.424974
$120$	3.31671	0.302773
$121$	25.0490	2.27718
$122$	−3.17813	−0.287734
$123$	−3.06340	−0.276218
$124$	−3.42411	−0.307494
$125$	−1.00000	−0.0894427
$126$	11.4837	1.02305
$127$	8.17858	0.725731	0.362866	−	0.931841i	$-0.381798\pi$
$127$	8.17858	0.725731	0.362866	+	0.931841i	$0.381798\pi$
$128$	19.8071	1.75072
$129$	1.58879	0.139885
$130$	2.32897	0.204264
$131$	−3.41088	−0.298010	−0.149005	−	0.988836i	$-0.547607\pi$
$131$	−3.41088	−0.298010	−0.149005	+	0.988836i	$0.547607\pi$
$132$	−20.5586	−1.78940
$133$	−38.8170	−3.36586
$134$	−22.1905	−1.91697
$135$	−1.00000	−0.0860663
$136$	3.11837	0.267398
$137$	−20.8888	−1.78465	−0.892327	−	0.451390i	$-0.850928\pi$
$137$	−20.8888	−1.78465	−0.892327	+	0.451390i	$0.850928\pi$
$138$	−17.0108	−1.44805
$139$	−0.355010	−0.0301116	−0.0150558	−	0.999887i	$-0.504793\pi$
$139$	−0.355010	−0.0301116	−0.0150558	+	0.999887i	$0.504793\pi$
$140$	16.8836	1.42692
$141$	6.21554	0.523443
$142$	13.8768	1.16452
$143$	−6.00408	−0.502086
$144$	0.876314	0.0730262
$145$	4.21777	0.350267
$146$	−19.2474	−1.59292
$147$	17.3127	1.42793
$148$	−16.5681	−1.36189
$149$	1.37226	0.112420	0.0562101	−	0.998419i	$-0.482098\pi$
$149$	1.37226	0.112420	0.0562101	+	0.998419i	$0.482098\pi$
$150$	−2.32897	−0.190160
$151$	−11.2875	−0.918560	−0.459280	−	0.888292i	$-0.651893\pi$
$151$	−11.2875	−0.918560	−0.459280	+	0.888292i	$0.651893\pi$
$152$	−26.1104	−2.11783
$153$	−0.940198	−0.0760105
$154$	−68.9490	−5.55606
$155$	1.00000	0.0803219
$156$	3.42411	0.274148
$157$	15.0194	1.19868	0.599340	−	0.800494i	$-0.295430\pi$
$157$	15.0194	1.19868	0.599340	+	0.800494i	$0.295430\pi$
$158$	21.0830	1.67727
$159$	−0.284478	−0.0225606
$160$	−4.59252	−0.363070
$161$	−36.0144	−2.83834
$162$	−2.32897	−0.182981
$163$	−11.7559	−0.920792	−0.460396	−	0.887714i	$-0.652293\pi$
$163$	−11.7559	−0.920792	−0.460396	+	0.887714i	$0.652293\pi$
$164$	−10.4894	−0.819087
$165$	6.00408	0.467417
$166$	−10.2380	−0.794621
$167$	12.0336	0.931191	0.465596	−	0.884998i	$-0.345840\pi$
$167$	12.0336	0.931191	0.465596	+	0.884998i	$0.345840\pi$
$168$	16.3540	1.26174
$169$	1.00000	0.0769231
$170$	−2.18969	−0.167942
$171$	7.87235	0.602014
$172$	5.44019	0.414811
$173$	4.65346	0.353796	0.176898	−	0.984229i	$-0.443394\pi$
$173$	4.65346	0.353796	0.176898	+	0.984229i	$0.443394\pi$
$174$	9.82307	0.744685
$175$	−4.93080	−0.372733
$176$	−5.26146	−0.396597
$177$	−9.05335	−0.680492
$178$	3.96788	0.297405
$179$	−13.7254	−1.02589	−0.512944	−	0.858422i	$-0.671445\pi$
$179$	−13.7254	−1.02589	−0.512944	+	0.858422i	$0.671445\pi$
$180$	−3.42411	−0.255218
$181$	−11.0790	−0.823494	−0.411747	−	0.911298i	$-0.635081\pi$
$181$	−11.0790	−0.823494	−0.411747	+	0.911298i	$0.635081\pi$
$182$	11.4837	0.851227
$183$	1.36460	0.100874
$184$	−24.2252	−1.78591
$185$	4.83866	0.355745
$186$	2.32897	0.170769
$187$	5.64502	0.412805
$188$	21.2827	1.55220
$189$	−4.93080	−0.358663
$190$	18.3345	1.33012
$191$	−12.7826	−0.924914	−0.462457	−	0.886642i	$-0.653032\pi$
$191$	−12.7826	−0.924914	−0.462457	+	0.886642i	$0.653032\pi$
$192$	−12.4485	−0.898391
$193$	−16.9992	−1.22363	−0.611815	−	0.791001i	$-0.709560\pi$
$193$	−16.9992	−1.22363	−0.611815	+	0.791001i	$0.709560\pi$
$194$	0.420733	0.0302068
$195$	−1.00000	−0.0716115
$196$	59.2807	4.23434
$197$	−25.6285	−1.82595	−0.912977	−	0.408012i	$-0.866222\pi$
$197$	−25.6285	−1.82595	−0.912977	+	0.408012i	$0.866222\pi$
$198$	13.9833	0.993752
$199$	3.38769	0.240147	0.120073	−	0.992765i	$-0.461687\pi$
$199$	3.38769	0.240147	0.120073	+	0.992765i	$0.461687\pi$
$200$	−3.31671	−0.234527
$201$	9.52801	0.672054
$202$	−26.9168	−1.89386
$203$	20.7970	1.45966
$204$	−3.21934	−0.225399
$205$	3.06340	0.213957
$206$	−25.8742	−1.80274
$207$	7.30398	0.507662
$208$	0.876314	0.0607615
$209$	−47.2662	−3.26947
$210$	−11.4837	−0.792450
$211$	−3.77438	−0.259839	−0.129919	−	0.991525i	$-0.541472\pi$
$211$	−3.77438	−0.259839	−0.129919	+	0.991525i	$0.541472\pi$
$212$	−0.974084	−0.0669004
$213$	−5.95836	−0.408260
$214$	−40.0233	−2.73593
$215$	−1.58879	−0.108355
$216$	−3.31671	−0.225674
$217$	4.93080	0.334724
$218$	−38.1197	−2.58179
$219$	8.26432	0.558451
$220$	20.5586	1.38606
$221$	−0.940198	−0.0632446
$222$	11.2691	0.756332
$223$	18.9882	1.27155	0.635773	−	0.771876i	$-0.280682\pi$
$223$	18.9882	1.27155	0.635773	+	0.771876i	$0.280682\pi$
$224$	−22.6448	−1.51302
$225$	1.00000	0.0666667
$226$	45.7485	3.04314
$227$	−5.53568	−0.367416	−0.183708	−	0.982981i	$-0.558810\pi$
$227$	−5.53568	−0.367416	−0.183708	+	0.982981i	$0.558810\pi$
$228$	26.9558	1.78519
$229$	15.1755	1.00282	0.501412	−	0.865209i	$-0.332814\pi$
$229$	15.1755	1.00282	0.501412	+	0.865209i	$0.332814\pi$
$230$	17.0108	1.12166
$231$	29.6049	1.94786
$232$	13.9891	0.918432
$233$	23.8868	1.56488	0.782438	−	0.622728i	$-0.213976\pi$
$233$	23.8868	1.56488	0.782438	+	0.622728i	$0.213976\pi$
$234$	−2.32897	−0.152250
$235$	−6.21554	−0.405457
$236$	−30.9997	−2.01791
$237$	−9.05249	−0.588022
$238$	−10.7969	−0.699861
$239$	−0.171715	−0.0111073	−0.00555365	−	0.999985i	$-0.501768\pi$
$239$	−0.171715	−0.0111073	−0.00555365	+	0.999985i	$0.501768\pi$
$240$	−0.876314	−0.0565658
$241$	−8.42128	−0.542462	−0.271231	−	0.962514i	$-0.587431\pi$
$241$	−8.42128	−0.542462	−0.271231	+	0.962514i	$0.587431\pi$
$242$	−58.3383	−3.75013
$243$	1.00000	0.0641500
$244$	4.67256	0.299130
$245$	−17.3127	−1.10607
$246$	7.13458	0.454884
$247$	7.87235	0.500906
$248$	3.31671	0.210612
$249$	4.39592	0.278580
$250$	2.32897	0.147297
$251$	−7.36219	−0.464698	−0.232349	−	0.972633i	$-0.574641\pi$
$251$	−7.36219	−0.464698	−0.232349	+	0.972633i	$0.574641\pi$
$252$	−16.8836	−1.06357
$253$	−43.8537	−2.75706
$254$	−19.0477	−1.19516
$255$	0.940198	0.0588775
$256$	−21.2333	−1.32708
$257$	−23.1387	−1.44335	−0.721675	−	0.692232i	$-0.756627\pi$
$257$	−23.1387	−1.44335	−0.721675	+	0.692232i	$0.756627\pi$
$258$	−3.70025	−0.230367
$259$	23.8584	1.48249
$260$	−3.42411	−0.212354
$261$	−4.21777	−0.261073
$262$	7.94385	0.490773
$263$	−12.8305	−0.791164	−0.395582	−	0.918431i	$-0.629457\pi$
$263$	−12.8305	−0.791164	−0.395582	+	0.918431i	$0.629457\pi$
$264$	19.9138	1.22561
$265$	0.284478	0.0174753
$266$	90.4036	5.54300
$267$	−1.70370	−0.104265
$268$	32.6250	1.99289
$269$	4.05366	0.247156	0.123578	−	0.992335i	$-0.460563\pi$
$269$	4.05366	0.247156	0.123578	+	0.992335i	$0.460563\pi$
$270$	2.32897	0.141737
$271$	−18.5373	−1.12606	−0.563030	−	0.826436i	$-0.690364\pi$
$271$	−18.5373	−1.12606	−0.563030	+	0.826436i	$0.690364\pi$
$272$	−0.823909	−0.0499568
$273$	−4.93080	−0.298425
$274$	48.6495	2.93903
$275$	−6.00408	−0.362060
$276$	25.0096	1.50540
$277$	−8.22400	−0.494132	−0.247066	−	0.968999i	$-0.579466\pi$
$277$	−8.22400	−0.494132	−0.247066	+	0.968999i	$0.579466\pi$
$278$	0.826809	0.0495887
$279$	−1.00000	−0.0598684
$280$	−16.3540	−0.977341
$281$	−15.0241	−0.896261	−0.448131	−	0.893968i	$-0.647910\pi$
$281$	−15.0241	−0.896261	−0.448131	+	0.893968i	$0.647910\pi$
$282$	−14.4758	−0.862023
$283$	−22.3873	−1.33078	−0.665392	−	0.746494i	$-0.731736\pi$
$283$	−22.3873	−1.33078	−0.665392	+	0.746494i	$0.731736\pi$
$284$	−20.4021	−1.21064
$285$	−7.87235	−0.466318
$286$	13.9833	0.826852
$287$	15.1050	0.891621
$288$	4.59252	0.270617
$289$	−16.1160	−0.948002
$290$	−9.82307	−0.576831
$291$	−0.180652	−0.0105900
$292$	28.2980	1.65601
$293$	24.4996	1.43128	0.715641	−	0.698468i	$-0.246135\pi$
$293$	24.4996	1.43128	0.715641	+	0.698468i	$0.246135\pi$
$294$	−40.3209	−2.35156
$295$	9.05335	0.527107
$296$	16.0484	0.932797
$297$	−6.00408	−0.348392
$298$	−3.19596	−0.185137
$299$	7.30398	0.422400
$300$	3.42411	0.197691
$301$	−7.83400	−0.451544
$302$	26.2882	1.51271
$303$	11.5574	0.663955
$304$	6.89866	0.395665
$305$	−1.36460	−0.0781370
$306$	2.18969	0.125177
$307$	2.25448	0.128670	0.0643349	−	0.997928i	$-0.479507\pi$
$307$	2.25448	0.128670	0.0643349	+	0.997928i	$0.479507\pi$
$308$	101.370	5.77612
$309$	11.1097	0.632009
$310$	−2.32897	−0.132277
$311$	−26.8090	−1.52020	−0.760100	−	0.649806i	$-0.774850\pi$
$311$	−26.8090	−1.52020	−0.760100	+	0.649806i	$0.774850\pi$
$312$	−3.31671	−0.187772
$313$	13.9494	0.788468	0.394234	−	0.919010i	$-0.371010\pi$
$313$	13.9494	0.788468	0.394234	+	0.919010i	$0.371010\pi$
$314$	−34.9798	−1.97403
$315$	4.93080	0.277819
$316$	−30.9967	−1.74370
$317$	9.17058	0.515071	0.257536	−	0.966269i	$-0.417090\pi$
$317$	9.17058	0.515071	0.257536	+	0.966269i	$0.417090\pi$
$318$	0.662541	0.0371535
$319$	25.3238	1.41786
$320$	12.4485	0.695891
$321$	17.1849	0.959170
$322$	83.8766	4.67426
$323$	−7.40157	−0.411834
$324$	3.42411	0.190228
$325$	1.00000	0.0554700
$326$	27.3791	1.51639
$327$	16.3676	0.905131
$328$	10.1604	0.561016
$329$	−30.6476	−1.68966
$330$	−13.9833	−0.769757
$331$	−25.7926	−1.41769	−0.708846	−	0.705363i	$-0.750784\pi$
$331$	−25.7926	−1.41769	−0.708846	+	0.705363i	$0.750784\pi$
$332$	15.0521	0.826093
$333$	−4.83866	−0.265157
$334$	−28.0260	−1.53352
$335$	−9.52801	−0.520571
$336$	−4.32093	−0.235726
$337$	−17.0223	−0.927263	−0.463632	−	0.886028i	$-0.653454\pi$
$337$	−17.0223	−0.927263	−0.463632	+	0.886028i	$0.653454\pi$
$338$	−2.32897	−0.126679
$339$	−19.6432	−1.06687
$340$	3.21934	0.174593
$341$	6.00408	0.325139
$342$	−18.3345	−0.991416
$343$	−50.8500	−2.74564
$344$	−5.26956	−0.284116
$345$	−7.30398	−0.393233
$346$	−10.8378	−0.582643
$347$	−19.4947	−1.04653	−0.523265	−	0.852170i	$-0.675286\pi$
$347$	−19.4947	−1.04653	−0.523265	+	0.852170i	$0.675286\pi$
$348$	−14.4421	−0.774179
$349$	−18.7154	−1.00181	−0.500906	−	0.865502i	$-0.667000\pi$
$349$	−18.7154	−1.00181	−0.500906	+	0.865502i	$0.667000\pi$
$350$	11.4837	0.613829
$351$	1.00000	0.0533761
$352$	−27.5738	−1.46969
$353$	18.2047	0.968937	0.484468	−	0.874809i	$-0.339013\pi$
$353$	18.2047	0.968937	0.484468	+	0.874809i	$0.339013\pi$
$354$	21.0850	1.12066
$355$	5.95836	0.316237
$356$	−5.83367	−0.309184
$357$	4.63592	0.245359
$358$	31.9662	1.68947
$359$	12.4715	0.658222	0.329111	−	0.944291i	$-0.393251\pi$
$359$	12.4715	0.658222	0.329111	+	0.944291i	$0.393251\pi$
$360$	3.31671	0.174806
$361$	42.9739	2.26179
$362$	25.8026	1.35616
$363$	25.0490	1.31473
$364$	−16.8836	−0.884941
$365$	−8.26432	−0.432574
$366$	−3.17813	−0.166123
$367$	34.7669	1.81482	0.907409	−	0.420248i	$-0.138057\pi$
$367$	34.7669	1.81482	0.907409	+	0.420248i	$0.138057\pi$
$368$	6.40058	0.333654
$369$	−3.06340	−0.159474
$370$	−11.2691	−0.585852
$371$	1.40270	0.0728247
$372$	−3.42411	−0.177532
$373$	28.2091	1.46061	0.730306	−	0.683120i	$-0.239377\pi$
$373$	28.2091	1.46061	0.730306	+	0.683120i	$0.239377\pi$
$374$	−13.1471	−0.679820
$375$	−1.00000	−0.0516398
$376$	−20.6152	−1.06315
$377$	−4.21777	−0.217226
$378$	11.4837	0.590657
$379$	−31.5983	−1.62310	−0.811549	−	0.584285i	$-0.801375\pi$
$379$	−31.5983	−1.62310	−0.811549	+	0.584285i	$0.801375\pi$
$380$	−26.9558	−1.38280
$381$	8.17858	0.419001
$382$	29.7702	1.52318
$383$	−33.9836	−1.73648	−0.868239	−	0.496146i	$-0.834748\pi$
$383$	−33.9836	−1.73648	−0.868239	+	0.496146i	$0.834748\pi$
$384$	19.8071	1.01078
$385$	−29.6049	−1.50880
$386$	39.5907	2.01511
$387$	1.58879	0.0807627
$388$	−0.618571	−0.0314032
$389$	−11.9841	−0.607620	−0.303810	−	0.952733i	$-0.598259\pi$
$389$	−11.9841	−0.607620	−0.303810	+	0.952733i	$0.598259\pi$
$390$	2.32897	0.117932
$391$	−6.86719	−0.347289
$392$	−57.4214	−2.90022
$393$	−3.41088	−0.172056
$394$	59.6880	3.00704
$395$	9.05249	0.455480
$396$	−20.5586	−1.03311
$397$	−14.7848	−0.742027	−0.371013	−	0.928628i	$-0.620990\pi$
$397$	−14.7848	−0.742027	−0.371013	+	0.928628i	$0.620990\pi$
$398$	−7.88984	−0.395482
$399$	−38.8170	−1.94328
$400$	0.876314	0.0438157
$401$	−11.0732	−0.552970	−0.276485	−	0.961018i	$-0.589170\pi$
$401$	−11.0732	−0.552970	−0.276485	+	0.961018i	$0.589170\pi$
$402$	−22.1905	−1.10676
$403$	−1.00000	−0.0498135
$404$	39.5738	1.96887
$405$	−1.00000	−0.0496904
$406$	−48.4355	−2.40382
$407$	29.0517	1.44004
$408$	3.11837	0.154382
$409$	29.1670	1.44222	0.721109	−	0.692822i	$-0.243633\pi$
$409$	29.1670	1.44222	0.721109	+	0.692822i	$0.243633\pi$
$410$	−7.13458	−0.352352
$411$	−20.8888	−1.03037
$412$	38.0408	1.87414
$413$	44.6402	2.19660
$414$	−17.0108	−0.836034
$415$	−4.39592	−0.215787
$416$	4.59252	0.225167
$417$	−0.355010	−0.0173849
$418$	110.082	5.38427
$419$	−8.64216	−0.422197	−0.211099	−	0.977465i	$-0.567704\pi$
$419$	−8.64216	−0.422197	−0.211099	+	0.977465i	$0.567704\pi$
$420$	16.8836	0.823835
$421$	22.2643	1.08509	0.542547	−	0.840025i	$-0.317460\pi$
$421$	22.2643	1.08509	0.542547	+	0.840025i	$0.317460\pi$
$422$	8.79042	0.427911
$423$	6.21554	0.302210
$424$	0.943532	0.0458220
$425$	−0.940198	−0.0456063
$426$	13.8768	0.672335
$427$	−6.72859	−0.325619
$428$	58.8432	2.84429
$429$	−6.00408	−0.289880
$430$	3.70025	0.178442
$431$	−15.6719	−0.754890	−0.377445	−	0.926032i	$-0.623197\pi$
$431$	−15.6719	−0.754890	−0.377445	+	0.926032i	$0.623197\pi$
$432$	0.876314	0.0421617
$433$	−12.5051	−0.600958	−0.300479	−	0.953788i	$-0.597147\pi$
$433$	−12.5051	−0.600958	−0.300479	+	0.953788i	$0.597147\pi$
$434$	−11.4837	−0.551234
$435$	4.21777	0.202227
$436$	56.0445	2.68405
$437$	57.4995	2.75058
$438$	−19.2474	−0.919675
$439$	30.0701	1.43517	0.717583	−	0.696473i	$-0.245248\pi$
$439$	30.0701	1.43517	0.717583	+	0.696473i	$0.245248\pi$
$440$	−19.9138	−0.949354
$441$	17.3127	0.824416
$442$	2.18969	0.104153
$443$	−15.7942	−0.750405	−0.375203	−	0.926943i	$-0.622427\pi$
$443$	−15.7942	−0.750405	−0.375203	+	0.926943i	$0.622427\pi$
$444$	−16.5681	−0.786287
$445$	1.70370	0.0807633
$446$	−44.2230	−2.09402
$447$	1.37226	0.0649058
$448$	61.3809	2.89997
$449$	30.0137	1.41643	0.708217	−	0.705995i	$-0.249500\pi$
$449$	30.0137	1.41643	0.708217	+	0.705995i	$0.249500\pi$
$450$	−2.32897	−0.109789
$451$	18.3929	0.866089
$452$	−67.2605	−3.16367
$453$	−11.2875	−0.530331
$454$	12.8924	0.605072
$455$	4.93080	0.231159
$456$	−26.1104	−1.22273
$457$	8.02711	0.375492	0.187746	−	0.982218i	$-0.439882\pi$
$457$	8.02711	0.375492	0.187746	+	0.982218i	$0.439882\pi$
$458$	−35.3433	−1.65148
$459$	−0.940198	−0.0438847
$460$	−25.0096	−1.16608
$461$	7.64018	0.355839	0.177919	−	0.984045i	$-0.443063\pi$
$461$	7.64018	0.355839	0.177919	+	0.984045i	$0.443063\pi$
$462$	−68.9490	−3.20780
$463$	−9.56152	−0.444362	−0.222181	−	0.975005i	$-0.571318\pi$
$463$	−9.56152	−0.444362	−0.222181	+	0.975005i	$0.571318\pi$
$464$	−3.69609	−0.171587
$465$	1.00000	0.0463739
$466$	−55.6317	−2.57709
$467$	−15.8447	−0.733207	−0.366604	−	0.930377i	$-0.619479\pi$
$467$	−15.8447	−0.733207	−0.366604	+	0.930377i	$0.619479\pi$
$468$	3.42411	0.158280
$469$	−46.9807	−2.16937
$470$	14.4758	0.667720
$471$	15.0194	0.692058
$472$	30.0274	1.38212
$473$	−9.53922	−0.438614
$474$	21.0830	0.968374
$475$	7.87235	0.361208
$476$	15.8739	0.727580
$477$	−0.284478	−0.0130254
$478$	0.399918	0.0182918
$479$	5.92547	0.270742	0.135371	−	0.990795i	$-0.456777\pi$
$479$	5.92547	0.270742	0.135371	+	0.990795i	$0.456777\pi$
$480$	−4.59252	−0.209619
$481$	−4.83866	−0.220624
$482$	19.6129	0.893344
$483$	−36.0144	−1.63871
$484$	85.7704	3.89866
$485$	0.180652	0.00820297
$486$	−2.32897	−0.105644
$487$	−16.1941	−0.733824	−0.366912	−	0.930256i	$-0.619585\pi$
$487$	−16.1941	−0.733824	−0.366912	+	0.930256i	$0.619585\pi$
$488$	−4.52601	−0.204883
$489$	−11.7559	−0.531620
$490$	40.3209	1.82151
$491$	−14.6589	−0.661547	−0.330773	−	0.943710i	$-0.607310\pi$
$491$	−14.6589	−0.661547	−0.330773	+	0.943710i	$0.607310\pi$
$492$	−10.4894	−0.472900
$493$	3.96554	0.178599
$494$	−18.3345	−0.824908
$495$	6.00408	0.269863
$496$	−0.876314	−0.0393477
$497$	29.3794	1.31785
$498$	−10.2380	−0.458775
$499$	−23.4805	−1.05113	−0.525565	−	0.850753i	$-0.676146\pi$
$499$	−23.4805	−1.05113	−0.525565	+	0.850753i	$0.676146\pi$
$500$	−3.42411	−0.153131
$501$	12.0336	0.537623
$502$	17.1463	0.765279
$503$	5.24297	0.233772	0.116886	−	0.993145i	$-0.462709\pi$
$503$	5.24297	0.233772	0.116886	+	0.993145i	$0.462709\pi$
$504$	16.3540	0.728467
$505$	−11.5574	−0.514297
$506$	102.134	4.54041
$507$	1.00000	0.0444116
$508$	28.0044	1.24249
$509$	26.8647	1.19076	0.595379	−	0.803445i	$-0.297002\pi$
$509$	26.8647	1.19076	0.595379	+	0.803445i	$0.297002\pi$
$510$	−2.18969	−0.0969613
$511$	−40.7497	−1.80266
$512$	9.83746	0.434758
$513$	7.87235	0.347573
$514$	53.8893	2.37695
$515$	−11.1097	−0.489552
$516$	5.44019	0.239491
$517$	−37.3186	−1.64127
$518$	−55.5656	−2.44141
$519$	4.65346	0.204264
$520$	3.31671	0.145448
$521$	−33.5932	−1.47174	−0.735872	−	0.677121i	$-0.763227\pi$
$521$	−33.5932	−1.47174	−0.735872	+	0.677121i	$0.763227\pi$
$522$	9.82307	0.429944
$523$	18.7221	0.818660	0.409330	−	0.912386i	$-0.365763\pi$
$523$	18.7221	0.818660	0.409330	+	0.912386i	$0.365763\pi$
$524$	−11.6792	−0.510210
$525$	−4.93080	−0.215198
$526$	29.8819	1.30291
$527$	0.940198	0.0409557
$528$	−5.26146	−0.228976
$529$	30.3482	1.31949
$530$	−0.662541	−0.0287790
$531$	−9.05335	−0.392882
$532$	−132.914	−5.76253
$533$	−3.06340	−0.132691
$534$	3.96788	0.171707
$535$	−17.1849	−0.742970
$536$	−31.6017	−1.36499
$537$	−13.7254	−0.592297
$538$	−9.44086	−0.407024
$539$	−103.947	−4.47732
$540$	−3.42411	−0.147350
$541$	15.1500	0.651351	0.325675	−	0.945482i	$-0.394408\pi$
$541$	15.1500	0.651351	0.325675	+	0.945482i	$0.394408\pi$
$542$	43.1728	1.85443
$543$	−11.0790	−0.475445
$544$	−4.31788	−0.185127
$545$	−16.3676	−0.701111
$546$	11.4837	0.491456
$547$	−14.7862	−0.632214	−0.316107	−	0.948724i	$-0.602376\pi$
$547$	−14.7862	−0.632214	−0.316107	+	0.948724i	$0.602376\pi$
$548$	−71.5257	−3.05543
$549$	1.36460	0.0582399
$550$	13.9833	0.596251
$551$	−33.2038	−1.41453
$552$	−24.2252	−1.03109
$553$	44.6360	1.89811
$554$	19.1535	0.813752
$555$	4.83866	0.205389
$556$	−1.21560	−0.0515527
$557$	−7.79925	−0.330465	−0.165232	−	0.986255i	$-0.552837\pi$
$557$	−7.79925	−0.330465	−0.165232	+	0.986255i	$0.552837\pi$
$558$	2.32897	0.0985933
$559$	1.58879	0.0671986
$560$	4.32093	0.182592
$561$	5.64502	0.238333
$562$	34.9907	1.47599
$563$	32.7281	1.37933	0.689663	−	0.724131i	$-0.257759\pi$
$563$	32.7281	1.37933	0.689663	+	0.724131i	$0.257759\pi$
$564$	21.2827	0.896164
$565$	19.6432	0.826396
$566$	52.1393	2.19158
$567$	−4.93080	−0.207074
$568$	19.7622	0.829202
$569$	−44.7417	−1.87567	−0.937835	−	0.347082i	$-0.887172\pi$
$569$	−44.7417	−1.87567	−0.937835	+	0.347082i	$0.887172\pi$
$570$	18.3345	0.767947
$571$	−25.7331	−1.07690	−0.538448	−	0.842659i	$-0.680989\pi$
$571$	−25.7331	−1.07690	−0.538448	+	0.842659i	$0.680989\pi$
$572$	−20.5586	−0.859600
$573$	−12.7826	−0.533999
$574$	−35.1792	−1.46835
$575$	7.30398	0.304597
$576$	−12.4485	−0.518686
$577$	36.4076	1.51567	0.757834	−	0.652448i	$-0.226258\pi$
$577$	36.4076	1.51567	0.757834	+	0.652448i	$0.226258\pi$
$578$	37.5338	1.56120
$579$	−16.9992	−0.706463
$580$	14.4421	0.599676
$581$	−21.6754	−0.899247
$582$	0.420733	0.0174399
$583$	1.70803	0.0707393
$584$	−27.4104	−1.13425
$585$	−1.00000	−0.0413449
$586$	−57.0589	−2.35708
$587$	−32.5036	−1.34157	−0.670783	−	0.741654i	$-0.734042\pi$
$587$	−32.5036	−1.34157	−0.670783	+	0.741654i	$0.734042\pi$
$588$	59.2807	2.44470
$589$	−7.87235	−0.324375
$590$	−21.0850	−0.868056
$591$	−25.6285	−1.05421
$592$	−4.24018	−0.174270
$593$	−27.9381	−1.14728	−0.573641	−	0.819107i	$-0.694470\pi$
$593$	−27.9381	−1.14728	−0.573641	+	0.819107i	$0.694470\pi$
$594$	13.9833	0.573743
$595$	−4.63592	−0.190054
$596$	4.69878	0.192470
$597$	3.38769	0.138649
$598$	−17.0108	−0.695622
$599$	11.7291	0.479238	0.239619	−	0.970867i	$-0.422978\pi$
$599$	11.7291	0.479238	0.239619	+	0.970867i	$0.422978\pi$
$600$	−3.31671	−0.135404
$601$	22.5212	0.918659	0.459330	−	0.888266i	$-0.348090\pi$
$601$	22.5212	0.918659	0.459330	+	0.888266i	$0.348090\pi$
$602$	18.2452	0.743617
$603$	9.52801	0.388011
$604$	−38.6495	−1.57263
$605$	−25.0490	−1.01839
$606$	−26.9168	−1.09342
$607$	−12.3035	−0.499385	−0.249692	−	0.968325i	$-0.580329\pi$
$607$	−12.3035	−0.499385	−0.249692	+	0.968325i	$0.580329\pi$
$608$	36.1539	1.46624
$609$	20.7970	0.842735
$610$	3.17813	0.128679
$611$	6.21554	0.251454
$612$	−3.21934	−0.130134
$613$	−3.18032	−0.128452	−0.0642259	−	0.997935i	$-0.520458\pi$
$613$	−3.18032	−0.128452	−0.0642259	+	0.997935i	$0.520458\pi$
$614$	−5.25061	−0.211897
$615$	3.06340	0.123528
$616$	−98.1910	−3.95623
$617$	28.0504	1.12927	0.564634	−	0.825342i	$-0.309017\pi$
$617$	28.0504	1.12927	0.564634	+	0.825342i	$0.309017\pi$
$618$	−25.8742	−1.04081
$619$	23.7151	0.953192	0.476596	−	0.879122i	$-0.341870\pi$
$619$	23.7151	0.953192	0.476596	+	0.879122i	$0.341870\pi$
$620$	3.42411	0.137516
$621$	7.30398	0.293099
$622$	62.4374	2.50351
$623$	8.40061	0.336564
$624$	0.876314	0.0350806
$625$	1.00000	0.0400000
$626$	−32.4878	−1.29847
$627$	−47.2662	−1.88763
$628$	51.4282	2.05221
$629$	4.54929	0.181392
$630$	−11.4837	−0.457521
$631$	−2.31798	−0.0922771	−0.0461386	−	0.998935i	$-0.514692\pi$
$631$	−2.31798	−0.0922771	−0.0461386	+	0.998935i	$0.514692\pi$
$632$	30.0245	1.19431
$633$	−3.77438	−0.150018
$634$	−21.3580	−0.848236
$635$	−8.17858	−0.324557
$636$	−0.974084	−0.0386250
$637$	17.3127	0.685956
$638$	−58.9785	−2.33498
$639$	−5.95836	−0.235709
$640$	−19.8071	−0.782945
$641$	−6.76468	−0.267189	−0.133594	−	0.991036i	$-0.542652\pi$
$641$	−6.76468	−0.267189	−0.133594	+	0.991036i	$0.542652\pi$
$642$	−40.0233	−1.57959
$643$	15.4034	0.607450	0.303725	−	0.952760i	$-0.401770\pi$
$643$	15.4034	0.607450	0.303725	+	0.952760i	$0.401770\pi$
$644$	−123.317	−4.85939
$645$	−1.58879	−0.0625585
$646$	17.2380	0.678222
$647$	−36.2530	−1.42525	−0.712626	−	0.701544i	$-0.752494\pi$
$647$	−36.2530	−1.42525	−0.712626	+	0.701544i	$0.752494\pi$
$648$	−3.31671	−0.130293
$649$	54.3571	2.13370
$650$	−2.32897	−0.0913498
$651$	4.93080	0.193253
$652$	−40.2535	−1.57645
$653$	−23.1421	−0.905619	−0.452810	−	0.891607i	$-0.649578\pi$
$653$	−23.1421	−0.905619	−0.452810	+	0.891607i	$0.649578\pi$
$654$	−38.1197	−1.49060
$655$	3.41088	0.133274
$656$	−2.68450	−0.104812
$657$	8.26432	0.322422
$658$	71.3773	2.78258
$659$	29.6571	1.15528	0.577638	−	0.816293i	$-0.303975\pi$
$659$	29.6571	1.15528	0.577638	+	0.816293i	$0.303975\pi$
$660$	20.5586	0.800244
$661$	4.94602	0.192378	0.0961889	−	0.995363i	$-0.469335\pi$
$661$	4.94602	0.192378	0.0961889	+	0.995363i	$0.469335\pi$
$662$	60.0704	2.33470
$663$	−0.940198	−0.0365143
$664$	−14.5800	−0.565814
$665$	38.8170	1.50526
$666$	11.2691	0.436668
$667$	−30.8065	−1.19283
$668$	41.2045	1.59425
$669$	18.9882	0.734127
$670$	22.1905	0.857293
$671$	−8.19320	−0.316295
$672$	−22.6448	−0.873541
$673$	−25.6229	−0.987691	−0.493845	−	0.869550i	$-0.664409\pi$
$673$	−25.6229	−0.987691	−0.493845	+	0.869550i	$0.664409\pi$
$674$	39.6444	1.52705
$675$	1.00000	0.0384900
$676$	3.42411	0.131697
$677$	−21.7622	−0.836391	−0.418195	−	0.908357i	$-0.637337\pi$
$677$	−21.7622	−0.836391	−0.418195	+	0.908357i	$0.637337\pi$
$678$	45.7485	1.75696
$679$	0.890756	0.0341841
$680$	−3.11837	−0.119584
$681$	−5.53568	−0.212128
$682$	−13.9833	−0.535450
$683$	−24.3361	−0.931197	−0.465598	−	0.884996i	$-0.654161\pi$
$683$	−24.3361	−0.931197	−0.465598	+	0.884996i	$0.654161\pi$
$684$	26.9558	1.03068
$685$	20.8888	0.798121
$686$	118.428	4.52161
$687$	15.1755	0.578981
$688$	1.39228	0.0530801
$689$	−0.284478	−0.0108377
$690$	17.0108	0.647589
$691$	−23.0793	−0.877976	−0.438988	−	0.898493i	$-0.644663\pi$
$691$	−23.0793	−0.877976	−0.438988	+	0.898493i	$0.644663\pi$
$692$	15.9340	0.605719
$693$	29.6049	1.12460
$694$	45.4026	1.72346
$695$	0.355010	0.0134663
$696$	13.9891	0.530257
$697$	2.88021	0.109096
$698$	43.5876	1.64981
$699$	23.8868	0.903482
$700$	−16.8836	−0.638140
$701$	40.4746	1.52871	0.764353	−	0.644798i	$-0.223058\pi$
$701$	40.4746	1.52871	0.764353	+	0.644798i	$0.223058\pi$
$702$	−2.32897	−0.0879014
$703$	−38.0916	−1.43665
$704$	74.7416	2.81693
$705$	−6.21554	−0.234091
$706$	−42.3982	−1.59568
$707$	−56.9871	−2.14322
$708$	−30.9997	−1.16504
$709$	−7.09355	−0.266404	−0.133202	−	0.991089i	$-0.542526\pi$
$709$	−7.09355	−0.266404	−0.133202	+	0.991089i	$0.542526\pi$
$710$	−13.8768	−0.520789
$711$	−9.05249	−0.339495
$712$	5.65070	0.211769
$713$	−7.30398	−0.273536
$714$	−10.7969	−0.404065
$715$	6.00408	0.224540
$716$	−46.9975	−1.75638
$717$	−0.171715	−0.00641280
$718$	−29.0458	−1.08398
$719$	−39.0481	−1.45625	−0.728124	−	0.685445i	$-0.759608\pi$
$719$	−39.0481	−1.45625	−0.728124	+	0.685445i	$0.759608\pi$
$720$	−0.876314	−0.0326583
$721$	−54.7796	−2.04010
$722$	−100.085	−3.72478
$723$	−8.42128	−0.313191
$724$	−37.9357	−1.40987
$725$	−4.21777	−0.156644
$726$	−58.3383	−2.16514
$727$	−33.7265	−1.25085	−0.625423	−	0.780286i	$-0.715074\pi$
$727$	−33.7265	−1.25085	−0.625423	+	0.780286i	$0.715074\pi$
$728$	16.3540	0.606121
$729$	1.00000	0.0370370
$730$	19.2474	0.712377
$731$	−1.49378	−0.0552493
$732$	4.67256	0.172703
$733$	9.83922	0.363420	0.181710	−	0.983352i	$-0.441837\pi$
$733$	9.83922	0.363420	0.181710	+	0.983352i	$0.441837\pi$
$734$	−80.9712	−2.98870
$735$	−17.3127	−0.638590
$736$	33.5437	1.23644
$737$	−57.2070	−2.10725
$738$	7.13458	0.262628
$739$	−40.1915	−1.47847	−0.739235	−	0.673448i	$-0.764813\pi$
$739$	−40.1915	−1.47847	−0.739235	+	0.673448i	$0.764813\pi$
$740$	16.5681	0.609055
$741$	7.87235	0.289198
$742$	−3.26686	−0.119930
$743$	0.598355	0.0219515	0.0109758	−	0.999940i	$-0.496506\pi$
$743$	0.598355	0.0219515	0.0109758	+	0.999940i	$0.496506\pi$
$744$	3.31671	0.121597
$745$	−1.37226	−0.0502758
$746$	−65.6982	−2.40538
$747$	4.39592	0.160838
$748$	19.3292	0.706745
$749$	−84.7354	−3.09617
$750$	2.32897	0.0850420
$751$	5.89978	0.215286	0.107643	−	0.994190i	$-0.465670\pi$
$751$	5.89978	0.215286	0.107643	+	0.994190i	$0.465670\pi$
$752$	5.44677	0.198623
$753$	−7.36219	−0.268293
$754$	9.82307	0.357735
$755$	11.2875	0.410792
$756$	−16.8836	−0.614050
$757$	−31.6169	−1.14914	−0.574568	−	0.818457i	$-0.694830\pi$
$757$	−31.6169	−1.14914	−0.574568	+	0.818457i	$0.694830\pi$
$758$	73.5916	2.67297
$759$	−43.8537	−1.59179
$760$	26.1104	0.947122
$761$	23.7843	0.862179	0.431089	−	0.902309i	$-0.358129\pi$
$761$	23.7843	0.862179	0.431089	+	0.902309i	$0.358129\pi$
$762$	−19.0477	−0.690025
$763$	−80.7053	−2.92173
$764$	−43.7689	−1.58350
$765$	0.940198	0.0339929
$766$	79.1467	2.85969
$767$	−9.05335	−0.326898
$768$	−21.2333	−0.766190
$769$	−34.1887	−1.23288	−0.616438	−	0.787404i	$-0.711425\pi$
$769$	−34.1887	−1.23288	−0.616438	+	0.787404i	$0.711425\pi$
$770$	68.9490	2.48475
$771$	−23.1387	−0.833319
$772$	−58.2071	−2.09492
$773$	29.6662	1.06702	0.533510	−	0.845794i	$-0.320873\pi$
$773$	29.6662	1.06702	0.533510	+	0.845794i	$0.320873\pi$
$774$	−3.70025	−0.133003
$775$	−1.00000	−0.0359211
$776$	0.599170	0.0215089
$777$	23.8584	0.855916
$778$	27.9107	1.00065
$779$	−24.1162	−0.864052
$780$	−3.42411	−0.122603
$781$	35.7744	1.28011
$782$	15.9935	0.571926
$783$	−4.21777	−0.150731
$784$	15.1714	0.541836
$785$	−15.0194	−0.536066
$786$	7.94385	0.283348
$787$	−42.4572	−1.51343	−0.756717	−	0.653743i	$-0.773198\pi$
$787$	−42.4572	−1.51343	−0.756717	+	0.653743i	$0.773198\pi$
$788$	−87.7547	−3.12613
$789$	−12.8305	−0.456779
$790$	−21.0830	−0.750099
$791$	96.8566	3.44382
$792$	19.9138	0.707607
$793$	1.36460	0.0484585
$794$	34.4333	1.22199
$795$	0.284478	0.0100894
$796$	11.5998	0.411145
$797$	−8.14106	−0.288371	−0.144186	−	0.989551i	$-0.546056\pi$
$797$	−8.14106	−0.288371	−0.144186	+	0.989551i	$0.546056\pi$
$798$	90.4036	3.20025
$799$	−5.84384	−0.206740
$800$	4.59252	0.162370
$801$	−1.70370	−0.0601974
$802$	25.7892	0.910649
$803$	−49.6196	−1.75104
$804$	32.6250	1.15059
$805$	36.0144	1.26934
$806$	2.32897	0.0820345
$807$	4.05366	0.142696
$808$	−38.3326	−1.34853
$809$	33.1605	1.16586	0.582931	−	0.812522i	$-0.301906\pi$
$809$	33.1605	1.16586	0.582931	+	0.812522i	$0.301906\pi$
$810$	2.32897	0.0818317
$811$	17.4852	0.613989	0.306995	−	0.951711i	$-0.400677\pi$
$811$	17.4852	0.613989	0.306995	+	0.951711i	$0.400677\pi$
$812$	71.2111	2.49902
$813$	−18.5373	−0.650131
$814$	−67.6605	−2.37150
$815$	11.7559	0.411791
$816$	−0.823909	−0.0288426
$817$	12.5075	0.437582
$818$	−67.9292	−2.37509
$819$	−4.93080	−0.172296
$820$	10.4894	0.366307
$821$	−38.0342	−1.32740	−0.663701	−	0.747998i	$-0.731015\pi$
$821$	−38.0342	−1.32740	−0.663701	+	0.747998i	$0.731015\pi$
$822$	48.6495	1.69685
$823$	−22.8701	−0.797202	−0.398601	−	0.917124i	$-0.630504\pi$
$823$	−22.8701	−0.797202	−0.398601	+	0.917124i	$0.630504\pi$
$824$	−36.8477	−1.28365
$825$	−6.00408	−0.209035
$826$	−103.966	−3.61744
$827$	21.3849	0.743625	0.371812	−	0.928308i	$-0.378736\pi$
$827$	21.3849	0.743625	0.371812	+	0.928308i	$0.378736\pi$
$828$	25.0096	0.869145
$829$	−47.9220	−1.66440	−0.832200	−	0.554476i	$-0.812919\pi$
$829$	−47.9220	−1.66440	−0.832200	+	0.554476i	$0.812919\pi$
$830$	10.2380	0.355365
$831$	−8.22400	−0.285287
$832$	−12.4485	−0.431573
$833$	−16.2774	−0.563979
$834$	0.826809	0.0286301
$835$	−12.0336	−0.416441
$836$	−161.845	−5.59752
$837$	−1.00000	−0.0345651
$838$	20.1274	0.695288
$839$	−42.9649	−1.48331	−0.741657	−	0.670779i	$-0.765960\pi$
$839$	−42.9649	−1.48331	−0.741657	+	0.670779i	$0.765960\pi$
$840$	−16.3540	−0.564268
$841$	−11.2104	−0.386566
$842$	−51.8529	−1.78697
$843$	−15.0241	−0.517457
$844$	−12.9239	−0.444859
$845$	−1.00000	−0.0344010
$846$	−14.4758	−0.497689
$847$	−123.511	−4.24390
$848$	−0.249292	−0.00856073
$849$	−22.3873	−0.768328
$850$	2.18969	0.0751059
$851$	−35.3415	−1.21149
$852$	−20.4021	−0.698964
$853$	5.86345	0.200761	0.100380	−	0.994949i	$-0.467994\pi$
$853$	5.86345	0.200761	0.100380	+	0.994949i	$0.467994\pi$
$854$	15.6707	0.536240
$855$	−7.87235	−0.269229
$856$	−56.9976	−1.94814
$857$	46.1208	1.57546	0.787728	−	0.616024i	$-0.211257\pi$
$857$	46.1208	1.57546	0.787728	+	0.616024i	$0.211257\pi$
$858$	13.9833	0.477383
$859$	−41.8305	−1.42724	−0.713619	−	0.700534i	$-0.752945\pi$
$859$	−41.8305	−1.42724	−0.713619	+	0.700534i	$0.752945\pi$
$860$	−5.44019	−0.185509
$861$	15.1050	0.514777
$862$	36.4995	1.24318
$863$	1.06557	0.0362726	0.0181363	−	0.999836i	$-0.494227\pi$
$863$	1.06557	0.0362726	0.0181363	+	0.999836i	$0.494227\pi$
$864$	4.59252	0.156241
$865$	−4.65346	−0.158222
$866$	29.1241	0.989677
$867$	−16.1160	−0.547329
$868$	16.8836	0.573066
$869$	54.3518	1.84376
$870$	−9.82307	−0.333033
$871$	9.52801	0.322844
$872$	−54.2867	−1.83838
$873$	−0.180652	−0.00611413
$874$	−133.915	−4.52974
$875$	4.93080	0.166691
$876$	28.2980	0.956099
$877$	16.3685	0.552725	0.276362	−	0.961054i	$-0.410871\pi$
$877$	16.3685	0.552725	0.276362	+	0.961054i	$0.410871\pi$
$878$	−70.0324	−2.36348
$879$	24.4996	0.826352
$880$	5.26146	0.177364
$881$	16.1422	0.543846	0.271923	−	0.962319i	$-0.412340\pi$
$881$	16.1422	0.543846	0.271923	+	0.962319i	$0.412340\pi$
$882$	−40.3209	−1.35767
$883$	−29.0207	−0.976623	−0.488312	−	0.872669i	$-0.662387\pi$
$883$	−29.0207	−0.976623	−0.488312	+	0.872669i	$0.662387\pi$
$884$	−3.21934	−0.108278
$885$	9.05335	0.304325
$886$	36.7843	1.23579
$887$	12.6760	0.425617	0.212809	−	0.977094i	$-0.431739\pi$
$887$	12.6760	0.425617	0.212809	+	0.977094i	$0.431739\pi$
$888$	16.0484	0.538550
$889$	−40.3269	−1.35252
$890$	−3.96788	−0.133004
$891$	−6.00408	−0.201144
$892$	65.0178	2.17696
$893$	48.9310	1.63741
$894$	−3.19596	−0.106889
$895$	13.7254	0.458791
$896$	−97.6648	−3.26275
$897$	7.30398	0.243873
$898$	−69.9011	−2.33263
$899$	4.21777	0.140671
$900$	3.42411	0.114137
$901$	0.267466	0.00891057
$902$	−42.8366	−1.42630
$903$	−7.83400	−0.260699
$904$	65.1509	2.16689
$905$	11.0790	0.368278
$906$	26.2882	0.873366
$907$	−38.2122	−1.26882	−0.634408	−	0.772999i	$-0.718756\pi$
$907$	−38.2122	−1.26882	−0.634408	+	0.772999i	$0.718756\pi$
$908$	−18.9548	−0.629036
$909$	11.5574	0.383334
$910$	−11.4837	−0.380680
$911$	43.1651	1.43012	0.715062	−	0.699061i	$-0.246398\pi$
$911$	43.1651	1.43012	0.715062	+	0.699061i	$0.246398\pi$
$912$	6.89866	0.228437
$913$	−26.3935	−0.873496
$914$	−18.6949	−0.618373
$915$	−1.36460	−0.0451124
$916$	51.9625	1.71689
$917$	16.8184	0.555391
$918$	2.18969	0.0722707
$919$	3.90732	0.128891	0.0644453	−	0.997921i	$-0.479472\pi$
$919$	3.90732	0.128891	0.0644453	+	0.997921i	$0.479472\pi$
$920$	24.2252	0.798682
$921$	2.25448	0.0742875
$922$	−17.7938	−0.586007
$923$	−5.95836	−0.196122
$924$	101.370	3.33484
$925$	−4.83866	−0.159094
$926$	22.2685	0.731789
$927$	11.1097	0.364890
$928$	−19.3702	−0.635858
$929$	4.84353	0.158911	0.0794555	−	0.996838i	$-0.474682\pi$
$929$	4.84353	0.158911	0.0794555	+	0.996838i	$0.474682\pi$
$930$	−2.32897	−0.0763700
$931$	136.292	4.46679
$932$	81.7911	2.67916
$933$	−26.8090	−0.877688
$934$	36.9020	1.20747
$935$	−5.64502	−0.184612
$936$	−3.31671	−0.108410
$937$	45.2321	1.47767	0.738833	−	0.673888i	$-0.235377\pi$
$937$	45.2321	1.47767	0.738833	+	0.673888i	$0.235377\pi$
$938$	109.417	3.57258
$939$	13.9494	0.455222
$940$	−21.2827	−0.694165
$941$	6.98163	0.227595	0.113797	−	0.993504i	$-0.463699\pi$
$941$	6.98163	0.227595	0.113797	+	0.993504i	$0.463699\pi$
$942$	−34.9798	−1.13970
$943$	−22.3750	−0.728632
$944$	−7.93358	−0.258216
$945$	4.93080	0.160399
$946$	22.2166	0.722323
$947$	−19.3270	−0.628042	−0.314021	−	0.949416i	$-0.601676\pi$
$947$	−19.3270	−0.628042	−0.314021	+	0.949416i	$0.601676\pi$
$948$	−30.9967	−1.00673
$949$	8.26432	0.268271
$950$	−18.3345	−0.594849
$951$	9.17058	0.297377
$952$	−15.3760	−0.498340
$953$	13.2534	0.429321	0.214660	−	0.976689i	$-0.431136\pi$
$953$	13.2534	0.429321	0.214660	+	0.976689i	$0.431136\pi$
$954$	0.662541	0.0214506
$955$	12.7826	0.413634
$956$	−0.587970	−0.0190163
$957$	25.3238	0.818603
$958$	−13.8003	−0.445866
$959$	102.999	3.32600
$960$	12.4485	0.401773
$961$	1.00000	0.0322581
$962$	11.2691	0.363330
$963$	17.1849	0.553777
$964$	−28.8354	−0.928725
$965$	16.9992	0.547224
$966$	83.8766	2.69869
$967$	46.6818	1.50118	0.750592	−	0.660766i	$-0.229768\pi$
$967$	46.6818	1.50118	0.750592	+	0.660766i	$0.229768\pi$
$968$	−83.0803	−2.67030
$969$	−7.40157	−0.237773
$970$	−0.420733	−0.0135089
$971$	−3.57683	−0.114786	−0.0573929	−	0.998352i	$-0.518279\pi$
$971$	−3.57683	−0.114786	−0.0573929	+	0.998352i	$0.518279\pi$
$972$	3.42411	0.109828
$973$	1.75048	0.0561179
$974$	37.7156	1.20849
$975$	1.00000	0.0320256
$976$	1.19582	0.0382773
$977$	41.6214	1.33159	0.665793	−	0.746136i	$-0.268093\pi$
$977$	41.6214	1.33159	0.665793	+	0.746136i	$0.268093\pi$
$978$	27.3791	0.875488
$979$	10.2292	0.326926
$980$	−59.2807	−1.89365
$981$	16.3676	0.522578
$982$	34.1402	1.08946
$983$	49.2844	1.57193	0.785964	−	0.618272i	$-0.212167\pi$
$983$	49.2844	1.57193	0.785964	+	0.618272i	$0.212167\pi$
$984$	10.1604	0.323903
$985$	25.6285	0.816591
$986$	−9.23563	−0.294122
$987$	−30.6476	−0.975523
$988$	26.9558	0.857579
$989$	11.6045	0.369001
$990$	−13.9833	−0.444420
$991$	10.0751	0.320046	0.160023	−	0.987113i	$-0.448843\pi$
$991$	10.0751	0.320046	0.160023	+	0.987113i	$0.448843\pi$
$992$	−4.59252	−0.145813
$993$	−25.7926	−0.818505
$994$	−68.4239	−2.17027
$995$	−3.38769	−0.107397
$996$	15.0521	0.476945
$997$	−57.3383	−1.81592	−0.907961	−	0.419054i	$-0.862362\pi$
$997$	−57.3383	−1.81592	−0.907961	+	0.419054i	$0.862362\pi$
$998$	54.6854	1.73103
$999$	−4.83866	−0.153088

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6045.2.a.y.1.3	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6045.2.a.y.1.3	✓	12	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)

\(f(q)\)	\(=\)	\(q-2.32897 q^{2} +1.00000 q^{3} +3.42411 q^{4} -1.00000 q^{5} -2.32897 q^{6} -4.93080 q^{7} -3.31671 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.32897 q^{2} +1.00000 q^{3} +3.42411 q^{4} -1.00000 q^{5} -2.32897 q^{6} -4.93080 q^{7} -3.31671 q^{8} +1.00000 q^{9} +2.32897 q^{10} -6.00408 q^{11} +3.42411 q^{12} +1.00000 q^{13} +11.4837 q^{14} -1.00000 q^{15} +0.876314 q^{16} -0.940198 q^{17} -2.32897 q^{18} +7.87235 q^{19} -3.42411 q^{20} -4.93080 q^{21} +13.9833 q^{22} +7.30398 q^{23} -3.31671 q^{24} +1.00000 q^{25} -2.32897 q^{26} +1.00000 q^{27} -16.8836 q^{28} -4.21777 q^{29} +2.32897 q^{30} -1.00000 q^{31} +4.59252 q^{32} -6.00408 q^{33} +2.18969 q^{34} +4.93080 q^{35} +3.42411 q^{36} -4.83866 q^{37} -18.3345 q^{38} +1.00000 q^{39} +3.31671 q^{40} -3.06340 q^{41} +11.4837 q^{42} +1.58879 q^{43} -20.5586 q^{44} -1.00000 q^{45} -17.0108 q^{46} +6.21554 q^{47} +0.876314 q^{48} +17.3127 q^{49} -2.32897 q^{50} -0.940198 q^{51} +3.42411 q^{52} -0.284478 q^{53} -2.32897 q^{54} +6.00408 q^{55} +16.3540 q^{56} +7.87235 q^{57} +9.82307 q^{58} -9.05335 q^{59} -3.42411 q^{60} +1.36460 q^{61} +2.32897 q^{62} -4.93080 q^{63} -12.4485 q^{64} -1.00000 q^{65} +13.9833 q^{66} +9.52801 q^{67} -3.21934 q^{68} +7.30398 q^{69} -11.4837 q^{70} -5.95836 q^{71} -3.31671 q^{72} +8.26432 q^{73} +11.2691 q^{74} +1.00000 q^{75} +26.9558 q^{76} +29.6049 q^{77} -2.32897 q^{78} -9.05249 q^{79} -0.876314 q^{80} +1.00000 q^{81} +7.13458 q^{82} +4.39592 q^{83} -16.8836 q^{84} +0.940198 q^{85} -3.70025 q^{86} -4.21777 q^{87} +19.9138 q^{88} -1.70370 q^{89} +2.32897 q^{90} -4.93080 q^{91} +25.0096 q^{92} -1.00000 q^{93} -14.4758 q^{94} -7.87235 q^{95} +4.59252 q^{96} -0.180652 q^{97} -40.3209 q^{98} -6.00408 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 3 q^{2} + 12 q^{3} + 15 q^{4} - 12 q^{5} - 3 q^{6} + 2 q^{7} - 12 q^{8} + 12 q^{9}+O(q^{10}) \) 12 * q - 3 * q^2 + 12 * q^3 + 15 * q^4 - 12 * q^5 - 3 * q^6 + 2 * q^7 - 12 * q^8 + 12 * q^9	\( 12 q - 3 q^{2} + 12 q^{3} + 15 q^{4} - 12 q^{5} - 3 q^{6} + 2 q^{7} - 12 q^{8} + 12 q^{9} + 3 q^{10} - 24 q^{11} + 15 q^{12} + 12 q^{13} - 13 q^{14} - 12 q^{15} + 21 q^{16} - 9 q^{17} - 3 q^{18} - 2 q^{19} - 15 q^{20} + 2 q^{21} + 9 q^{22} + 7 q^{23} - 12 q^{24} + 12 q^{25} - 3 q^{26} + 12 q^{27} - 16 q^{28} - 24 q^{29} + 3 q^{30} - 12 q^{31} - 57 q^{32} - 24 q^{33} - 15 q^{34} - 2 q^{35} + 15 q^{36} - 13 q^{37} - 22 q^{38} + 12 q^{39} + 12 q^{40} - 46 q^{41} - 13 q^{42} - 5 q^{43} - 49 q^{44} - 12 q^{45} - 2 q^{46} - 39 q^{47} + 21 q^{48} - 3 q^{50} - 9 q^{51} + 15 q^{52} - 12 q^{53} - 3 q^{54} + 24 q^{55} + q^{56} - 2 q^{57} - 8 q^{58} - 31 q^{59} - 15 q^{60} + 2 q^{61} + 3 q^{62} + 2 q^{63} + 18 q^{64} - 12 q^{65} + 9 q^{66} + 5 q^{67} - 5 q^{68} + 7 q^{69} + 13 q^{70} - 31 q^{71} - 12 q^{72} + 3 q^{73} + 16 q^{74} + 12 q^{75} - 2 q^{76} + 2 q^{77} - 3 q^{78} + 5 q^{79} - 21 q^{80} + 12 q^{81} + 25 q^{82} + 2 q^{83} - 16 q^{84} + 9 q^{85} - 26 q^{86} - 24 q^{87} + 27 q^{88} - 13 q^{89} + 3 q^{90} + 2 q^{91} + 5 q^{92} - 12 q^{93} - 12 q^{94} + 2 q^{95} - 57 q^{96} - 28 q^{97} - 29 q^{98} - 24 q^{99}+O(q^{100}) \) 12 * q - 3 * q^2 + 12 * q^3 + 15 * q^4 - 12 * q^5 - 3 * q^6 + 2 * q^7 - 12 * q^8 + 12 * q^9 + 3 * q^10 - 24 * q^11 + 15 * q^12 + 12 * q^13 - 13 * q^14 - 12 * q^15 + 21 * q^16 - 9 * q^17 - 3 * q^18 - 2 * q^19 - 15 * q^20 + 2 * q^21 + 9 * q^22 + 7 * q^23 - 12 * q^24 + 12 * q^25 - 3 * q^26 + 12 * q^27 - 16 * q^28 - 24 * q^29 + 3 * q^30 - 12 * q^31 - 57 * q^32 - 24 * q^33 - 15 * q^34 - 2 * q^35 + 15 * q^36 - 13 * q^37 - 22 * q^38 + 12 * q^39 + 12 * q^40 - 46 * q^41 - 13 * q^42 - 5 * q^43 - 49 * q^44 - 12 * q^45 - 2 * q^46 - 39 * q^47 + 21 * q^48 - 3 * q^50 - 9 * q^51 + 15 * q^52 - 12 * q^53 - 3 * q^54 + 24 * q^55 + q^56 - 2 * q^57 - 8 * q^58 - 31 * q^59 - 15 * q^60 + 2 * q^61 + 3 * q^62 + 2 * q^63 + 18 * q^64 - 12 * q^65 + 9 * q^66 + 5 * q^67 - 5 * q^68 + 7 * q^69 + 13 * q^70 - 31 * q^71 - 12 * q^72 + 3 * q^73 + 16 * q^74 + 12 * q^75 - 2 * q^76 + 2 * q^77 - 3 * q^78 + 5 * q^79 - 21 * q^80 + 12 * q^81 + 25 * q^82 + 2 * q^83 - 16 * q^84 + 9 * q^85 - 26 * q^86 - 24 * q^87 + 27 * q^88 - 13 * q^89 + 3 * q^90 + 2 * q^91 + 5 * q^92 - 12 * q^93 - 12 * q^94 + 2 * q^95 - 57 * q^96 - 28 * q^97 - 29 * q^98 - 24 * q^99

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