LMFDB - Embedded newform 1045.2.a.i.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1045,2,Mod(1,1045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1045, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1045.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1045 = 5 \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1045.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:	$8.34436701122$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 9x^{6} + 12x^{5} + 28x^{4} - 17x^{3} - 28x^{2} + 6x + 8 $ x^8 - 2x^7 - 9x^6 + 12x^5 + 28x^4 - 17x^3 - 28x^2 + 6*x + 8
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.714778$ of defining polynomial
Character	$\chi$	$=$	1045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.285222 q^{2} -1.61587 q^{3} -1.91865 q^{4} +1.00000 q^{5} +0.460881 q^{6} -1.06724 q^{7} +1.11768 q^{8} -0.388965 q^{9} +O(q^{10})$	\(q-0.285222 q^{2} -1.61587 q^{3} -1.91865 q^{4} +1.00000 q^{5} +0.460881 q^{6} -1.06724 q^{7} +1.11768 q^{8} -0.388965 q^{9} -0.285222 q^{10} +1.00000 q^{11} +3.10029 q^{12} +2.90648 q^{13} +0.304400 q^{14} -1.61587 q^{15} +3.51851 q^{16} +4.79291 q^{17} +0.110941 q^{18} -1.00000 q^{19} -1.91865 q^{20} +1.72452 q^{21} -0.285222 q^{22} -9.15842 q^{23} -1.80603 q^{24} +1.00000 q^{25} -0.828993 q^{26} +5.47613 q^{27} +2.04766 q^{28} +3.15727 q^{29} +0.460881 q^{30} -6.05100 q^{31} -3.23892 q^{32} -1.61587 q^{33} -1.36704 q^{34} -1.06724 q^{35} +0.746287 q^{36} -4.32974 q^{37} +0.285222 q^{38} -4.69650 q^{39} +1.11768 q^{40} +2.53697 q^{41} -0.491871 q^{42} -6.45078 q^{43} -1.91865 q^{44} -0.388965 q^{45} +2.61218 q^{46} -2.00319 q^{47} -5.68545 q^{48} -5.86100 q^{49} -0.285222 q^{50} -7.74471 q^{51} -5.57652 q^{52} -8.33604 q^{53} -1.56191 q^{54} +1.00000 q^{55} -1.19284 q^{56} +1.61587 q^{57} -0.900523 q^{58} +9.66844 q^{59} +3.10029 q^{60} +5.37576 q^{61} +1.72588 q^{62} +0.415119 q^{63} -6.11321 q^{64} +2.90648 q^{65} +0.460881 q^{66} -0.940037 q^{67} -9.19590 q^{68} +14.7988 q^{69} +0.304400 q^{70} -2.06230 q^{71} -0.434740 q^{72} -4.63606 q^{73} +1.23494 q^{74} -1.61587 q^{75} +1.91865 q^{76} -1.06724 q^{77} +1.33954 q^{78} -9.36965 q^{79} +3.51851 q^{80} -7.68181 q^{81} -0.723600 q^{82} -13.2853 q^{83} -3.30875 q^{84} +4.79291 q^{85} +1.83990 q^{86} -5.10174 q^{87} +1.11768 q^{88} -14.4450 q^{89} +0.110941 q^{90} -3.10192 q^{91} +17.5718 q^{92} +9.77763 q^{93} +0.571354 q^{94} -1.00000 q^{95} +5.23368 q^{96} -14.9033 q^{97} +1.67168 q^{98} -0.388965 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 6 q^{2} - 7 q^{3} + 10 q^{4} + 8 q^{5} - 11 q^{7} - 18 q^{8} + 11 q^{9}+O(q^{10}) $ 8 * q - 6 * q^2 - 7 * q^3 + 10 * q^4 + 8 * q^5 - 11 * q^7 - 18 * q^8 + 11 * q^9	\( 8 q - 6 q^{2} - 7 q^{3} + 10 q^{4} + 8 q^{5} - 11 q^{7} - 18 q^{8} + 11 q^{9} - 6 q^{10} + 8 q^{11} - 7 q^{12} - 17 q^{13} + 12 q^{14} - 7 q^{15} + 18 q^{16} - 9 q^{17} - 2 q^{18} - 8 q^{19} + 10 q^{20} + q^{21} - 6 q^{22} - 8 q^{23} + q^{24} + 8 q^{25} + 10 q^{26} - 34 q^{27} - 22 q^{28} - 3 q^{29} - q^{31} - 37 q^{32} - 7 q^{33} - 8 q^{34} - 11 q^{35} + 30 q^{36} - 17 q^{37} + 6 q^{38} + 14 q^{39} - 18 q^{40} - 5 q^{41} + 15 q^{42} - 21 q^{43} + 10 q^{44} + 11 q^{45} - 2 q^{46} - 8 q^{47} + 10 q^{48} + 19 q^{49} - 6 q^{50} - 16 q^{51} + 9 q^{52} - 19 q^{53} - 3 q^{54} + 8 q^{55} + 24 q^{56} + 7 q^{57} + 37 q^{58} - 33 q^{59} - 7 q^{60} - q^{61} - 42 q^{62} - 20 q^{63} + 48 q^{64} - 17 q^{65} - 18 q^{67} - 37 q^{68} + 16 q^{69} + 12 q^{70} - 18 q^{71} + 13 q^{72} - 18 q^{73} + 15 q^{74} - 7 q^{75} - 10 q^{76} - 11 q^{77} - 51 q^{78} - 5 q^{79} + 18 q^{80} + 32 q^{81} + 12 q^{82} - 33 q^{83} - 51 q^{84} - 9 q^{85} - 16 q^{86} - 26 q^{87} - 18 q^{88} - 20 q^{89} - 2 q^{90} + 6 q^{91} - 3 q^{92} + 18 q^{93} + 30 q^{94} - 8 q^{95} + 21 q^{96} - 69 q^{98} + 11 q^{99}+O(q^{100}) \) 8 * q - 6 * q^2 - 7 * q^3 + 10 * q^4 + 8 * q^5 - 11 * q^7 - 18 * q^8 + 11 * q^9 - 6 * q^10 + 8 * q^11 - 7 * q^12 - 17 * q^13 + 12 * q^14 - 7 * q^15 + 18 * q^16 - 9 * q^17 - 2 * q^18 - 8 * q^19 + 10 * q^20 + q^21 - 6 * q^22 - 8 * q^23 + q^24 + 8 * q^25 + 10 * q^26 - 34 * q^27 - 22 * q^28 - 3 * q^29 - q^31 - 37 * q^32 - 7 * q^33 - 8 * q^34 - 11 * q^35 + 30 * q^36 - 17 * q^37 + 6 * q^38 + 14 * q^39 - 18 * q^40 - 5 * q^41 + 15 * q^42 - 21 * q^43 + 10 * q^44 + 11 * q^45 - 2 * q^46 - 8 * q^47 + 10 * q^48 + 19 * q^49 - 6 * q^50 - 16 * q^51 + 9 * q^52 - 19 * q^53 - 3 * q^54 + 8 * q^55 + 24 * q^56 + 7 * q^57 + 37 * q^58 - 33 * q^59 - 7 * q^60 - q^61 - 42 * q^62 - 20 * q^63 + 48 * q^64 - 17 * q^65 - 18 * q^67 - 37 * q^68 + 16 * q^69 + 12 * q^70 - 18 * q^71 + 13 * q^72 - 18 * q^73 + 15 * q^74 - 7 * q^75 - 10 * q^76 - 11 * q^77 - 51 * q^78 - 5 * q^79 + 18 * q^80 + 32 * q^81 + 12 * q^82 - 33 * q^83 - 51 * q^84 - 9 * q^85 - 16 * q^86 - 26 * q^87 - 18 * q^88 - 20 * q^89 - 2 * q^90 + 6 * q^91 - 3 * q^92 + 18 * q^93 + 30 * q^94 - 8 * q^95 + 21 * q^96 - 69 * q^98 + 11 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−0.285222	−0.201682	−0.100841	−	0.994903i	$-0.532153\pi$
$2$	−0.285222	−0.201682	−0.100841	+	0.994903i	$0.532153\pi$
$3$	−1.61587	−0.932923	−0.466461	−	0.884542i	$-0.654471\pi$
$3$	−1.61587	−0.932923	−0.466461	+	0.884542i	$0.654471\pi$
$4$	−1.91865	−0.959324
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0.460881	0.188154
$7$	−1.06724	−0.403379	−0.201689	−	0.979450i	$-0.564643\pi$
$7$	−1.06724	−0.403379	−0.201689	+	0.979450i	$0.564643\pi$
$8$	1.11768	0.395161
$9$	−0.388965	−0.129655
$10$	−0.285222	−0.0901951
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	3.10029	0.894975
$13$	2.90648	0.806114	0.403057	−	0.915175i	$-0.367948\pi$
$13$	2.90648	0.806114	0.403057	+	0.915175i	$0.367948\pi$
$14$	0.304400	0.0813544
$15$	−1.61587	−0.417216
$16$	3.51851	0.879627
$17$	4.79291	1.16245	0.581225	−	0.813743i	$-0.302573\pi$
$17$	4.79291	1.16245	0.581225	+	0.813743i	$0.302573\pi$
$18$	0.110941	0.0261491
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−1.91865	−0.429023
$21$	1.72452	0.376321
$22$	−0.285222	−0.0608095
$23$	−9.15842	−1.90966	−0.954831	−	0.297150i	$-0.903964\pi$
$23$	−9.15842	−1.90966	−0.954831	+	0.297150i	$0.903964\pi$
$24$	−1.80603	−0.368655
$25$	1.00000	0.200000
$26$	−0.828993	−0.162579
$27$	5.47613	1.05388
$28$	2.04766	0.386971
$29$	3.15727	0.586290	0.293145	−	0.956068i	$-0.405298\pi$
$29$	3.15727	0.586290	0.293145	+	0.956068i	$0.405298\pi$
$30$	0.460881	0.0841450
$31$	−6.05100	−1.08679	−0.543396	−	0.839477i	$-0.682862\pi$
$31$	−6.05100	−1.08679	−0.543396	+	0.839477i	$0.682862\pi$
$32$	−3.23892	−0.572566
$33$	−1.61587	−0.281287
$34$	−1.36704	−0.234446
$35$	−1.06724	−0.180397
$36$	0.746287	0.124381
$37$	−4.32974	−0.711805	−0.355903	−	0.934523i	$-0.615827\pi$
$37$	−4.32974	−0.711805	−0.355903	+	0.934523i	$0.615827\pi$
$38$	0.285222	0.0462691
$39$	−4.69650	−0.752042
$40$	1.11768	0.176721
$41$	2.53697	0.396208	0.198104	−	0.980181i	$-0.436522\pi$
$41$	2.53697	0.396208	0.198104	+	0.980181i	$0.436522\pi$
$42$	−0.491871	−0.0758974
$43$	−6.45078	−0.983735	−0.491868	−	0.870670i	$-0.663686\pi$
$43$	−6.45078	−0.983735	−0.491868	+	0.870670i	$0.663686\pi$
$44$	−1.91865	−0.289247
$45$	−0.388965	−0.0579835
$46$	2.61218	0.385145
$47$	−2.00319	−0.292196	−0.146098	−	0.989270i	$-0.546671\pi$
$47$	−2.00319	−0.292196	−0.146098	+	0.989270i	$0.546671\pi$
$48$	−5.68545	−0.820624
$49$	−5.86100	−0.837285
$50$	−0.285222	−0.0403365
$51$	−7.74471	−1.08448
$52$	−5.57652	−0.773324
$53$	−8.33604	−1.14504	−0.572522	−	0.819890i	$-0.694035\pi$
$53$	−8.33604	−1.14504	−0.572522	+	0.819890i	$0.694035\pi$
$54$	−1.56191	−0.212549
$55$	1.00000	0.134840
$56$	−1.19284	−0.159400
$57$	1.61587	0.214027
$58$	−0.900523	−0.118244
$59$	9.66844	1.25872	0.629362	−	0.777112i	$-0.283316\pi$
$59$	9.66844	1.25872	0.629362	+	0.777112i	$0.283316\pi$
$60$	3.10029	0.400245
$61$	5.37576	0.688296	0.344148	−	0.938915i	$-0.388168\pi$
$61$	5.37576	0.688296	0.344148	+	0.938915i	$0.388168\pi$
$62$	1.72588	0.219187
$63$	0.415119	0.0523001
$64$	−6.11321	−0.764151
$65$	2.90648	0.360505
$66$	0.460881	0.0567306
$67$	−0.940037	−0.114844	−0.0574219	−	0.998350i	$-0.518288\pi$
$67$	−0.940037	−0.114844	−0.0574219	+	0.998350i	$0.518288\pi$
$68$	−9.19590	−1.11517
$69$	14.7988	1.78157
$70$	0.304400	0.0363828
$71$	−2.06230	−0.244750	−0.122375	−	0.992484i	$-0.539051\pi$
$71$	−2.06230	−0.244750	−0.122375	+	0.992484i	$0.539051\pi$
$72$	−0.434740	−0.0512346
$73$	−4.63606	−0.542610	−0.271305	−	0.962493i	$-0.587455\pi$
$73$	−4.63606	−0.542610	−0.271305	+	0.962493i	$0.587455\pi$
$74$	1.23494	0.143559
$75$	−1.61587	−0.186585
$76$	1.91865	0.220084
$77$	−1.06724	−0.121623
$78$	1.33954	0.151673
$79$	−9.36965	−1.05417	−0.527084	−	0.849813i	$-0.676715\pi$
$79$	−9.36965	−1.05417	−0.527084	+	0.849813i	$0.676715\pi$
$80$	3.51851	0.393381
$81$	−7.68181	−0.853535
$82$	−0.723600	−0.0799082
$83$	−13.2853	−1.45825	−0.729125	−	0.684380i	$-0.760073\pi$
$83$	−13.2853	−1.45825	−0.729125	+	0.684380i	$0.760073\pi$
$84$	−3.30875	−0.361014
$85$	4.79291	0.519864
$86$	1.83990	0.198402
$87$	−5.10174	−0.546964
$88$	1.11768	0.119146
$89$	−14.4450	−1.53117	−0.765584	−	0.643336i	$-0.777550\pi$
$89$	−14.4450	−1.53117	−0.765584	+	0.643336i	$0.777550\pi$
$90$	0.110941	0.0116942
$91$	−3.10192	−0.325169
$92$	17.5718	1.83198
$93$	9.77763	1.01389
$94$	0.571354	0.0589307
$95$	−1.00000	−0.102598
$96$	5.23368	0.534160
$97$	−14.9033	−1.51320	−0.756600	−	0.653878i	$-0.773141\pi$
$97$	−14.9033	−1.51320	−0.756600	+	0.653878i	$0.773141\pi$
$98$	1.67168	0.168866
$99$	−0.388965	−0.0390925
$100$	−1.91865	−0.191865
$101$	13.6329	1.35653	0.678264	−	0.734818i	$-0.262733\pi$
$101$	13.6329	1.35653	0.678264	+	0.734818i	$0.262733\pi$
$102$	2.20896	0.218720
$103$	−0.766469	−0.0755224	−0.0377612	−	0.999287i	$-0.512023\pi$
$103$	−0.766469	−0.0755224	−0.0377612	+	0.999287i	$0.512023\pi$
$104$	3.24853	0.318545
$105$	1.72452	0.168296
$106$	2.37762	0.230935
$107$	−15.3064	−1.47973	−0.739865	−	0.672756i	$-0.765111\pi$
$107$	−15.3064	−1.47973	−0.739865	+	0.672756i	$0.765111\pi$
$108$	−10.5068	−1.01101
$109$	−5.95627	−0.570507	−0.285254	−	0.958452i	$-0.592078\pi$
$109$	−5.95627	−0.570507	−0.285254	+	0.958452i	$0.592078\pi$
$110$	−0.285222	−0.0271948
$111$	6.99630	0.664059
$112$	−3.75509	−0.354823
$113$	−0.202541	−0.0190534	−0.00952672	−	0.999955i	$-0.503032\pi$
$113$	−0.202541	−0.0190534	−0.00952672	+	0.999955i	$0.503032\pi$
$114$	−0.460881	−0.0431655
$115$	−9.15842	−0.854027
$116$	−6.05769	−0.562443
$117$	−1.13052	−0.104517
$118$	−2.75765	−0.253862
$119$	−5.11518	−0.468908
$120$	−1.80603	−0.164867
$121$	1.00000	0.0909091
$122$	−1.53328	−0.138817
$123$	−4.09941	−0.369632
$124$	11.6097	1.04259
$125$	1.00000	0.0894427
$126$	−0.118401	−0.0105480
$127$	−2.94270	−0.261123	−0.130561	−	0.991440i	$-0.541678\pi$
$127$	−2.94270	−0.261123	−0.130561	+	0.991440i	$0.541678\pi$
$128$	8.22147	0.726682
$129$	10.4236	0.917749
$130$	−0.828993	−0.0727075
$131$	21.9174	1.91493	0.957467	−	0.288541i	$-0.0931703\pi$
$131$	21.9174	1.91493	0.957467	+	0.288541i	$0.0931703\pi$
$132$	3.10029	0.269845
$133$	1.06724	0.0925415
$134$	0.268119	0.0231620
$135$	5.47613	0.471310
$136$	5.35696	0.459355
$137$	−3.98501	−0.340462	−0.170231	−	0.985404i	$-0.554451\pi$
$137$	−3.98501	−0.340462	−0.170231	+	0.985404i	$0.554451\pi$
$138$	−4.22094	−0.359310
$139$	−4.90030	−0.415638	−0.207819	−	0.978167i	$-0.566637\pi$
$139$	−4.90030	−0.415638	−0.207819	+	0.978167i	$0.566637\pi$
$140$	2.04766	0.173059
$141$	3.23690	0.272596
$142$	0.588214	0.0493618
$143$	2.90648	0.243052
$144$	−1.36858	−0.114048
$145$	3.15727	0.262197
$146$	1.32231	0.109435
$147$	9.47061	0.781123
$148$	8.30725	0.682852
$149$	−1.83424	−0.150267	−0.0751336	−	0.997173i	$-0.523938\pi$
$149$	−1.83424	−0.150267	−0.0751336	+	0.997173i	$0.523938\pi$
$150$	0.460881	0.0376308
$151$	18.6630	1.51877	0.759385	−	0.650642i	$-0.225500\pi$
$151$	18.6630	1.51877	0.759385	+	0.650642i	$0.225500\pi$
$152$	−1.11768	−0.0906561
$153$	−1.86427	−0.150718
$154$	0.304400	0.0245293
$155$	−6.05100	−0.486028
$156$	9.01093	0.721452
$157$	19.6048	1.56464	0.782318	−	0.622879i	$-0.214037\pi$
$157$	19.6048	1.56464	0.782318	+	0.622879i	$0.214037\pi$
$158$	2.67243	0.212607
$159$	13.4700	1.06824
$160$	−3.23892	−0.256059
$161$	9.77423	0.770317
$162$	2.19102	0.172143
$163$	−11.4258	−0.894936	−0.447468	−	0.894300i	$-0.647674\pi$
$163$	−11.4258	−0.894936	−0.447468	+	0.894300i	$0.647674\pi$
$164$	−4.86756	−0.380092
$165$	−1.61587	−0.125795
$166$	3.78925	0.294103
$167$	21.8471	1.69058	0.845291	−	0.534307i	$-0.179427\pi$
$167$	21.8471	1.69058	0.845291	+	0.534307i	$0.179427\pi$
$168$	1.92747	0.148708
$169$	−4.55235	−0.350181
$170$	−1.36704	−0.104847
$171$	0.388965	0.0297449
$172$	12.3768	0.943721
$173$	−16.8049	−1.27766	−0.638828	−	0.769350i	$-0.720580\pi$
$173$	−16.8049	−1.27766	−0.638828	+	0.769350i	$0.720580\pi$
$174$	1.45513	0.110313
$175$	−1.06724	−0.0806758
$176$	3.51851	0.265218
$177$	−15.6229	−1.17429
$178$	4.12003	0.308810
$179$	0.344505	0.0257495	0.0128748	−	0.999917i	$-0.495902\pi$
$179$	0.344505	0.0257495	0.0128748	+	0.999917i	$0.495902\pi$
$180$	0.746287	0.0556250
$181$	1.92884	0.143369	0.0716846	−	0.997427i	$-0.477162\pi$
$181$	1.92884	0.143369	0.0716846	+	0.997427i	$0.477162\pi$
$182$	0.884734	0.0655809
$183$	−8.68653	−0.642127
$184$	−10.2362	−0.754624
$185$	−4.32974	−0.318329
$186$	−2.78879	−0.204484
$187$	4.79291	0.350492
$188$	3.84342	0.280310
$189$	−5.84434	−0.425113
$190$	0.285222	0.0206922
$191$	−25.8158	−1.86797	−0.933983	−	0.357318i	$-0.883691\pi$
$191$	−25.8158	−1.86797	−0.933983	+	0.357318i	$0.883691\pi$
$192$	9.87815	0.712894
$193$	8.37666	0.602965	0.301483	−	0.953472i	$-0.402518\pi$
$193$	8.37666	0.602965	0.301483	+	0.953472i	$0.402518\pi$
$194$	4.25074	0.305186
$195$	−4.69650	−0.336323
$196$	11.2452	0.803228
$197$	−4.33431	−0.308807	−0.154403	−	0.988008i	$-0.549346\pi$
$197$	−4.33431	−0.308807	−0.154403	+	0.988008i	$0.549346\pi$
$198$	0.110941	0.00788426
$199$	18.4203	1.30578	0.652891	−	0.757452i	$-0.273556\pi$
$199$	18.4203	1.30578	0.652891	+	0.757452i	$0.273556\pi$
$200$	1.11768	0.0790322
$201$	1.51898	0.107140
$202$	−3.88841	−0.273588
$203$	−3.36957	−0.236497
$204$	14.8594	1.04036
$205$	2.53697	0.177190
$206$	0.218614	0.0152315
$207$	3.56230	0.247597
$208$	10.2265	0.709079
$209$	−1.00000	−0.0691714
$210$	−0.491871	−0.0339423
$211$	0.0183766	0.00126510	0.000632549	−	1.00000i	$-0.499799\pi$
$211$	0.0183766	0.00126510	0.000632549		1.00000i	$0.499799\pi$
$212$	15.9939	1.09847
$213$	3.33241	0.228333
$214$	4.36573	0.298435
$215$	−6.45078	−0.439940
$216$	6.12058	0.416453
$217$	6.45787	0.438389
$218$	1.69886	0.115061
$219$	7.49127	0.506213
$220$	−1.91865	−0.129355
$221$	13.9305	0.937067
$222$	−1.99550	−0.133929
$223$	4.60080	0.308092	0.154046	−	0.988064i	$-0.450770\pi$
$223$	4.60080	0.308092	0.154046	+	0.988064i	$0.450770\pi$
$224$	3.45671	0.230961
$225$	−0.388965	−0.0259310
$226$	0.0577691	0.00384274
$227$	−19.8111	−1.31491	−0.657454	−	0.753494i	$-0.728367\pi$
$227$	−19.8111	−1.31491	−0.657454	+	0.753494i	$0.728367\pi$
$228$	−3.10029	−0.205321
$229$	24.4330	1.61457	0.807287	−	0.590159i	$-0.200935\pi$
$229$	24.4330	1.61457	0.807287	+	0.590159i	$0.200935\pi$
$230$	2.61218	0.172242
$231$	1.72452	0.113465
$232$	3.52883	0.231679
$233$	8.96428	0.587269	0.293635	−	0.955918i	$-0.405135\pi$
$233$	8.96428	0.587269	0.293635	+	0.955918i	$0.405135\pi$
$234$	0.322449	0.0210792
$235$	−2.00319	−0.130674
$236$	−18.5503	−1.20752
$237$	15.1401	0.983457
$238$	1.45896	0.0945704
$239$	−3.32887	−0.215326	−0.107663	−	0.994187i	$-0.534337\pi$
$239$	−3.32887	−0.215326	−0.107663	+	0.994187i	$0.534337\pi$
$240$	−5.68545	−0.366994
$241$	13.2936	0.856316	0.428158	−	0.903704i	$-0.359163\pi$
$241$	13.2936	0.856316	0.428158	+	0.903704i	$0.359163\pi$
$242$	−0.285222	−0.0183348
$243$	−4.01557	−0.257599
$244$	−10.3142	−0.660299
$245$	−5.86100	−0.374445
$246$	1.16924	0.0745482
$247$	−2.90648	−0.184935
$248$	−6.76311	−0.429458
$249$	21.4673	1.36043
$250$	−0.285222	−0.0180390
$251$	−9.72927	−0.614106	−0.307053	−	0.951692i	$-0.599343\pi$
$251$	−9.72927	−0.614106	−0.307053	+	0.951692i	$0.599343\pi$
$252$	−0.796468	−0.0501728
$253$	−9.15842	−0.575785
$254$	0.839323	0.0526638
$255$	−7.74471	−0.484993
$256$	9.88147	0.617592
$257$	−22.1968	−1.38460	−0.692298	−	0.721612i	$-0.743402\pi$
$257$	−22.1968	−1.38460	−0.692298	+	0.721612i	$0.743402\pi$
$258$	−2.97305	−0.185094
$259$	4.62088	0.287127
$260$	−5.57652	−0.345841
$261$	−1.22807	−0.0760155
$262$	−6.25133	−0.386208
$263$	3.99313	0.246227	0.123113	−	0.992393i	$-0.460712\pi$
$263$	3.99313	0.246227	0.123113	+	0.992393i	$0.460712\pi$
$264$	−1.80603	−0.111154
$265$	−8.33604	−0.512079
$266$	−0.304400	−0.0186640
$267$	23.3413	1.42846
$268$	1.80360	0.110172
$269$	−5.48116	−0.334193	−0.167096	−	0.985941i	$-0.553439\pi$
$269$	−5.48116	−0.334193	−0.167096	+	0.985941i	$0.553439\pi$
$270$	−1.56191	−0.0950549
$271$	−30.1182	−1.82955	−0.914776	−	0.403962i	$-0.867633\pi$
$271$	−30.1182	−1.82955	−0.914776	+	0.403962i	$0.867633\pi$
$272$	16.8639	1.02252
$273$	5.01229	0.303358
$274$	1.13661	0.0686652
$275$	1.00000	0.0603023
$276$	−28.3937	−1.70910
$277$	−16.7720	−1.00773	−0.503866	−	0.863782i	$-0.668089\pi$
$277$	−16.7720	−1.00773	−0.503866	+	0.863782i	$0.668089\pi$
$278$	1.39767	0.0838269
$279$	2.35363	0.140908
$280$	−1.19284	−0.0712857
$281$	−7.09037	−0.422976	−0.211488	−	0.977381i	$-0.567831\pi$
$281$	−7.09037	−0.422976	−0.211488	+	0.977381i	$0.567831\pi$
$282$	−0.923234	−0.0549778
$283$	17.4001	1.03433	0.517165	−	0.855886i	$-0.326987\pi$
$283$	17.4001	1.03433	0.517165	+	0.855886i	$0.326987\pi$
$284$	3.95684	0.234795
$285$	1.61587	0.0957159
$286$	−0.828993	−0.0490194
$287$	−2.70756	−0.159822
$288$	1.25983	0.0742361
$289$	5.97195	0.351291
$290$	−0.900523	−0.0528805
$291$	24.0818	1.41170
$292$	8.89497	0.520539
$293$	−17.5131	−1.02313	−0.511563	−	0.859246i	$-0.670933\pi$
$293$	−17.5131	−1.02313	−0.511563	+	0.859246i	$0.670933\pi$
$294$	−2.70122	−0.157539
$295$	9.66844	0.562918
$296$	−4.83928	−0.281278
$297$	5.47613	0.317757
$298$	0.523166	0.0303062
$299$	−26.6188	−1.53940
$300$	3.10029	0.178995
$301$	6.88454	0.396818
$302$	−5.32308	−0.306309
$303$	−22.0291	−1.26554
$304$	−3.51851	−0.201800
$305$	5.37576	0.307815
$306$	0.531731	0.0303971
$307$	−28.2349	−1.61145	−0.805727	−	0.592288i	$-0.798225\pi$
$307$	−28.2349	−1.61145	−0.805727	+	0.592288i	$0.798225\pi$
$308$	2.04766	0.116676
$309$	1.23851	0.0704566
$310$	1.72588	0.0980233
$311$	−25.5874	−1.45093	−0.725464	−	0.688261i	$-0.758375\pi$
$311$	−25.5874	−1.45093	−0.725464	+	0.688261i	$0.758375\pi$
$312$	−5.24920	−0.297178
$313$	4.46648	0.252460	0.126230	−	0.992001i	$-0.459712\pi$
$313$	4.46648	0.252460	0.126230	+	0.992001i	$0.459712\pi$
$314$	−5.59173	−0.315559
$315$	0.415119	0.0233893
$316$	17.9771	1.01129
$317$	−11.9211	−0.669554	−0.334777	−	0.942297i	$-0.608661\pi$
$317$	−11.9211	−0.669554	−0.334777	+	0.942297i	$0.608661\pi$
$318$	−3.84192	−0.215444
$319$	3.15727	0.176773
$320$	−6.11321	−0.341739
$321$	24.7332	1.38047
$322$	−2.78782	−0.155359
$323$	−4.79291	−0.266684
$324$	14.7387	0.818816
$325$	2.90648	0.161223
$326$	3.25888	0.180493
$327$	9.62456	0.532239
$328$	2.83553	0.156566
$329$	2.13789	0.117866
$330$	0.460881	0.0253707
$331$	−12.5948	−0.692275	−0.346137	−	0.938184i	$-0.612507\pi$
$331$	−12.5948	−0.692275	−0.346137	+	0.938184i	$0.612507\pi$
$332$	25.4898	1.39893
$333$	1.68412	0.0922892
$334$	−6.23128	−0.340960
$335$	−0.940037	−0.0513597
$336$	6.06774	0.331023
$337$	4.84070	0.263690	0.131845	−	0.991270i	$-0.457910\pi$
$337$	4.84070	0.263690	0.131845	+	0.991270i	$0.457910\pi$
$338$	1.29843	0.0706253
$339$	0.327280	0.0177754
$340$	−9.19590	−0.498718
$341$	−6.05100	−0.327680
$342$	−0.110941	−0.00599902
$343$	13.7258	0.741122
$344$	−7.20994	−0.388734
$345$	14.7988	0.796741
$346$	4.79314	0.257681
$347$	−2.27861	−0.122322	−0.0611610	−	0.998128i	$-0.519480\pi$
$347$	−2.27861	−0.122322	−0.0611610	+	0.998128i	$0.519480\pi$
$348$	9.78844	0.524716
$349$	23.6100	1.26382	0.631908	−	0.775043i	$-0.282272\pi$
$349$	23.6100	1.26382	0.631908	+	0.775043i	$0.282272\pi$
$350$	0.304400	0.0162709
$351$	15.9163	0.849548
$352$	−3.23892	−0.172635
$353$	−10.7932	−0.574465	−0.287232	−	0.957861i	$-0.592735\pi$
$353$	−10.7932	−0.574465	−0.287232	+	0.957861i	$0.592735\pi$
$354$	4.45600	0.236834
$355$	−2.06230	−0.109456
$356$	27.7149	1.46889
$357$	8.26547	0.437455
$358$	−0.0982604	−0.00519322
$359$	−0.648722	−0.0342382	−0.0171191	−	0.999853i	$-0.505449\pi$
$359$	−0.648722	−0.0342382	−0.0171191	+	0.999853i	$0.505449\pi$
$360$	−0.434740	−0.0229128
$361$	1.00000	0.0526316
$362$	−0.550146	−0.0289150
$363$	−1.61587	−0.0848112
$364$	5.95149	0.311943
$365$	−4.63606	−0.242663
$366$	2.47759	0.129506
$367$	−8.61348	−0.449620	−0.224810	−	0.974403i	$-0.572176\pi$
$367$	−8.61348	−0.449620	−0.224810	+	0.974403i	$0.572176\pi$
$368$	−32.2240	−1.67979
$369$	−0.986793	−0.0513704
$370$	1.23494	0.0642013
$371$	8.89656	0.461886
$372$	−18.7598	−0.972652
$373$	−19.4360	−1.00636	−0.503179	−	0.864182i	$-0.667837\pi$
$373$	−19.4360	−1.00636	−0.503179	+	0.864182i	$0.667837\pi$
$374$	−1.36704	−0.0706880
$375$	−1.61587	−0.0834432
$376$	−2.23894	−0.115464
$377$	9.17656	0.472617
$378$	1.66693	0.0857378
$379$	12.1909	0.626203	0.313101	−	0.949720i	$-0.398632\pi$
$379$	12.1909	0.626203	0.313101	+	0.949720i	$0.398632\pi$
$380$	1.91865	0.0984246
$381$	4.75502	0.243607
$382$	7.36323	0.376736
$383$	−26.6129	−1.35985	−0.679927	−	0.733280i	$-0.737989\pi$
$383$	−26.6129	−1.35985	−0.679927	+	0.733280i	$0.737989\pi$
$384$	−13.2848	−0.677938
$385$	−1.06724	−0.0543916
$386$	−2.38921	−0.121607
$387$	2.50913	0.127546
$388$	28.5942	1.45165
$389$	37.3509	1.89377	0.946883	−	0.321578i	$-0.104213\pi$
$389$	37.3509	1.89377	0.946883	+	0.321578i	$0.104213\pi$
$390$	1.33954	0.0678304
$391$	−43.8954	−2.21989
$392$	−6.55074	−0.330863
$393$	−35.4157	−1.78649
$394$	1.23624	0.0622808
$395$	−9.36965	−0.471438
$396$	0.746287	0.0375024
$397$	22.4086	1.12466	0.562329	−	0.826914i	$-0.309905\pi$
$397$	22.4086	1.12466	0.562329	+	0.826914i	$0.309905\pi$
$398$	−5.25388	−0.263353
$399$	−1.72452	−0.0863340
$400$	3.51851	0.175925
$401$	10.1117	0.504957	0.252478	−	0.967603i	$-0.418754\pi$
$401$	10.1117	0.504957	0.252478	+	0.967603i	$0.418754\pi$
$402$	−0.433246	−0.0216083
$403$	−17.5871	−0.876078
$404$	−26.1568	−1.30135
$405$	−7.68181	−0.381712
$406$	0.961074	0.0476973
$407$	−4.32974	−0.214617
$408$	−8.65614	−0.428543
$409$	3.82218	0.188995	0.0944974	−	0.995525i	$-0.469876\pi$
$409$	3.82218	0.188995	0.0944974	+	0.995525i	$0.469876\pi$
$410$	−0.723600	−0.0357360
$411$	6.43925	0.317625
$412$	1.47058	0.0724505
$413$	−10.3186	−0.507743
$414$	−1.01605	−0.0499360
$415$	−13.2853	−0.652149
$416$	−9.41388	−0.461553
$417$	7.91825	0.387758
$418$	0.285222	0.0139507
$419$	−3.26513	−0.159512	−0.0797560	−	0.996814i	$-0.525414\pi$
$419$	−3.26513	−0.159512	−0.0797560	+	0.996814i	$0.525414\pi$
$420$	−3.30875	−0.161450
$421$	−1.82764	−0.0890738	−0.0445369	−	0.999008i	$-0.514181\pi$
$421$	−1.82764	−0.0890738	−0.0445369	+	0.999008i	$0.514181\pi$
$422$	−0.00524141	−0.000255148 0
$423$	0.779172	0.0378847
$424$	−9.31706	−0.452476
$425$	4.79291	0.232490
$426$	−0.950477	−0.0460508
$427$	−5.73723	−0.277644
$428$	29.3677	1.41954
$429$	−4.69650	−0.226749
$430$	1.83990	0.0887281
$431$	−30.7384	−1.48062	−0.740309	−	0.672267i	$-0.765321\pi$
$431$	−30.7384	−1.48062	−0.740309	+	0.672267i	$0.765321\pi$
$432$	19.2678	0.927022
$433$	25.3598	1.21872	0.609358	−	0.792895i	$-0.291427\pi$
$433$	25.3598	1.21872	0.609358	+	0.792895i	$0.291427\pi$
$434$	−1.84193	−0.0884153
$435$	−5.10174	−0.244610
$436$	11.4280	0.547301
$437$	9.15842	0.438106
$438$	−2.13667	−0.102094
$439$	31.0248	1.48073	0.740367	−	0.672203i	$-0.234652\pi$
$439$	31.0248	1.48073	0.740367	+	0.672203i	$0.234652\pi$
$440$	1.11768	0.0532835
$441$	2.27972	0.108558
$442$	−3.97328	−0.188990
$443$	15.8397	0.752565	0.376282	−	0.926505i	$-0.377202\pi$
$443$	15.8397	0.752565	0.376282	+	0.926505i	$0.377202\pi$
$444$	−13.4234	−0.637048
$445$	−14.4450	−0.684759
$446$	−1.31225	−0.0621367
$447$	2.96390	0.140188
$448$	6.52426	0.308242
$449$	−10.8784	−0.513386	−0.256693	−	0.966493i	$-0.582633\pi$
$449$	−10.8784	−0.513386	−0.256693	+	0.966493i	$0.582633\pi$
$450$	0.110941	0.00522983
$451$	2.53697	0.119461
$452$	0.388605	0.0182784
$453$	−30.1569	−1.41689
$454$	5.65056	0.265194
$455$	−3.10192	−0.145420
$456$	1.80603	0.0845752
$457$	−27.2847	−1.27632	−0.638162	−	0.769902i	$-0.720305\pi$
$457$	−27.2847	−1.27632	−0.638162	+	0.769902i	$0.720305\pi$
$458$	−6.96881	−0.325631
$459$	26.2466	1.22508
$460$	17.5718	0.819289
$461$	15.1508	0.705644	0.352822	−	0.935691i	$-0.385222\pi$
$461$	15.1508	0.705644	0.352822	+	0.935691i	$0.385222\pi$
$462$	−0.491871	−0.0228839
$463$	10.7804	0.501009	0.250505	−	0.968115i	$-0.419403\pi$
$463$	10.7804	0.501009	0.250505	+	0.968115i	$0.419403\pi$
$464$	11.1089	0.515717
$465$	9.77763	0.453427
$466$	−2.55681	−0.118442
$467$	26.5591	1.22901	0.614505	−	0.788913i	$-0.289356\pi$
$467$	26.5591	1.22901	0.614505	+	0.788913i	$0.289356\pi$
$468$	2.16907	0.100265
$469$	1.00325	0.0463256
$470$	0.571354	0.0263546
$471$	−31.6789	−1.45968
$472$	10.8063	0.497399
$473$	−6.45078	−0.296607
$474$	−4.31830	−0.198346
$475$	−1.00000	−0.0458831
$476$	9.81424	0.449835
$477$	3.24243	0.148461
$478$	0.949465	0.0434275
$479$	−42.8193	−1.95646	−0.978232	−	0.207513i	$-0.933463\pi$
$479$	−42.8193	−1.95646	−0.978232	+	0.207513i	$0.933463\pi$
$480$	5.23368	0.238884
$481$	−12.5843	−0.573796
$482$	−3.79162	−0.172704
$483$	−15.7939	−0.718647
$484$	−1.91865	−0.0872113
$485$	−14.9033	−0.676723
$486$	1.14533	0.0519532
$487$	−2.29243	−0.103880	−0.0519400	−	0.998650i	$-0.516540\pi$
$487$	−2.29243	−0.103880	−0.0519400	+	0.998650i	$0.516540\pi$
$488$	6.00840	0.271988
$489$	18.4626	0.834906
$490$	1.67168	0.0755190
$491$	−1.10483	−0.0498604	−0.0249302	−	0.999689i	$-0.507936\pi$
$491$	−1.10483	−0.0498604	−0.0249302	+	0.999689i	$0.507936\pi$
$492$	7.86534	0.354597
$493$	15.1325	0.681534
$494$	0.828993	0.0372981
$495$	−0.388965	−0.0174827
$496$	−21.2905	−0.955972
$497$	2.20097	0.0987272
$498$	−6.12294	−0.274376
$499$	−13.4096	−0.600297	−0.300149	−	0.953892i	$-0.597036\pi$
$499$	−13.4096	−0.600297	−0.300149	+	0.953892i	$0.597036\pi$
$500$	−1.91865	−0.0858046
$501$	−35.3021	−1.57718
$502$	2.77500	0.123854
$503$	32.4285	1.44592	0.722958	−	0.690892i	$-0.242782\pi$
$503$	32.4285	1.44592	0.722958	+	0.690892i	$0.242782\pi$
$504$	0.463972	0.0206670
$505$	13.6329	0.606658
$506$	2.61218	0.116126
$507$	7.35601	0.326692
$508$	5.64601	0.250501
$509$	−25.8169	−1.14431	−0.572157	−	0.820144i	$-0.693893\pi$
$509$	−25.8169	−1.14431	−0.572157	+	0.820144i	$0.693893\pi$
$510$	2.20896	0.0978144
$511$	4.94779	0.218877
$512$	−19.2613	−0.851239
$513$	−5.47613	−0.241777
$514$	6.33100	0.279248
$515$	−0.766469	−0.0337746
$516$	−19.9993	−0.880419
$517$	−2.00319	−0.0881003
$518$	−1.31797	−0.0579085
$519$	27.1546	1.19195
$520$	3.24853	0.142457
$521$	30.6715	1.34374	0.671871	−	0.740668i	$-0.265491\pi$
$521$	30.6715	1.34374	0.671871	+	0.740668i	$0.265491\pi$
$522$	0.350272	0.0153310
$523$	−24.7541	−1.08242	−0.541211	−	0.840887i	$-0.682034\pi$
$523$	−24.7541	−1.08242	−0.541211	+	0.840887i	$0.682034\pi$
$524$	−42.0518	−1.83704
$525$	1.72452	0.0752643
$526$	−1.13893	−0.0496596
$527$	−29.0019	−1.26334
$528$	−5.68545	−0.247428
$529$	60.8766	2.64681
$530$	2.37762	0.103277
$531$	−3.76069	−0.163200
$532$	−2.04766	−0.0887773
$533$	7.37367	0.319389
$534$	−6.65744	−0.288095
$535$	−15.3064	−0.661755
$536$	−1.05066	−0.0453818
$537$	−0.556675	−0.0240223
$538$	1.56335	0.0674007
$539$	−5.86100	−0.252451
$540$	−10.5068	−0.452139
$541$	−6.09420	−0.262010	−0.131005	−	0.991382i	$-0.541820\pi$
$541$	−6.09420	−0.262010	−0.131005	+	0.991382i	$0.541820\pi$
$542$	8.59037	0.368988
$543$	−3.11675	−0.133752
$544$	−15.5239	−0.665580
$545$	−5.95627	−0.255139
$546$	−1.42962	−0.0611819
$547$	7.33467	0.313608	0.156804	−	0.987630i	$-0.449881\pi$
$547$	7.33467	0.313608	0.156804	+	0.987630i	$0.449881\pi$
$548$	7.64583	0.326614
$549$	−2.09098	−0.0892410
$550$	−0.285222	−0.0121619
$551$	−3.15727	−0.134504
$552$	16.5404	0.704006
$553$	9.99967	0.425229
$554$	4.78374	0.203242
$555$	6.99630	0.296976
$556$	9.40196	0.398732
$557$	−19.8753	−0.842143	−0.421072	−	0.907027i	$-0.638346\pi$
$557$	−19.8753	−0.842143	−0.421072	+	0.907027i	$0.638346\pi$
$558$	−0.671306	−0.0284187
$559$	−18.7491	−0.793003
$560$	−3.75509	−0.158682
$561$	−7.74471	−0.326982
$562$	2.02233	0.0853067
$563$	−31.0075	−1.30681	−0.653405	−	0.757008i	$-0.726660\pi$
$563$	−31.0075	−1.30681	−0.653405	+	0.757008i	$0.726660\pi$
$564$	−6.21047	−0.261508
$565$	−0.202541	−0.00852096
$566$	−4.96289	−0.208606
$567$	8.19834	0.344298
$568$	−2.30500	−0.0967158
$569$	−6.21085	−0.260373	−0.130186	−	0.991490i	$-0.541558\pi$
$569$	−6.21085	−0.260373	−0.130186	+	0.991490i	$0.541558\pi$
$570$	−0.460881	−0.0193042
$571$	33.0852	1.38457	0.692287	−	0.721622i	$-0.256603\pi$
$571$	33.0852	1.38457	0.692287	+	0.721622i	$0.256603\pi$
$572$	−5.57652	−0.233166
$573$	41.7150	1.74267
$574$	0.772255	0.0322333
$575$	−9.15842	−0.381932
$576$	2.37782	0.0990760
$577$	24.1448	1.00516	0.502580	−	0.864531i	$-0.332384\pi$
$577$	24.1448	1.00516	0.502580	+	0.864531i	$0.332384\pi$
$578$	−1.70333	−0.0708493
$579$	−13.5356	−0.562520
$580$	−6.05769	−0.251532
$581$	14.1786	0.588227
$582$	−6.86865	−0.284715
$583$	−8.33604	−0.345244
$584$	−5.18165	−0.214418
$585$	−1.13052	−0.0467413
$586$	4.99512	0.206347
$587$	−23.2025	−0.957668	−0.478834	−	0.877905i	$-0.658940\pi$
$587$	−23.2025	−0.957668	−0.478834	+	0.877905i	$0.658940\pi$
$588$	−18.1708	−0.749350
$589$	6.05100	0.249327
$590$	−2.75765	−0.113531
$591$	7.00368	0.288093
$592$	−15.2342	−0.626123
$593$	13.4485	0.552262	0.276131	−	0.961120i	$-0.410948\pi$
$593$	13.4485	0.552262	0.276131	+	0.961120i	$0.410948\pi$
$594$	−1.56191	−0.0640860
$595$	−5.11518	−0.209702
$596$	3.51927	0.144155
$597$	−29.7648	−1.21819
$598$	7.59226	0.310471
$599$	−21.5939	−0.882301	−0.441150	−	0.897433i	$-0.645429\pi$
$599$	−21.5939	−0.882301	−0.441150	+	0.897433i	$0.645429\pi$
$600$	−1.80603	−0.0737309
$601$	−45.1307	−1.84092	−0.920459	−	0.390839i	$-0.872185\pi$
$601$	−45.1307	−1.84092	−0.920459	+	0.390839i	$0.872185\pi$
$602$	−1.96362	−0.0800312
$603$	0.365642	0.0148901
$604$	−35.8076	−1.45699
$605$	1.00000	0.0406558
$606$	6.28317	0.255236
$607$	43.4545	1.76376	0.881882	−	0.471470i	$-0.156276\pi$
$607$	43.4545	1.76376	0.881882	+	0.471470i	$0.156276\pi$
$608$	3.23892	0.131356
$609$	5.44478	0.220634
$610$	−1.53328	−0.0620809
$611$	−5.82225	−0.235543
$612$	3.57689	0.144587
$613$	30.2175	1.22047	0.610237	−	0.792219i	$-0.291074\pi$
$613$	30.2175	1.22047	0.610237	+	0.792219i	$0.291074\pi$
$614$	8.05322	0.325002
$615$	−4.09941	−0.165304
$616$	−1.19284	−0.0480608
$617$	11.6138	0.467554	0.233777	−	0.972290i	$-0.424891\pi$
$617$	11.6138	0.467554	0.233777	+	0.972290i	$0.424891\pi$
$618$	−0.353251	−0.0142098
$619$	−22.0326	−0.885563	−0.442782	−	0.896629i	$-0.646008\pi$
$619$	−22.0326	−0.885563	−0.442782	+	0.896629i	$0.646008\pi$
$620$	11.6097	0.466259
$621$	−50.1526	−2.01256
$622$	7.29808	0.292626
$623$	15.4163	0.617641
$624$	−16.5247	−0.661516
$625$	1.00000	0.0400000
$626$	−1.27394	−0.0509168
$627$	1.61587	0.0645316
$628$	−37.6148	−1.50099
$629$	−20.7521	−0.827439
$630$	−0.118401	−0.00471721
$631$	23.7241	0.944441	0.472221	−	0.881480i	$-0.343453\pi$
$631$	23.7241	0.944441	0.472221	+	0.881480i	$0.343453\pi$
$632$	−10.4723	−0.416566
$633$	−0.0296942	−0.00118024
$634$	3.40015	0.135037
$635$	−2.94270	−0.116778
$636$	−25.8441	−1.02479
$637$	−17.0349	−0.674947
$638$	−0.900523	−0.0356520
$639$	0.802164	0.0317331
$640$	8.22147	0.324982
$641$	27.1667	1.07302	0.536510	−	0.843894i	$-0.319743\pi$
$641$	27.1667	1.07302	0.536510	+	0.843894i	$0.319743\pi$
$642$	−7.05445	−0.278417
$643$	24.7715	0.976893	0.488446	−	0.872594i	$-0.337564\pi$
$643$	24.7715	0.976893	0.488446	+	0.872594i	$0.337564\pi$
$644$	−18.7533	−0.738984
$645$	10.4236	0.410430
$646$	1.36704	0.0537855
$647$	22.6525	0.890563	0.445282	−	0.895391i	$-0.353104\pi$
$647$	22.6525	0.890563	0.445282	+	0.895391i	$0.353104\pi$
$648$	−8.58584	−0.337284
$649$	9.66844	0.379520
$650$	−0.828993	−0.0325158
$651$	−10.4351	−0.408983
$652$	21.9220	0.858534
$653$	−23.1640	−0.906476	−0.453238	−	0.891390i	$-0.649731\pi$
$653$	−23.1640	−0.906476	−0.453238	+	0.891390i	$0.649731\pi$
$654$	−2.74513	−0.107343
$655$	21.9174	0.856385
$656$	8.92636	0.348516
$657$	1.80327	0.0703521
$658$	−0.609772	−0.0237714
$659$	−30.3683	−1.18298	−0.591491	−	0.806312i	$-0.701460\pi$
$659$	−30.3683	−1.18298	−0.591491	+	0.806312i	$0.701460\pi$
$660$	3.10029	0.120678
$661$	6.87576	0.267436	0.133718	−	0.991019i	$-0.457308\pi$
$661$	6.87576	0.267436	0.133718	+	0.991019i	$0.457308\pi$
$662$	3.59232	0.139620
$663$	−22.5099	−0.874211
$664$	−14.8488	−0.576244
$665$	1.06724	0.0413858
$666$	−0.480347	−0.0186131
$667$	−28.9156	−1.11962
$668$	−41.9169	−1.62182
$669$	−7.43429	−0.287426
$670$	0.268119	0.0103583
$671$	5.37576	0.207529
$672$	−5.58559	−0.215469
$673$	−46.5687	−1.79509	−0.897546	−	0.440922i	$-0.854652\pi$
$673$	−46.5687	−1.79509	−0.897546	+	0.440922i	$0.854652\pi$
$674$	−1.38067	−0.0531816
$675$	5.47613	0.210776
$676$	8.73436	0.335937
$677$	−33.4277	−1.28473	−0.642366	−	0.766398i	$-0.722047\pi$
$677$	−33.4277	−1.28473	−0.642366	+	0.766398i	$0.722047\pi$
$678$	−0.0933473	−0.00358498
$679$	15.9054	0.610393
$680$	5.35696	0.205430
$681$	32.0121	1.22671
$682$	1.72588	0.0660873
$683$	22.6625	0.867157	0.433579	−	0.901116i	$-0.357251\pi$
$683$	22.6625	0.867157	0.433579	+	0.901116i	$0.357251\pi$
$684$	−0.746287	−0.0285350
$685$	−3.98501	−0.152259
$686$	−3.91489	−0.149471
$687$	−39.4805	−1.50627
$688$	−22.6971	−0.865321
$689$	−24.2286	−0.923035
$690$	−4.22094	−0.160689
$691$	−19.0700	−0.725457	−0.362728	−	0.931895i	$-0.618155\pi$
$691$	−19.0700	−0.725457	−0.362728	+	0.931895i	$0.618155\pi$
$692$	32.2428	1.22569
$693$	0.415119	0.0157691
$694$	0.649908	0.0246702
$695$	−4.90030	−0.185879
$696$	−5.70213	−0.216139
$697$	12.1595	0.460573
$698$	−6.73410	−0.254889
$699$	−14.4851	−0.547877
$700$	2.04766	0.0773942
$701$	6.74967	0.254932	0.127466	−	0.991843i	$-0.459316\pi$
$701$	6.74967	0.254932	0.127466	+	0.991843i	$0.459316\pi$
$702$	−4.53967	−0.171339
$703$	4.32974	0.163299
$704$	−6.11321	−0.230400
$705$	3.23690	0.121909
$706$	3.07846	0.115859
$707$	−14.5496	−0.547195
$708$	29.9749	1.12653
$709$	6.87245	0.258100	0.129050	−	0.991638i	$-0.458807\pi$
$709$	6.87245	0.258100	0.129050	+	0.991638i	$0.458807\pi$
$710$	0.588214	0.0220753
$711$	3.64447	0.136678
$712$	−16.1450	−0.605058
$713$	55.4176	2.07540
$714$	−2.35749	−0.0882269
$715$	2.90648	0.108696
$716$	−0.660984	−0.0247021
$717$	5.37901	0.200883
$718$	0.185030	0.00690525
$719$	44.8220	1.67158	0.835789	−	0.549051i	$-0.185011\pi$
$719$	44.8220	1.67158	0.835789	+	0.549051i	$0.185011\pi$
$720$	−1.36858	−0.0510039
$721$	0.818006	0.0304641
$722$	−0.285222	−0.0106149
$723$	−21.4807	−0.798876
$724$	−3.70076	−0.137538
$725$	3.15727	0.117258
$726$	0.460881	0.0171049
$727$	10.8328	0.401765	0.200882	−	0.979615i	$-0.435619\pi$
$727$	10.8328	0.401765	0.200882	+	0.979615i	$0.435619\pi$
$728$	−3.46696	−0.128494
$729$	29.5341	1.09385
$730$	1.32231	0.0489407
$731$	−30.9180	−1.14354
$732$	16.6664	0.616008
$733$	36.4329	1.34568	0.672840	−	0.739788i	$-0.265074\pi$
$733$	36.4329	1.34568	0.672840	+	0.739788i	$0.265074\pi$
$734$	2.45675	0.0906804
$735$	9.47061	0.349329
$736$	29.6634	1.09341
$737$	−0.940037	−0.0346267
$738$	0.281455	0.0103605
$739$	−34.7649	−1.27885	−0.639424	−	0.768854i	$-0.720827\pi$
$739$	−34.7649	−1.27885	−0.639424	+	0.768854i	$0.720827\pi$
$740$	8.30725	0.305381
$741$	4.69650	0.172530
$742$	−2.53749	−0.0931543
$743$	47.9481	1.75904	0.879522	−	0.475858i	$-0.157862\pi$
$743$	47.9481	1.75904	0.879522	+	0.475858i	$0.157862\pi$
$744$	10.9283	0.400651
$745$	−1.83424	−0.0672015
$746$	5.54358	0.202965
$747$	5.16752	0.189069
$748$	−9.19590	−0.336236
$749$	16.3356	0.596892
$750$	0.460881	0.0168290
$751$	−12.8844	−0.470157	−0.235079	−	0.971976i	$-0.575535\pi$
$751$	−12.8844	−0.470157	−0.235079	+	0.971976i	$0.575535\pi$
$752$	−7.04825	−0.257023
$753$	15.7212	0.572913
$754$	−2.61735	−0.0953184
$755$	18.6630	0.679214
$756$	11.2132	0.407822
$757$	0.193982	0.00705039	0.00352519	−	0.999994i	$-0.498878\pi$
$757$	0.193982	0.00705039	0.00352519	+	0.999994i	$0.498878\pi$
$758$	−3.47710	−0.126294
$759$	14.7988	0.537163
$760$	−1.11768	−0.0405427
$761$	−0.714043	−0.0258840	−0.0129420	−	0.999916i	$-0.504120\pi$
$761$	−0.714043	−0.0258840	−0.0129420	+	0.999916i	$0.504120\pi$
$762$	−1.35624	−0.0491312
$763$	6.35677	0.230131
$764$	49.5314	1.79198
$765$	−1.86427	−0.0674030
$766$	7.59057	0.274258
$767$	28.1012	1.01467
$768$	−15.9672	−0.576166
$769$	4.51509	0.162818	0.0814091	−	0.996681i	$-0.474058\pi$
$769$	4.51509	0.162818	0.0814091	+	0.996681i	$0.474058\pi$
$770$	0.304400	0.0109698
$771$	35.8671	1.29172
$772$	−16.0719	−0.578439
$773$	−0.377760	−0.0135871	−0.00679355	−	0.999977i	$-0.502162\pi$
$773$	−0.377760	−0.0135871	−0.00679355	+	0.999977i	$0.502162\pi$
$774$	−0.715659	−0.0257238
$775$	−6.05100	−0.217358
$776$	−16.6572	−0.597957
$777$	−7.46673	−0.267868
$778$	−10.6533	−0.381939
$779$	−2.53697	−0.0908964
$780$	9.01093	0.322643
$781$	−2.06230	−0.0737950
$782$	12.5199	0.447712
$783$	17.2896	0.617880
$784$	−20.6220	−0.736499
$785$	19.6048	0.699727
$786$	10.1013	0.360303
$787$	−15.9223	−0.567569	−0.283785	−	0.958888i	$-0.591590\pi$
$787$	−15.9223	−0.567569	−0.283785	+	0.958888i	$0.591590\pi$
$788$	8.31601	0.296246
$789$	−6.45237	−0.229711
$790$	2.67243	0.0950807
$791$	0.216160	0.00768576
$792$	−0.434740	−0.0154478
$793$	15.6246	0.554844
$794$	−6.39144	−0.226824
$795$	13.4700	0.477730
$796$	−35.3421	−1.25267
$797$	34.3945	1.21832	0.609158	−	0.793049i	$-0.291508\pi$
$797$	34.3945	1.21832	0.609158	+	0.793049i	$0.291508\pi$
$798$	0.491871	0.0174120
$799$	−9.60112	−0.339663
$800$	−3.23892	−0.114513
$801$	5.61861	0.198524
$802$	−2.88409	−0.101841
$803$	−4.63606	−0.163603
$804$	−2.91438	−0.102782
$805$	9.77423	0.344496
$806$	5.01623	0.176689
$807$	8.85685	0.311776
$808$	15.2373	0.536047
$809$	−5.46814	−0.192250	−0.0961248	−	0.995369i	$-0.530645\pi$
$809$	−5.46814	−0.192250	−0.0961248	+	0.995369i	$0.530645\pi$
$810$	2.19102	0.0769846
$811$	28.2165	0.990816	0.495408	−	0.868660i	$-0.335019\pi$
$811$	28.2165	0.990816	0.495408	+	0.868660i	$0.335019\pi$
$812$	6.46501	0.226878
$813$	48.6671	1.70683
$814$	1.23494	0.0432845
$815$	−11.4258	−0.400227
$816$	−27.2498	−0.953935
$817$	6.45078	0.225684
$818$	−1.09017	−0.0381169
$819$	1.20654	0.0421598
$820$	−4.86756	−0.169982
$821$	−26.0249	−0.908275	−0.454137	−	0.890932i	$-0.650052\pi$
$821$	−26.0249	−0.908275	−0.454137	+	0.890932i	$0.650052\pi$
$822$	−1.83661	−0.0640593
$823$	41.1592	1.43472	0.717359	−	0.696703i	$-0.245351\pi$
$823$	41.1592	1.43472	0.717359	+	0.696703i	$0.245351\pi$
$824$	−0.856670	−0.0298435
$825$	−1.61587	−0.0562574
$826$	2.94308	0.102403
$827$	15.7775	0.548638	0.274319	−	0.961639i	$-0.411548\pi$
$827$	15.7775	0.548638	0.274319	+	0.961639i	$0.411548\pi$
$828$	−6.83481	−0.237526
$829$	−28.7891	−0.999886	−0.499943	−	0.866058i	$-0.666646\pi$
$829$	−28.7891	−0.999886	−0.499943	+	0.866058i	$0.666646\pi$
$830$	3.78925	0.131527
$831$	27.1014	0.940136
$832$	−17.7679	−0.615992
$833$	−28.0912	−0.973303
$834$	−2.25846	−0.0782040
$835$	21.8471	0.756051
$836$	1.91865	0.0663578
$837$	−33.1360	−1.14535
$838$	0.931286	0.0321708
$839$	−18.3610	−0.633892	−0.316946	−	0.948444i	$-0.602657\pi$
$839$	−18.3610	−0.633892	−0.316946	+	0.948444i	$0.602657\pi$
$840$	1.92747	0.0665040
$841$	−19.0316	−0.656263
$842$	0.521284	0.0179646
$843$	11.4571	0.394604
$844$	−0.0352583	−0.00121364
$845$	−4.55235	−0.156606
$846$	−0.222237	−0.00764066
$847$	−1.06724	−0.0366708
$848$	−29.3304	−1.00721
$849$	−28.1163	−0.964950
$850$	−1.36704	−0.0468891
$851$	39.6536	1.35931
$852$	−6.39373	−0.219046
$853$	14.3634	0.491794	0.245897	−	0.969296i	$-0.420917\pi$
$853$	14.3634	0.491794	0.245897	+	0.969296i	$0.420917\pi$
$854$	1.63638	0.0559959
$855$	0.388965	0.0133023
$856$	−17.1078	−0.584731
$857$	−37.9711	−1.29707	−0.648534	−	0.761186i	$-0.724618\pi$
$857$	−37.9711	−1.29707	−0.648534	+	0.761186i	$0.724618\pi$
$858$	1.33954	0.0457313
$859$	55.6028	1.89714	0.948572	−	0.316562i	$-0.102529\pi$
$859$	55.6028	1.89714	0.948572	+	0.316562i	$0.102529\pi$
$860$	12.3768	0.422045
$861$	4.37506	0.149102
$862$	8.76727	0.298614
$863$	−34.6789	−1.18048	−0.590242	−	0.807226i	$-0.700968\pi$
$863$	−34.6789	−1.18048	−0.590242	+	0.807226i	$0.700968\pi$
$864$	−17.7368	−0.603417
$865$	−16.8049	−0.571385
$866$	−7.23318	−0.245794
$867$	−9.64990	−0.327728
$868$	−12.3904	−0.420557
$869$	−9.36965	−0.317844
$870$	1.45513	0.0493334
$871$	−2.73220	−0.0925772
$872$	−6.65723	−0.225442
$873$	5.79686	0.196194
$874$	−2.61218	−0.0883583
$875$	−1.06724	−0.0360793
$876$	−14.3731	−0.485623
$877$	16.1999	0.547033	0.273516	−	0.961867i	$-0.411813\pi$
$877$	16.1999	0.547033	0.273516	+	0.961867i	$0.411813\pi$
$878$	−8.84896	−0.298638
$879$	28.2989	0.954498
$880$	3.51851	0.118609
$881$	10.5799	0.356447	0.178223	−	0.983990i	$-0.442965\pi$
$881$	10.5799	0.356447	0.178223	+	0.983990i	$0.442965\pi$
$882$	−0.650227	−0.0218943
$883$	46.3714	1.56052	0.780260	−	0.625455i	$-0.215087\pi$
$883$	46.3714	1.56052	0.780260	+	0.625455i	$0.215087\pi$
$884$	−26.7277	−0.898951
$885$	−15.6229	−0.525159
$886$	−4.51782	−0.151779
$887$	−4.09085	−0.137357	−0.0686786	−	0.997639i	$-0.521878\pi$
$887$	−4.09085	−0.137357	−0.0686786	+	0.997639i	$0.521878\pi$
$888$	7.81965	0.262410
$889$	3.14057	0.105331
$890$	4.12003	0.138104
$891$	−7.68181	−0.257350
$892$	−8.82731	−0.295560
$893$	2.00319	0.0670343
$894$	−0.845369	−0.0282734
$895$	0.344505	0.0115155
$896$	−8.77428	−0.293128
$897$	43.0125	1.43615
$898$	3.10277	0.103541
$899$	−19.1047	−0.637176
$900$	0.746287	0.0248762
$901$	−39.9539	−1.33106
$902$	−0.723600	−0.0240932
$903$	−11.1245	−0.370201
$904$	−0.226377	−0.00752918
$905$	1.92884	0.0641167
$906$	8.60141	0.285763
$907$	−27.2913	−0.906192	−0.453096	−	0.891462i	$-0.649680\pi$
$907$	−27.2913	−0.906192	−0.453096	+	0.891462i	$0.649680\pi$
$908$	38.0105	1.26142
$909$	−5.30274	−0.175881
$910$	0.884734	0.0293287
$911$	57.2181	1.89572	0.947860	−	0.318687i	$-0.103242\pi$
$911$	57.2181	1.89572	0.947860	+	0.318687i	$0.103242\pi$
$912$	5.68545	0.188264
$913$	−13.2853	−0.439679
$914$	7.78218	0.257412
$915$	−8.68653	−0.287168
$916$	−46.8782	−1.54890
$917$	−23.3912	−0.772444
$918$	−7.48609	−0.247078
$919$	29.1281	0.960846	0.480423	−	0.877037i	$-0.340483\pi$
$919$	29.1281	0.960846	0.480423	+	0.877037i	$0.340483\pi$
$920$	−10.2362	−0.337478
$921$	45.6240	1.50336
$922$	−4.32134	−0.142316
$923$	−5.99405	−0.197297
$924$	−3.30875	−0.108850
$925$	−4.32974	−0.142361
$926$	−3.07481	−0.101045
$927$	0.298130	0.00979186
$928$	−10.2262	−0.335690
$929$	45.8765	1.50516	0.752579	−	0.658501i	$-0.228809\pi$
$929$	45.8765	1.50516	0.752579	+	0.658501i	$0.228809\pi$
$930$	−2.78879	−0.0914481
$931$	5.86100	0.192086
$932$	−17.1993	−0.563382
$933$	41.3459	1.35360
$934$	−7.57524	−0.247869
$935$	4.79291	0.156745
$936$	−1.26357	−0.0413009
$937$	34.6099	1.13066	0.565328	−	0.824866i	$-0.308750\pi$
$937$	34.6099	1.13066	0.565328	+	0.824866i	$0.308750\pi$
$938$	−0.286148	−0.00934305
$939$	−7.21725	−0.235526
$940$	3.84342	0.125359
$941$	−16.2629	−0.530156	−0.265078	−	0.964227i	$-0.585398\pi$
$941$	−16.2629	−0.530156	−0.265078	+	0.964227i	$0.585398\pi$
$942$	9.03550	0.294393
$943$	−23.2346	−0.756624
$944$	34.0185	1.10721
$945$	−5.84434	−0.190116
$946$	1.83990	0.0598205
$947$	42.9091	1.39436	0.697179	−	0.716897i	$-0.254438\pi$
$947$	42.9091	1.39436	0.697179	+	0.716897i	$0.254438\pi$
$948$	−29.0486	−0.943454
$949$	−13.4746	−0.437405
$950$	0.285222	0.00925382
$951$	19.2629	0.624642
$952$	−5.71716	−0.185294
$953$	52.5829	1.70333	0.851663	−	0.524090i	$-0.175594\pi$
$953$	52.5829	1.70333	0.851663	+	0.524090i	$0.175594\pi$
$954$	−0.924811	−0.0299419
$955$	−25.8158	−0.835380
$956$	6.38692	0.206568
$957$	−5.10174	−0.164916
$958$	12.2130	0.394584
$959$	4.25296	0.137335
$960$	9.87815	0.318816
$961$	5.61462	0.181117
$962$	3.58932	0.115724
$963$	5.95367	0.191854
$964$	−25.5057	−0.821484
$965$	8.37666	0.269654
$966$	4.50476	0.144938
$967$	−13.3701	−0.429953	−0.214976	−	0.976619i	$-0.568967\pi$
$967$	−13.3701	−0.429953	−0.214976	+	0.976619i	$0.568967\pi$
$968$	1.11768	0.0359237
$969$	7.74471	0.248796
$970$	4.25074	0.136483
$971$	−41.7945	−1.34125	−0.670625	−	0.741797i	$-0.733974\pi$
$971$	−41.7945	−1.34125	−0.670625	+	0.741797i	$0.733974\pi$
$972$	7.70447	0.247121
$973$	5.22980	0.167660
$974$	0.653851	0.0209507
$975$	−4.69650	−0.150408
$976$	18.9147	0.605444
$977$	−49.3269	−1.57811	−0.789053	−	0.614325i	$-0.789428\pi$
$977$	−49.3269	−1.57811	−0.789053	+	0.614325i	$0.789428\pi$
$978$	−5.26593	−0.168386
$979$	−14.4450	−0.461665
$980$	11.2452	0.359215
$981$	2.31678	0.0739691
$982$	0.315123	0.0100560
$983$	33.3920	1.06504	0.532520	−	0.846417i	$-0.321245\pi$
$983$	33.3920	1.06504	0.532520	+	0.846417i	$0.321245\pi$
$984$	−4.58185	−0.146064
$985$	−4.33431	−0.138103
$986$	−4.31612	−0.137453
$987$	−3.45455	−0.109960
$988$	5.57652	0.177413
$989$	59.0790	1.87860
$990$	0.110941	0.00352595
$991$	32.1270	1.02055	0.510275	−	0.860012i	$-0.329544\pi$
$991$	32.1270	1.02055	0.510275	+	0.860012i	$0.329544\pi$
$992$	19.5987	0.622260
$993$	20.3516	0.645839
$994$	−0.627766	−0.0199115
$995$	18.4203	0.583963
$996$	−41.1882	−1.30510
$997$	−18.4077	−0.582979	−0.291489	−	0.956574i	$-0.594151\pi$
$997$	−18.4077	−0.582979	−0.291489	+	0.956574i	$0.594151\pi$
$998$	3.82472	0.121069
$999$	−23.7102	−0.750158

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.2.a.i.1.5	✓	8
3.2	odd	2		9405.2.a.bf.1.4		8
5.4	even	2		5225.2.a.o.1.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.2.a.i.1.5	✓	8	1.1	even	1	trivial
5225.2.a.o.1.4		8	5.4	even	2
9405.2.a.bf.1.4		8	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1045 = 5 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1045.a (trivial)

\(f(q)\)	\(=\)	\(q-0.285222 q^{2} -1.61587 q^{3} -1.91865 q^{4} +1.00000 q^{5} +0.460881 q^{6} -1.06724 q^{7} +1.11768 q^{8} -0.388965 q^{9} +O(q^{10})\)	\(q-0.285222 q^{2} -1.61587 q^{3} -1.91865 q^{4} +1.00000 q^{5} +0.460881 q^{6} -1.06724 q^{7} +1.11768 q^{8} -0.388965 q^{9} -0.285222 q^{10} +1.00000 q^{11} +3.10029 q^{12} +2.90648 q^{13} +0.304400 q^{14} -1.61587 q^{15} +3.51851 q^{16} +4.79291 q^{17} +0.110941 q^{18} -1.00000 q^{19} -1.91865 q^{20} +1.72452 q^{21} -0.285222 q^{22} -9.15842 q^{23} -1.80603 q^{24} +1.00000 q^{25} -0.828993 q^{26} +5.47613 q^{27} +2.04766 q^{28} +3.15727 q^{29} +0.460881 q^{30} -6.05100 q^{31} -3.23892 q^{32} -1.61587 q^{33} -1.36704 q^{34} -1.06724 q^{35} +0.746287 q^{36} -4.32974 q^{37} +0.285222 q^{38} -4.69650 q^{39} +1.11768 q^{40} +2.53697 q^{41} -0.491871 q^{42} -6.45078 q^{43} -1.91865 q^{44} -0.388965 q^{45} +2.61218 q^{46} -2.00319 q^{47} -5.68545 q^{48} -5.86100 q^{49} -0.285222 q^{50} -7.74471 q^{51} -5.57652 q^{52} -8.33604 q^{53} -1.56191 q^{54} +1.00000 q^{55} -1.19284 q^{56} +1.61587 q^{57} -0.900523 q^{58} +9.66844 q^{59} +3.10029 q^{60} +5.37576 q^{61} +1.72588 q^{62} +0.415119 q^{63} -6.11321 q^{64} +2.90648 q^{65} +0.460881 q^{66} -0.940037 q^{67} -9.19590 q^{68} +14.7988 q^{69} +0.304400 q^{70} -2.06230 q^{71} -0.434740 q^{72} -4.63606 q^{73} +1.23494 q^{74} -1.61587 q^{75} +1.91865 q^{76} -1.06724 q^{77} +1.33954 q^{78} -9.36965 q^{79} +3.51851 q^{80} -7.68181 q^{81} -0.723600 q^{82} -13.2853 q^{83} -3.30875 q^{84} +4.79291 q^{85} +1.83990 q^{86} -5.10174 q^{87} +1.11768 q^{88} -14.4450 q^{89} +0.110941 q^{90} -3.10192 q^{91} +17.5718 q^{92} +9.77763 q^{93} +0.571354 q^{94} -1.00000 q^{95} +5.23368 q^{96} -14.9033 q^{97} +1.67168 q^{98} -0.388965 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 6 q^{2} - 7 q^{3} + 10 q^{4} + 8 q^{5} - 11 q^{7} - 18 q^{8} + 11 q^{9}+O(q^{10}) \) 8 * q - 6 * q^2 - 7 * q^3 + 10 * q^4 + 8 * q^5 - 11 * q^7 - 18 * q^8 + 11 * q^9	\( 8 q - 6 q^{2} - 7 q^{3} + 10 q^{4} + 8 q^{5} - 11 q^{7} - 18 q^{8} + 11 q^{9} - 6 q^{10} + 8 q^{11} - 7 q^{12} - 17 q^{13} + 12 q^{14} - 7 q^{15} + 18 q^{16} - 9 q^{17} - 2 q^{18} - 8 q^{19} + 10 q^{20} + q^{21} - 6 q^{22} - 8 q^{23} + q^{24} + 8 q^{25} + 10 q^{26} - 34 q^{27} - 22 q^{28} - 3 q^{29} - q^{31} - 37 q^{32} - 7 q^{33} - 8 q^{34} - 11 q^{35} + 30 q^{36} - 17 q^{37} + 6 q^{38} + 14 q^{39} - 18 q^{40} - 5 q^{41} + 15 q^{42} - 21 q^{43} + 10 q^{44} + 11 q^{45} - 2 q^{46} - 8 q^{47} + 10 q^{48} + 19 q^{49} - 6 q^{50} - 16 q^{51} + 9 q^{52} - 19 q^{53} - 3 q^{54} + 8 q^{55} + 24 q^{56} + 7 q^{57} + 37 q^{58} - 33 q^{59} - 7 q^{60} - q^{61} - 42 q^{62} - 20 q^{63} + 48 q^{64} - 17 q^{65} - 18 q^{67} - 37 q^{68} + 16 q^{69} + 12 q^{70} - 18 q^{71} + 13 q^{72} - 18 q^{73} + 15 q^{74} - 7 q^{75} - 10 q^{76} - 11 q^{77} - 51 q^{78} - 5 q^{79} + 18 q^{80} + 32 q^{81} + 12 q^{82} - 33 q^{83} - 51 q^{84} - 9 q^{85} - 16 q^{86} - 26 q^{87} - 18 q^{88} - 20 q^{89} - 2 q^{90} + 6 q^{91} - 3 q^{92} + 18 q^{93} + 30 q^{94} - 8 q^{95} + 21 q^{96} - 69 q^{98} + 11 q^{99}+O(q^{100}) \) 8 * q - 6 * q^2 - 7 * q^3 + 10 * q^4 + 8 * q^5 - 11 * q^7 - 18 * q^8 + 11 * q^9 - 6 * q^10 + 8 * q^11 - 7 * q^12 - 17 * q^13 + 12 * q^14 - 7 * q^15 + 18 * q^16 - 9 * q^17 - 2 * q^18 - 8 * q^19 + 10 * q^20 + q^21 - 6 * q^22 - 8 * q^23 + q^24 + 8 * q^25 + 10 * q^26 - 34 * q^27 - 22 * q^28 - 3 * q^29 - q^31 - 37 * q^32 - 7 * q^33 - 8 * q^34 - 11 * q^35 + 30 * q^36 - 17 * q^37 + 6 * q^38 + 14 * q^39 - 18 * q^40 - 5 * q^41 + 15 * q^42 - 21 * q^43 + 10 * q^44 + 11 * q^45 - 2 * q^46 - 8 * q^47 + 10 * q^48 + 19 * q^49 - 6 * q^50 - 16 * q^51 + 9 * q^52 - 19 * q^53 - 3 * q^54 + 8 * q^55 + 24 * q^56 + 7 * q^57 + 37 * q^58 - 33 * q^59 - 7 * q^60 - q^61 - 42 * q^62 - 20 * q^63 + 48 * q^64 - 17 * q^65 - 18 * q^67 - 37 * q^68 + 16 * q^69 + 12 * q^70 - 18 * q^71 + 13 * q^72 - 18 * q^73 + 15 * q^74 - 7 * q^75 - 10 * q^76 - 11 * q^77 - 51 * q^78 - 5 * q^79 + 18 * q^80 + 32 * q^81 + 12 * q^82 - 33 * q^83 - 51 * q^84 - 9 * q^85 - 16 * q^86 - 26 * q^87 - 18 * q^88 - 20 * q^89 - 2 * q^90 + 6 * q^91 - 3 * q^92 + 18 * q^93 + 30 * q^94 - 8 * q^95 + 21 * q^96 - 69 * q^98 + 11 * q^99

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