LMFDB - Embedded newform 8041.2.a.g.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8041,2,Mod(1,8041)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8041, 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("8041.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8041 = 11 \cdot 17 \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8041.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:	$64.2077082653$
Analytic rank:	$1$
Dimension:	$69$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Character	$\chi$	$=$	8041.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.87945 q^{2} -2.61165 q^{3} +1.53233 q^{4} -1.23543 q^{5} +4.90847 q^{6} -3.71692 q^{7} +0.878969 q^{8} +3.82074 q^{9} +O(q^{10})$	\(q-1.87945 q^{2} -2.61165 q^{3} +1.53233 q^{4} -1.23543 q^{5} +4.90847 q^{6} -3.71692 q^{7} +0.878969 q^{8} +3.82074 q^{9} +2.32194 q^{10} -1.00000 q^{11} -4.00191 q^{12} -4.16760 q^{13} +6.98576 q^{14} +3.22653 q^{15} -4.71663 q^{16} +1.00000 q^{17} -7.18088 q^{18} -5.54379 q^{19} -1.89309 q^{20} +9.70731 q^{21} +1.87945 q^{22} -0.942351 q^{23} -2.29556 q^{24} -3.47370 q^{25} +7.83280 q^{26} -2.14348 q^{27} -5.69553 q^{28} -2.45710 q^{29} -6.06409 q^{30} -3.17078 q^{31} +7.10672 q^{32} +2.61165 q^{33} -1.87945 q^{34} +4.59201 q^{35} +5.85461 q^{36} +9.01121 q^{37} +10.4193 q^{38} +10.8843 q^{39} -1.08591 q^{40} +1.96599 q^{41} -18.2444 q^{42} +1.00000 q^{43} -1.53233 q^{44} -4.72027 q^{45} +1.77110 q^{46} -4.72284 q^{47} +12.3182 q^{48} +6.81550 q^{49} +6.52864 q^{50} -2.61165 q^{51} -6.38613 q^{52} -7.71299 q^{53} +4.02855 q^{54} +1.23543 q^{55} -3.26706 q^{56} +14.4785 q^{57} +4.61799 q^{58} +9.28151 q^{59} +4.94409 q^{60} -14.6966 q^{61} +5.95932 q^{62} -14.2014 q^{63} -3.92346 q^{64} +5.14880 q^{65} -4.90847 q^{66} -10.9358 q^{67} +1.53233 q^{68} +2.46109 q^{69} -8.63045 q^{70} -1.56038 q^{71} +3.35831 q^{72} +0.378831 q^{73} -16.9361 q^{74} +9.07211 q^{75} -8.49490 q^{76} +3.71692 q^{77} -20.4565 q^{78} -4.75181 q^{79} +5.82709 q^{80} -5.86419 q^{81} -3.69497 q^{82} +17.6463 q^{83} +14.8748 q^{84} -1.23543 q^{85} -1.87945 q^{86} +6.41709 q^{87} -0.878969 q^{88} -1.26378 q^{89} +8.87150 q^{90} +15.4907 q^{91} -1.44399 q^{92} +8.28098 q^{93} +8.87633 q^{94} +6.84899 q^{95} -18.5603 q^{96} +4.07807 q^{97} -12.8094 q^{98} -3.82074 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 69 q - 11 q^{2} - 3 q^{3} + 65 q^{4} - 6 q^{5} - 10 q^{6} - 11 q^{7} - 33 q^{8} + 56 q^{9}+O(q^{10}) $ 69 * q - 11 * q^2 - 3 * q^3 + 65 * q^4 - 6 * q^5 - 10 * q^6 - 11 * q^7 - 33 * q^8 + 56 * q^9	\( 69 q - 11 q^{2} - 3 q^{3} + 65 q^{4} - 6 q^{5} - 10 q^{6} - 11 q^{7} - 33 q^{8} + 56 q^{9} - q^{10} - 69 q^{11} - 3 q^{12} - 28 q^{13} - 15 q^{14} - 45 q^{15} + 53 q^{16} + 69 q^{17} - 17 q^{18} - 32 q^{19} - 21 q^{20} - 38 q^{21} + 11 q^{22} - 41 q^{23} - 11 q^{24} + 67 q^{25} - 6 q^{26} - 3 q^{27} - 21 q^{28} - 22 q^{29} - 22 q^{30} - 27 q^{31} - 87 q^{32} + 3 q^{33} - 11 q^{34} - 44 q^{35} + 59 q^{36} + 24 q^{37} - 22 q^{38} - 59 q^{39} + q^{40} - 43 q^{41} - 15 q^{42} + 69 q^{43} - 65 q^{44} - 12 q^{45} - 21 q^{46} - 99 q^{47} + 2 q^{48} + 64 q^{49} - 78 q^{50} - 3 q^{51} - 57 q^{52} - 50 q^{53} + 20 q^{54} + 6 q^{55} - 59 q^{56} - 15 q^{57} + 22 q^{58} - 82 q^{59} - 86 q^{60} - 24 q^{61} + 15 q^{62} - 63 q^{63} + 63 q^{64} - 23 q^{65} + 10 q^{66} - 54 q^{67} + 65 q^{68} + 36 q^{69} + 9 q^{70} - 128 q^{71} - 69 q^{72} + 2 q^{73} - 58 q^{74} - 31 q^{75} - 76 q^{76} + 11 q^{77} - 19 q^{78} - 43 q^{79} - 19 q^{80} + 49 q^{81} - 2 q^{82} - 62 q^{83} - 82 q^{84} - 6 q^{85} - 11 q^{86} - 62 q^{87} + 33 q^{88} - 49 q^{89} - 37 q^{90} - 2 q^{91} - 96 q^{92} - 29 q^{93} - 75 q^{94} - 133 q^{95} - 86 q^{96} + 5 q^{97} - 72 q^{98} - 56 q^{99}+O(q^{100}) \) 69 * q - 11 * q^2 - 3 * q^3 + 65 * q^4 - 6 * q^5 - 10 * q^6 - 11 * q^7 - 33 * q^8 + 56 * q^9 - q^10 - 69 * q^11 - 3 * q^12 - 28 * q^13 - 15 * q^14 - 45 * q^15 + 53 * q^16 + 69 * q^17 - 17 * q^18 - 32 * q^19 - 21 * q^20 - 38 * q^21 + 11 * q^22 - 41 * q^23 - 11 * q^24 + 67 * q^25 - 6 * q^26 - 3 * q^27 - 21 * q^28 - 22 * q^29 - 22 * q^30 - 27 * q^31 - 87 * q^32 + 3 * q^33 - 11 * q^34 - 44 * q^35 + 59 * q^36 + 24 * q^37 - 22 * q^38 - 59 * q^39 + q^40 - 43 * q^41 - 15 * q^42 + 69 * q^43 - 65 * q^44 - 12 * q^45 - 21 * q^46 - 99 * q^47 + 2 * q^48 + 64 * q^49 - 78 * q^50 - 3 * q^51 - 57 * q^52 - 50 * q^53 + 20 * q^54 + 6 * q^55 - 59 * q^56 - 15 * q^57 + 22 * q^58 - 82 * q^59 - 86 * q^60 - 24 * q^61 + 15 * q^62 - 63 * q^63 + 63 * q^64 - 23 * q^65 + 10 * q^66 - 54 * q^67 + 65 * q^68 + 36 * q^69 + 9 * q^70 - 128 * q^71 - 69 * q^72 + 2 * q^73 - 58 * q^74 - 31 * q^75 - 76 * q^76 + 11 * q^77 - 19 * q^78 - 43 * q^79 - 19 * q^80 + 49 * q^81 - 2 * q^82 - 62 * q^83 - 82 * q^84 - 6 * q^85 - 11 * q^86 - 62 * q^87 + 33 * q^88 - 49 * q^89 - 37 * q^90 - 2 * q^91 - 96 * q^92 - 29 * q^93 - 75 * q^94 - 133 * q^95 - 86 * q^96 + 5 * q^97 - 72 * q^98 - 56 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.87945	−1.32897	−0.664485	−	0.747301i	$-0.731349\pi$
$2$	−1.87945	−1.32897	−0.664485	+	0.747301i	$0.731349\pi$
$3$	−2.61165	−1.50784	−0.753920	−	0.656967i	$-0.771839\pi$
$3$	−2.61165	−1.50784	−0.753920	+	0.656967i	$0.771839\pi$
$4$	1.53233	0.766163
$5$	−1.23543	−0.552503	−0.276252	−	0.961085i	$-0.589092\pi$
$5$	−1.23543	−0.552503	−0.276252	+	0.961085i	$0.589092\pi$
$6$	4.90847	2.00387
$7$	−3.71692	−1.40486	−0.702432	−	0.711751i	$-0.747902\pi$
$7$	−3.71692	−1.40486	−0.702432	+	0.711751i	$0.747902\pi$
$8$	0.878969	0.310762
$9$	3.82074	1.27358
$10$	2.32194	0.734260
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−4.00191	−1.15525
$13$	−4.16760	−1.15589	−0.577943	−	0.816077i	$-0.696144\pi$
$13$	−4.16760	−1.15589	−0.577943	+	0.816077i	$0.696144\pi$
$14$	6.98576	1.86702
$15$	3.22653	0.833086
$16$	−4.71663	−1.17916
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	−7.18088	−1.69255
$19$	−5.54379	−1.27183	−0.635916	−	0.771758i	$-0.719378\pi$
$19$	−5.54379	−1.27183	−0.635916	+	0.771758i	$0.719378\pi$
$20$	−1.89309	−0.423307
$21$	9.70731	2.11831
$22$	1.87945	0.400700
$23$	−0.942351	−0.196494	−0.0982468	−	0.995162i	$-0.531323\pi$
$23$	−0.942351	−0.196494	−0.0982468	+	0.995162i	$0.531323\pi$
$24$	−2.29556	−0.468580
$25$	−3.47370	−0.694740
$26$	7.83280	1.53614
$27$	−2.14348	−0.412512
$28$	−5.69553	−1.07635
$29$	−2.45710	−0.456271	−0.228136	−	0.973629i	$-0.573263\pi$
$29$	−2.45710	−0.456271	−0.228136	+	0.973629i	$0.573263\pi$
$30$	−6.06409	−1.10715
$31$	−3.17078	−0.569489	−0.284745	−	0.958603i	$-0.591909\pi$
$31$	−3.17078	−0.569489	−0.284745	+	0.958603i	$0.591909\pi$
$32$	7.10672	1.25630
$33$	2.61165	0.454631
$34$	−1.87945	−0.322323
$35$	4.59201	0.776192
$36$	5.85461	0.975769
$37$	9.01121	1.48143	0.740717	−	0.671818i	$-0.234486\pi$
$37$	9.01121	1.48143	0.740717	+	0.671818i	$0.234486\pi$
$38$	10.4193	1.69023
$39$	10.8843	1.74289
$40$	−1.08591	−0.171697
$41$	1.96599	0.307036	0.153518	−	0.988146i	$-0.450940\pi$
$41$	1.96599	0.307036	0.153518	+	0.988146i	$0.450940\pi$
$42$	−18.2444	−2.81517
$43$	1.00000	0.152499
$43$	1.00000	0.152499
$44$	−1.53233	−0.231007
$45$	−4.72027	−0.703656
$46$	1.77110	0.261134
$47$	−4.72284	−0.688897	−0.344448	−	0.938805i	$-0.611934\pi$
$47$	−4.72284	−0.688897	−0.344448	+	0.938805i	$0.611934\pi$
$48$	12.3182	1.77798
$49$	6.81550	0.973642
$50$	6.52864	0.923289
$51$	−2.61165	−0.365705
$52$	−6.38613	−0.885597
$53$	−7.71299	−1.05946	−0.529730	−	0.848166i	$-0.677707\pi$
$53$	−7.71299	−1.05946	−0.529730	+	0.848166i	$0.677707\pi$
$54$	4.02855	0.548217
$55$	1.23543	0.166586
$56$	−3.26706	−0.436579
$57$	14.4785	1.91772
$58$	4.61799	0.606371
$59$	9.28151	1.20835	0.604175	−	0.796852i	$-0.293503\pi$
$59$	9.28151	1.20835	0.604175	+	0.796852i	$0.293503\pi$
$60$	4.94409	0.638280
$61$	−14.6966	−1.88171	−0.940855	−	0.338809i	$-0.889976\pi$
$61$	−14.6966	−1.88171	−0.940855	+	0.338809i	$0.889976\pi$
$62$	5.95932	0.756834
$63$	−14.2014	−1.78920
$64$	−3.92346	−0.490432
$65$	5.14880	0.638630
$66$	−4.90847	−0.604191
$67$	−10.9358	−1.33602	−0.668010	−	0.744152i	$-0.732854\pi$
$67$	−10.9358	−1.33602	−0.668010	+	0.744152i	$0.732854\pi$
$68$	1.53233	0.185822
$69$	2.46109	0.296281
$70$	−8.63045	−1.03154
$71$	−1.56038	−0.185184	−0.0925918	−	0.995704i	$-0.529515\pi$
$71$	−1.56038	−0.185184	−0.0925918	+	0.995704i	$0.529515\pi$
$72$	3.35831	0.395780
$73$	0.378831	0.0443388	0.0221694	−	0.999754i	$-0.492943\pi$
$73$	0.378831	0.0443388	0.0221694	+	0.999754i	$0.492943\pi$
$74$	−16.9361	−1.96878
$75$	9.07211	1.04756
$76$	−8.49490	−0.974431
$77$	3.71692	0.423582
$78$	−20.4565	−2.31625
$79$	−4.75181	−0.534620	−0.267310	−	0.963611i	$-0.586135\pi$
$79$	−4.75181	−0.534620	−0.267310	+	0.963611i	$0.586135\pi$
$80$	5.82709	0.651488
$81$	−5.86419	−0.651576
$82$	−3.69497	−0.408042
$83$	17.6463	1.93693	0.968466	−	0.249147i	$-0.0801501\pi$
$83$	17.6463	1.93693	0.968466	+	0.249147i	$0.0801501\pi$
$84$	14.8748	1.62297
$85$	−1.23543	−0.134002
$86$	−1.87945	−0.202666
$87$	6.41709	0.687984
$88$	−0.878969	−0.0936984
$89$	−1.26378	−0.133960	−0.0669802	−	0.997754i	$-0.521336\pi$
$89$	−1.26378	−0.133960	−0.0669802	+	0.997754i	$0.521336\pi$
$90$	8.87150	0.935138
$91$	15.4907	1.62386
$92$	−1.44399	−0.150546
$93$	8.28098	0.858698
$94$	8.87633	0.915524
$95$	6.84899	0.702692
$96$	−18.5603	−1.89430
$97$	4.07807	0.414065	0.207033	−	0.978334i	$-0.433619\pi$
$97$	4.07807	0.414065	0.207033	+	0.978334i	$0.433619\pi$
$98$	−12.8094	−1.29394
$99$	−3.82074	−0.383998
$100$	−5.32284	−0.532284
$101$	−4.19158	−0.417078	−0.208539	−	0.978014i	$-0.566871\pi$
$101$	−4.19158	−0.417078	−0.208539	+	0.978014i	$0.566871\pi$
$102$	4.90847	0.486011
$103$	6.79926	0.669951	0.334975	−	0.942227i	$-0.391272\pi$
$103$	6.79926	0.669951	0.334975	+	0.942227i	$0.391272\pi$
$104$	−3.66319	−0.359206
$105$	−11.9927	−1.17037
$106$	14.4962	1.40799
$107$	5.08133	0.491230	0.245615	−	0.969367i	$-0.421010\pi$
$107$	5.08133	0.491230	0.245615	+	0.969367i	$0.421010\pi$
$108$	−3.28451	−0.316052
$109$	−12.7028	−1.21671	−0.608353	−	0.793666i	$-0.708170\pi$
$109$	−12.7028	−1.21671	−0.608353	+	0.793666i	$0.708170\pi$
$110$	−2.32194	−0.221388
$111$	−23.5342	−2.23376
$112$	17.5313	1.65656
$113$	−1.29250	−0.121588	−0.0607941	−	0.998150i	$-0.519363\pi$
$113$	−1.29250	−0.121588	−0.0607941	+	0.998150i	$0.519363\pi$
$114$	−27.2115	−2.54859
$115$	1.16421	0.108563
$116$	−3.76507	−0.349578
$117$	−15.9233	−1.47211
$118$	−17.4441	−1.60586
$119$	−3.71692	−0.340730
$120$	2.83602	0.258892
$121$	1.00000	0.0909091
$122$	27.6216	2.50074
$123$	−5.13448	−0.462961
$124$	−4.85867	−0.436322
$125$	10.4687	0.936349
$126$	26.6907	2.37780
$127$	−15.5239	−1.37752	−0.688762	−	0.724987i	$-0.741846\pi$
$127$	−15.5239	−1.37752	−0.688762	+	0.724987i	$0.741846\pi$
$128$	−6.83951	−0.604533
$129$	−2.61165	−0.229943
$130$	−9.67691	−0.848721
$131$	−0.248366	−0.0216999	−0.0108499	−	0.999941i	$-0.503454\pi$
$131$	−0.248366	−0.0216999	−0.0108499	+	0.999941i	$0.503454\pi$
$132$	4.00191	0.348321
$133$	20.6058	1.78675
$134$	20.5533	1.77553
$135$	2.64813	0.227914
$136$	0.878969	0.0753710
$137$	10.0029	0.854603	0.427302	−	0.904109i	$-0.359464\pi$
$137$	10.0029	0.854603	0.427302	+	0.904109i	$0.359464\pi$
$138$	−4.62550	−0.393749
$139$	−20.1663	−1.71048	−0.855241	−	0.518231i	$-0.826591\pi$
$139$	−20.1663	−1.71048	−0.855241	+	0.518231i	$0.826591\pi$
$140$	7.03646	0.594689
$141$	12.3344	1.03875
$142$	2.93266	0.246104
$143$	4.16760	0.348513
$144$	−18.0210	−1.50175
$145$	3.03558	0.252091
$146$	−0.711993	−0.0589249
$147$	−17.7997	−1.46810
$148$	13.8081	1.13502
$149$	22.7000	1.85966	0.929830	−	0.367989i	$-0.119954\pi$
$149$	22.7000	1.85966	0.929830	+	0.367989i	$0.119954\pi$
$150$	−17.0506	−1.39217
$151$	15.4469	1.25705	0.628524	−	0.777790i	$-0.283659\pi$
$151$	15.4469	1.25705	0.628524	+	0.777790i	$0.283659\pi$
$152$	−4.87282	−0.395238
$153$	3.82074	0.308888
$154$	−6.98576	−0.562929
$155$	3.91729	0.314645
$156$	16.6784	1.33534
$157$	13.6005	1.08544	0.542719	−	0.839914i	$-0.317395\pi$
$157$	13.6005	1.08544	0.542719	+	0.839914i	$0.317395\pi$
$158$	8.93077	0.710494
$159$	20.1437	1.59750
$160$	−8.77989	−0.694111
$161$	3.50264	0.276047
$162$	11.0214	0.865926
$163$	−6.91743	−0.541815	−0.270907	−	0.962605i	$-0.587324\pi$
$163$	−6.91743	−0.541815	−0.270907	+	0.962605i	$0.587324\pi$
$164$	3.01254	0.235240
$165$	−3.22653	−0.251185
$166$	−33.1653	−2.57413
$167$	4.95994	0.383811	0.191906	−	0.981413i	$-0.438533\pi$
$167$	4.95994	0.383811	0.191906	+	0.981413i	$0.438533\pi$
$168$	8.53242	0.658291
$169$	4.36892	0.336071
$170$	2.32194	0.178084
$171$	−21.1814	−1.61978
$172$	1.53233	0.116839
$173$	20.9972	1.59639	0.798193	−	0.602402i	$-0.205789\pi$
$173$	20.9972	1.59639	0.798193	+	0.602402i	$0.205789\pi$
$174$	−12.0606	−0.914310
$175$	12.9115	0.976016
$176$	4.71663	0.355529
$177$	−24.2401	−1.82200
$178$	2.37521	0.178030
$179$	−2.74331	−0.205045	−0.102522	−	0.994731i	$-0.532691\pi$
$179$	−2.74331	−0.205045	−0.102522	+	0.994731i	$0.532691\pi$
$180$	−7.23299	−0.539115
$181$	24.6365	1.83122	0.915610	−	0.402068i	$-0.131708\pi$
$181$	24.6365	1.83122	0.915610	+	0.402068i	$0.131708\pi$
$182$	−29.1139	−2.15806
$183$	38.3825	2.83732
$184$	−0.828297	−0.0610629
$185$	−11.1328	−0.818496
$186$	−15.5637	−1.14118
$187$	−1.00000	−0.0731272
$188$	−7.23693	−0.527807
$189$	7.96713	0.579524
$190$	−12.8723	−0.933857
$191$	12.6808	0.917549	0.458775	−	0.888553i	$-0.348289\pi$
$191$	12.6808	0.917549	0.458775	+	0.888553i	$0.348289\pi$
$192$	10.2467	0.739493
$193$	13.8459	0.996649	0.498325	−	0.866991i	$-0.333949\pi$
$193$	13.8459	0.996649	0.498325	+	0.866991i	$0.333949\pi$
$194$	−7.66452	−0.550280
$195$	−13.4469	−0.962952
$196$	10.4436	0.745969
$197$	13.8384	0.985942	0.492971	−	0.870046i	$-0.335911\pi$
$197$	13.8384	0.985942	0.492971	+	0.870046i	$0.335911\pi$
$198$	7.18088	0.510323
$199$	7.08749	0.502419	0.251210	−	0.967933i	$-0.419172\pi$
$199$	7.08749	0.502419	0.251210	+	0.967933i	$0.419172\pi$
$200$	−3.05328	−0.215899
$201$	28.5605	2.01450
$202$	7.87786	0.554284
$203$	9.13283	0.640999
$204$	−4.00191	−0.280189
$205$	−2.42885	−0.169638
$206$	−12.7789	−0.890345
$207$	−3.60047	−0.250250
$208$	19.6570	1.36297
$209$	5.54379	0.383472
$210$	22.5397	1.55539
$211$	23.4891	1.61706	0.808528	−	0.588458i	$-0.200265\pi$
$211$	23.4891	1.61706	0.808528	+	0.588458i	$0.200265\pi$
$212$	−11.8188	−0.811719
$213$	4.07518	0.279227
$214$	−9.55009	−0.652831
$215$	−1.23543	−0.0842559
$216$	−1.88405	−0.128193
$217$	11.7855	0.800055
$218$	23.8742	1.61697
$219$	−0.989374	−0.0668557
$220$	1.89309	0.127632
$221$	−4.16760	−0.280343
$222$	44.2312	2.96861
$223$	5.71676	0.382823	0.191411	−	0.981510i	$-0.438694\pi$
$223$	5.71676	0.382823	0.191411	+	0.981510i	$0.438694\pi$
$224$	−26.4151	−1.76493
$225$	−13.2721	−0.884806
$226$	2.42919	0.161587
$227$	−9.48892	−0.629802	−0.314901	−	0.949125i	$-0.601971\pi$
$227$	−9.48892	−0.629802	−0.314901	+	0.949125i	$0.601971\pi$
$228$	22.1857	1.46929
$229$	−1.20499	−0.0796278	−0.0398139	−	0.999207i	$-0.512677\pi$
$229$	−1.20499	−0.0796278	−0.0398139	+	0.999207i	$0.512677\pi$
$230$	−2.18808	−0.144278
$231$	−9.70731	−0.638694
$232$	−2.15971	−0.141792
$233$	5.85795	0.383767	0.191884	−	0.981418i	$-0.438540\pi$
$233$	5.85795	0.383767	0.191884	+	0.981418i	$0.438540\pi$
$234$	29.9270	1.95639
$235$	5.83476	0.380618
$236$	14.2223	0.925792
$237$	12.4101	0.806121
$238$	6.98576	0.452820
$239$	−13.3843	−0.865756	−0.432878	−	0.901452i	$-0.642502\pi$
$239$	−13.3843	−0.865756	−0.432878	+	0.901452i	$0.642502\pi$
$240$	−15.2183	−0.982339
$241$	−13.5993	−0.876010	−0.438005	−	0.898973i	$-0.644315\pi$
$241$	−13.5993	−0.876010	−0.438005	+	0.898973i	$0.644315\pi$
$242$	−1.87945	−0.120816
$243$	21.7457	1.39498
$244$	−22.5200	−1.44170
$245$	−8.42010	−0.537940
$246$	9.64999	0.615261
$247$	23.1043	1.47009
$248$	−2.78702	−0.176976
$249$	−46.0860	−2.92058
$250$	−19.6754	−1.24438
$251$	−24.7495	−1.56217	−0.781087	−	0.624422i	$-0.785335\pi$
$251$	−24.7495	−1.56217	−0.781087	+	0.624422i	$0.785335\pi$
$252$	−21.7611	−1.37082
$253$	0.942351	0.0592451
$254$	29.1764	1.83069
$255$	3.22653	0.202053
$256$	20.7014	1.29384
$257$	19.8535	1.23842	0.619212	−	0.785224i	$-0.287452\pi$
$257$	19.8535	1.23842	0.619212	+	0.785224i	$0.287452\pi$
$258$	4.90847	0.305588
$259$	−33.4939	−2.08121
$260$	7.88964	0.489295
$261$	−9.38792	−0.581097
$262$	0.466792	0.0288385
$263$	−27.7326	−1.71007	−0.855034	−	0.518572i	$-0.826464\pi$
$263$	−27.7326	−1.71007	−0.855034	+	0.518572i	$0.826464\pi$
$264$	2.29556	0.141282
$265$	9.52889	0.585355
$266$	−38.7276	−2.37454
$267$	3.30056	0.201991
$268$	−16.7572	−1.02361
$269$	7.62916	0.465158	0.232579	−	0.972577i	$-0.425284\pi$
$269$	7.62916	0.465158	0.232579	+	0.972577i	$0.425284\pi$
$270$	−4.97702	−0.302892
$271$	−17.1206	−1.04000	−0.520002	−	0.854165i	$-0.674069\pi$
$271$	−17.1206	−1.04000	−0.520002	+	0.854165i	$0.674069\pi$
$272$	−4.71663	−0.285988
$273$	−40.4562	−2.44852
$274$	−18.7999	−1.13574
$275$	3.47370	0.209472
$276$	3.77120	0.226999
$277$	1.54921	0.0930827	0.0465414	−	0.998916i	$-0.485180\pi$
$277$	1.54921	0.0930827	0.0465414	+	0.998916i	$0.485180\pi$
$278$	37.9015	2.27318
$279$	−12.1147	−0.725289
$280$	4.03624	0.241211
$281$	−6.98042	−0.416417	−0.208209	−	0.978084i	$-0.566763\pi$
$281$	−6.98042	−0.416417	−0.208209	+	0.978084i	$0.566763\pi$
$282$	−23.1819	−1.38046
$283$	9.21245	0.547623	0.273812	−	0.961783i	$-0.411715\pi$
$283$	9.21245	0.547623	0.273812	+	0.961783i	$0.411715\pi$
$284$	−2.39102	−0.141881
$285$	−17.8872	−1.05955
$286$	−7.83280	−0.463163
$287$	−7.30742	−0.431344
$288$	27.1529	1.60000
$289$	1.00000	0.0588235
$290$	−5.70522	−0.335022
$291$	−10.6505	−0.624343
$292$	0.580492	0.0339707
$293$	29.7459	1.73777	0.868887	−	0.495011i	$-0.164836\pi$
$293$	29.7459	1.73777	0.868887	+	0.495011i	$0.164836\pi$
$294$	33.4537	1.95106
$295$	−11.4667	−0.667617
$296$	7.92057	0.460374
$297$	2.14348	0.124377
$298$	−42.6636	−2.47143
$299$	3.92734	0.227124
$300$	13.9014	0.802599
$301$	−3.71692	−0.214240
$302$	−29.0316	−1.67058
$303$	10.9470	0.628886
$304$	26.1480	1.49969
$305$	18.1567	1.03965
$306$	−7.18088	−0.410503
$307$	−5.66235	−0.323167	−0.161584	−	0.986859i	$-0.551660\pi$
$307$	−5.66235	−0.323167	−0.161584	+	0.986859i	$0.551660\pi$
$308$	5.69553	0.324533
$309$	−17.7573	−1.01018
$310$	−7.36235	−0.418153
$311$	5.05158	0.286449	0.143224	−	0.989690i	$-0.454253\pi$
$311$	5.05158	0.286449	0.143224	+	0.989690i	$0.454253\pi$
$312$	9.56700	0.541624
$313$	20.3204	1.14858	0.574290	−	0.818652i	$-0.305278\pi$
$313$	20.3204	1.14858	0.574290	+	0.818652i	$0.305278\pi$
$314$	−25.5614	−1.44252
$315$	17.5449	0.988541
$316$	−7.28132	−0.409606
$317$	−0.490531	−0.0275510	−0.0137755	−	0.999905i	$-0.504385\pi$
$317$	−0.490531	−0.0275510	−0.0137755	+	0.999905i	$0.504385\pi$
$318$	−37.8590	−2.12303
$319$	2.45710	0.137571
$320$	4.84718	0.270965
$321$	−13.2707	−0.740696
$322$	−6.58303	−0.366858
$323$	−5.54379	−0.308465
$324$	−8.98585	−0.499214
$325$	14.4770	0.803040
$326$	13.0009	0.720056
$327$	33.1753	1.83460
$328$	1.72804	0.0954152
$329$	17.5544	0.967806
$330$	6.06409	0.333817
$331$	14.6330	0.804303	0.402151	−	0.915573i	$-0.368263\pi$
$331$	14.6330	0.804303	0.402151	+	0.915573i	$0.368263\pi$
$332$	27.0399	1.48401
$333$	34.4294	1.88672
$334$	−9.32194	−0.510074
$335$	13.5105	0.738155
$336$	−45.7858	−2.49782
$337$	−2.49971	−0.136168	−0.0680840	−	0.997680i	$-0.521689\pi$
$337$	−2.49971	−0.136168	−0.0680840	+	0.997680i	$0.521689\pi$
$338$	−8.21116	−0.446628
$339$	3.37556	0.183335
$340$	−1.89309	−0.102667
$341$	3.17078	0.171707
$342$	39.8093	2.15264
$343$	0.685784	0.0370289
$344$	0.878969	0.0473908
$345$	−3.04052	−0.163696
$346$	−39.4631	−2.12155
$347$	12.7918	0.686698	0.343349	−	0.939208i	$-0.388439\pi$
$347$	12.7918	0.686698	0.343349	+	0.939208i	$0.388439\pi$
$348$	9.83307	0.527108
$349$	10.2188	0.546999	0.273499	−	0.961872i	$-0.411819\pi$
$349$	10.2188	0.546999	0.273499	+	0.961872i	$0.411819\pi$
$350$	−24.2664	−1.29710
$351$	8.93316	0.476817
$352$	−7.10672	−0.378790
$353$	18.6457	0.992413	0.496206	−	0.868205i	$-0.334726\pi$
$353$	18.6457	0.992413	0.496206	+	0.868205i	$0.334726\pi$
$354$	45.5580	2.42138
$355$	1.92775	0.102314
$356$	−1.93652	−0.102636
$357$	9.70731	0.513765
$358$	5.15592	0.272499
$359$	−12.1451	−0.640996	−0.320498	−	0.947249i	$-0.603850\pi$
$359$	−12.1451	−0.640996	−0.320498	+	0.947249i	$0.603850\pi$
$360$	−4.14897	−0.218670
$361$	11.7336	0.617559
$362$	−46.3031	−2.43364
$363$	−2.61165	−0.137076
$364$	23.7367	1.24414
$365$	−0.468020	−0.0244973
$366$	−72.1379	−3.77071
$367$	−6.71172	−0.350349	−0.175175	−	0.984537i	$-0.556049\pi$
$367$	−6.71172	−0.350349	−0.175175	+	0.984537i	$0.556049\pi$
$368$	4.44472	0.231697
$369$	7.51152	0.391034
$370$	20.9234	1.08776
$371$	28.6686	1.48840
$372$	12.6892	0.657903
$373$	−19.3351	−1.00113	−0.500566	−	0.865699i	$-0.666875\pi$
$373$	−19.3351	−1.00113	−0.500566	+	0.865699i	$0.666875\pi$
$374$	1.87945	0.0971840
$375$	−27.3406	−1.41186
$376$	−4.15123	−0.214083
$377$	10.2402	0.527397
$378$	−14.9738	−0.770170
$379$	−5.49769	−0.282397	−0.141199	−	0.989981i	$-0.545096\pi$
$379$	−5.49769	−0.282397	−0.141199	+	0.989981i	$0.545096\pi$
$380$	10.4949	0.538376
$381$	40.5431	2.07709
$382$	−23.8329	−1.21940
$383$	−19.2787	−0.985098	−0.492549	−	0.870285i	$-0.663935\pi$
$383$	−19.2787	−0.985098	−0.492549	+	0.870285i	$0.663935\pi$
$384$	17.8624	0.911538
$385$	−4.59201	−0.234031
$386$	−26.0226	−1.32452
$387$	3.82074	0.194219
$388$	6.24893	0.317241
$389$	−12.6496	−0.641362	−0.320681	−	0.947187i	$-0.603912\pi$
$389$	−12.6496	−0.641362	−0.320681	+	0.947187i	$0.603912\pi$
$390$	25.2727	1.27973
$391$	−0.942351	−0.0476567
$392$	5.99061	0.302572
$393$	0.648647	0.0327199
$394$	−26.0085	−1.31029
$395$	5.87054	0.295379
$396$	−5.85461	−0.294205
$397$	11.4929	0.576812	0.288406	−	0.957508i	$-0.406875\pi$
$397$	11.4929	0.576812	0.288406	+	0.957508i	$0.406875\pi$
$398$	−13.3206	−0.667700
$399$	−53.8153	−2.69413
$400$	16.3842	0.819208
$401$	8.82987	0.440943	0.220471	−	0.975393i	$-0.429240\pi$
$401$	8.82987	0.440943	0.220471	+	0.975393i	$0.429240\pi$
$402$	−53.6780	−2.67722
$403$	13.2146	0.658264
$404$	−6.42287	−0.319550
$405$	7.24482	0.359998
$406$	−17.1647	−0.851869
$407$	−9.01121	−0.446669
$408$	−2.29556	−0.113647
$409$	−18.5658	−0.918020	−0.459010	−	0.888431i	$-0.651796\pi$
$409$	−18.5658	−0.918020	−0.459010	+	0.888431i	$0.651796\pi$
$410$	4.56490	0.225444
$411$	−26.1240	−1.28860
$412$	10.4187	0.513291
$413$	−34.4986	−1.69757
$414$	6.76690	0.332575
$415$	−21.8008	−1.07016
$416$	−29.6180	−1.45214
$417$	52.6673	2.57913
$418$	−10.4193	−0.509623
$419$	39.3049	1.92017	0.960085	−	0.279708i	$-0.0902376\pi$
$419$	39.3049	1.92017	0.960085	+	0.279708i	$0.0902376\pi$
$420$	−18.3768	−0.896696
$421$	27.7544	1.35267	0.676333	−	0.736596i	$-0.263568\pi$
$421$	27.7544	1.35267	0.676333	+	0.736596i	$0.263568\pi$
$422$	−44.1465	−2.14902
$423$	−18.0447	−0.877364
$424$	−6.77948	−0.329241
$425$	−3.47370	−0.168499
$426$	−7.65910	−0.371084
$427$	54.6262	2.64355
$428$	7.78625	0.376363
$429$	−10.8843	−0.525501
$430$	2.32194	0.111974
$431$	−25.4371	−1.22526	−0.612631	−	0.790369i	$-0.709889\pi$
$431$	−25.4371	−1.22526	−0.612631	+	0.790369i	$0.709889\pi$
$432$	10.1100	0.486417
$433$	19.4336	0.933921	0.466961	−	0.884278i	$-0.345349\pi$
$433$	19.4336	0.933921	0.466961	+	0.884278i	$0.345349\pi$
$434$	−22.1503	−1.06325
$435$	−7.92789	−0.380113
$436$	−19.4648	−0.932195
$437$	5.22419	0.249907
$438$	1.85948	0.0888493
$439$	−18.7491	−0.894847	−0.447423	−	0.894322i	$-0.647658\pi$
$439$	−18.7491	−0.894847	−0.447423	+	0.894322i	$0.647658\pi$
$440$	1.08591	0.0517687
$441$	26.0402	1.24001
$442$	7.83280	0.372568
$443$	−23.4840	−1.11576	−0.557880	−	0.829922i	$-0.688385\pi$
$443$	−23.4840	−1.11576	−0.557880	+	0.829922i	$0.688385\pi$
$444$	−36.0620	−1.71143
$445$	1.56132	0.0740136
$446$	−10.7444	−0.508760
$447$	−59.2847	−2.80407
$448$	14.5832	0.688991
$449$	23.5069	1.10936	0.554681	−	0.832063i	$-0.312840\pi$
$449$	23.5069	1.10936	0.554681	+	0.832063i	$0.312840\pi$
$450$	24.9442	1.17588
$451$	−1.96599	−0.0925748
$452$	−1.98053	−0.0931563
$453$	−40.3418	−1.89543
$454$	17.8339	0.836988
$455$	−19.1377	−0.897188
$456$	12.7261	0.595955
$457$	33.3341	1.55930	0.779651	−	0.626215i	$-0.215397\pi$
$457$	33.3341	1.55930	0.779651	+	0.626215i	$0.215397\pi$
$458$	2.26471	0.105823
$459$	−2.14348	−0.100049
$460$	1.78395	0.0831772
$461$	13.1322	0.611629	0.305815	−	0.952091i	$-0.401071\pi$
$461$	13.1322	0.611629	0.305815	+	0.952091i	$0.401071\pi$
$462$	18.2444	0.848806
$463$	−9.55467	−0.444043	−0.222022	−	0.975042i	$-0.571266\pi$
$463$	−9.55467	−0.444043	−0.222022	+	0.975042i	$0.571266\pi$
$464$	11.5892	0.538016
$465$	−10.2306	−0.474433
$466$	−11.0097	−0.510015
$467$	−31.1091	−1.43956	−0.719780	−	0.694202i	$-0.755757\pi$
$467$	−31.1091	−1.43956	−0.719780	+	0.694202i	$0.755757\pi$
$468$	−24.3997	−1.12788
$469$	40.6475	1.87693
$470$	−10.9661	−0.505830
$471$	−35.5198	−1.63667
$472$	8.15816	0.375510
$473$	−1.00000	−0.0459800
$474$	−23.3241	−1.07131
$475$	19.2575	0.883594
$476$	−5.69553	−0.261054
$477$	−29.4693	−1.34931
$478$	25.1550	1.15056
$479$	−7.96866	−0.364097	−0.182049	−	0.983290i	$-0.558273\pi$
$479$	−7.96866	−0.364097	−0.182049	+	0.983290i	$0.558273\pi$
$480$	22.9300	1.04661
$481$	−37.5551	−1.71237
$482$	25.5592	1.16419
$483$	−9.14769	−0.416234
$484$	1.53233	0.0696512
$485$	−5.03819	−0.228772
$486$	−40.8698	−1.85389
$487$	−0.595390	−0.0269797	−0.0134898	−	0.999909i	$-0.504294\pi$
$487$	−0.595390	−0.0269797	−0.0134898	+	0.999909i	$0.504294\pi$
$488$	−12.9179	−0.584765
$489$	18.0659	0.816970
$490$	15.8251	0.714907
$491$	−4.19963	−0.189527	−0.0947634	−	0.995500i	$-0.530209\pi$
$491$	−4.19963	−0.189527	−0.0947634	+	0.995500i	$0.530209\pi$
$492$	−7.86770	−0.354703
$493$	−2.45710	−0.110662
$494$	−43.4234	−1.95371
$495$	4.72027	0.212160
$496$	14.9554	0.671517
$497$	5.79982	0.260158
$498$	86.6162	3.88137
$499$	3.16092	0.141502	0.0707511	−	0.997494i	$-0.477460\pi$
$499$	3.16092	0.141502	0.0707511	+	0.997494i	$0.477460\pi$
$500$	16.0415	0.717396
$501$	−12.9536	−0.578726
$502$	46.5154	2.07608
$503$	−21.9706	−0.979621	−0.489810	−	0.871829i	$-0.662934\pi$
$503$	−21.9706	−0.979621	−0.489810	+	0.871829i	$0.662934\pi$
$504$	−12.4826	−0.556018
$505$	5.17842	0.230437
$506$	−1.77110	−0.0787350
$507$	−11.4101	−0.506740
$508$	−23.7877	−1.05541
$509$	10.2971	0.456413	0.228206	−	0.973613i	$-0.426714\pi$
$509$	10.2971	0.456413	0.228206	+	0.973613i	$0.426714\pi$
$510$	−6.06409	−0.268522
$511$	−1.40808	−0.0622899
$512$	−25.2282	−1.11494
$513$	11.8830	0.524647
$514$	−37.3136	−1.64583
$515$	−8.40004	−0.370150
$516$	−4.00191	−0.176174
$517$	4.72284	0.207710
$518$	62.9501	2.76587
$519$	−54.8374	−2.40709
$520$	4.52564	0.198462
$521$	3.34953	0.146746	0.0733729	−	0.997305i	$-0.476624\pi$
$521$	3.34953	0.146746	0.0733729	+	0.997305i	$0.476624\pi$
$522$	17.6441	0.772261
$523$	10.0729	0.440458	0.220229	−	0.975448i	$-0.429320\pi$
$523$	10.0729	0.440458	0.220229	+	0.975448i	$0.429320\pi$
$524$	−0.380578	−0.0166256
$525$	−33.7203	−1.47167
$526$	52.1221	2.27263
$527$	−3.17078	−0.138121
$528$	−12.3182	−0.536081
$529$	−22.1120	−0.961390
$530$	−17.9091	−0.777920
$531$	35.4622	1.53893
$532$	31.5748	1.36894
$533$	−8.19346	−0.354898
$534$	−6.20323	−0.268440
$535$	−6.27765	−0.271406
$536$	−9.61222	−0.415185
$537$	7.16458	0.309175
$538$	−14.3386	−0.618181
$539$	−6.81550	−0.293564
$540$	4.05779	0.174620
$541$	−42.1401	−1.81175	−0.905873	−	0.423549i	$-0.860784\pi$
$541$	−42.1401	−1.81175	−0.905873	+	0.423549i	$0.860784\pi$
$542$	32.1774	1.38214
$543$	−64.3421	−2.76118
$544$	7.10672	0.304698
$545$	15.6935	0.672234
$546$	76.0354	3.25401
$547$	38.6103	1.65086	0.825429	−	0.564506i	$-0.190933\pi$
$547$	38.6103	1.65086	0.825429	+	0.564506i	$0.190933\pi$
$548$	15.3277	0.654765
$549$	−56.1519	−2.39651
$550$	−6.52864	−0.278382
$551$	13.6216	0.580301
$552$	2.16322	0.0920730
$553$	17.6621	0.751068
$554$	−2.91165	−0.123704
$555$	29.0749	1.23416
$556$	−30.9013	−1.31051
$557$	22.9269	0.971444	0.485722	−	0.874113i	$-0.338557\pi$
$557$	22.9269	0.971444	0.485722	+	0.874113i	$0.338557\pi$
$558$	22.7690	0.963888
$559$	−4.16760	−0.176271
$560$	−21.6588	−0.915252
$561$	2.61165	0.110264
$562$	13.1193	0.553406
$563$	−22.0885	−0.930919	−0.465460	−	0.885069i	$-0.654111\pi$
$563$	−22.0885	−0.930919	−0.465460	+	0.885069i	$0.654111\pi$
$564$	18.9003	0.795848
$565$	1.59680	0.0671778
$566$	−17.3143	−0.727775
$567$	21.7967	0.915376
$568$	−1.37153	−0.0575481
$569$	−18.1352	−0.760265	−0.380133	−	0.924932i	$-0.624122\pi$
$569$	−18.1352	−0.760265	−0.380133	+	0.924932i	$0.624122\pi$
$570$	33.6181	1.40811
$571$	33.4079	1.39808	0.699039	−	0.715083i	$-0.253611\pi$
$571$	33.4079	1.39808	0.699039	+	0.715083i	$0.253611\pi$
$572$	6.38613	0.267017
$573$	−33.1178	−1.38352
$574$	13.7339	0.573243
$575$	3.27344	0.136512
$576$	−14.9905	−0.624604
$577$	35.7416	1.48794	0.743972	−	0.668211i	$-0.232940\pi$
$577$	35.7416	1.48794	0.743972	+	0.668211i	$0.232940\pi$
$578$	−1.87945	−0.0781747
$579$	−36.1607	−1.50279
$580$	4.65150	0.193143
$581$	−65.5898	−2.72113
$582$	20.0171	0.829734
$583$	7.71299	0.319439
$584$	0.332980	0.0137788
$585$	19.6722	0.813346
$586$	−55.9059	−2.30945
$587$	−46.7204	−1.92836	−0.964180	−	0.265250i	$-0.914545\pi$
$587$	−46.7204	−1.92836	−0.964180	+	0.265250i	$0.914545\pi$
$588$	−27.2750	−1.12480
$589$	17.5782	0.724295
$590$	21.5511	0.887243
$591$	−36.1410	−1.48664
$592$	−42.5025	−1.74684
$593$	−8.83285	−0.362722	−0.181361	−	0.983417i	$-0.558050\pi$
$593$	−8.83285	−0.362722	−0.181361	+	0.983417i	$0.558050\pi$
$594$	−4.02855	−0.165294
$595$	4.59201	0.188254
$596$	34.7839	1.42480
$597$	−18.5101	−0.757567
$598$	−7.38124	−0.301841
$599$	0.447343	0.0182779	0.00913897	−	0.999958i	$-0.497091\pi$
$599$	0.447343	0.0182779	0.00913897	+	0.999958i	$0.497091\pi$
$600$	7.97410	0.325541
$601$	32.5502	1.32775	0.663874	−	0.747844i	$-0.268911\pi$
$601$	32.5502	1.32775	0.663874	+	0.747844i	$0.268911\pi$
$602$	6.98576	0.284718
$603$	−41.7828	−1.70153
$604$	23.6696	0.963103
$605$	−1.23543	−0.0502276
$606$	−20.5742	−0.835771
$607$	45.5242	1.84777	0.923886	−	0.382669i	$-0.124995\pi$
$607$	45.5242	1.84777	0.923886	+	0.382669i	$0.124995\pi$
$608$	−39.3982	−1.59781
$609$	−23.8518	−0.966523
$610$	−34.1246	−1.38167
$611$	19.6829	0.796286
$612$	5.85461	0.236659
$613$	−31.9346	−1.28983	−0.644913	−	0.764256i	$-0.723106\pi$
$613$	−31.9346	−1.28983	−0.644913	+	0.764256i	$0.723106\pi$
$614$	10.6421	0.429480
$615$	6.34332	0.255787
$616$	3.26706	0.131634
$617$	−0.866433	−0.0348813	−0.0174406	−	0.999848i	$-0.505552\pi$
$617$	−0.866433	−0.0348813	−0.0174406	+	0.999848i	$0.505552\pi$
$618$	33.3739	1.34250
$619$	6.64791	0.267202	0.133601	−	0.991035i	$-0.457346\pi$
$619$	6.64791	0.267202	0.133601	+	0.991035i	$0.457346\pi$
$620$	6.00257	0.241069
$621$	2.01991	0.0810561
$622$	−9.49417	−0.380682
$623$	4.69737	0.188196
$624$	−51.3374	−2.05514
$625$	4.43511	0.177404
$626$	−38.1912	−1.52643
$627$	−14.4785	−0.578214
$628$	20.8404	0.831623
$629$	9.01121	0.359300
$630$	−32.9747	−1.31374
$631$	34.9282	1.39047	0.695235	−	0.718782i	$-0.255300\pi$
$631$	34.9282	1.39047	0.695235	+	0.718782i	$0.255300\pi$
$632$	−4.17669	−0.166140
$633$	−61.3454	−2.43826
$634$	0.921928	0.0366144
$635$	19.1788	0.761087
$636$	30.8666	1.22394
$637$	−28.4043	−1.12542
$638$	−4.61799	−0.182828
$639$	−5.96182	−0.235846
$640$	8.44976	0.334006
$641$	−43.0709	−1.70120	−0.850599	−	0.525815i	$-0.823760\pi$
$641$	−43.0709	−1.70120	−0.850599	+	0.525815i	$0.823760\pi$
$642$	24.9415	0.984364
$643$	−3.85352	−0.151968	−0.0759840	−	0.997109i	$-0.524210\pi$
$643$	−3.85352	−0.151968	−0.0759840	+	0.997109i	$0.524210\pi$
$644$	5.36719	0.211497
$645$	3.22653	0.127044
$646$	10.4193	0.409941
$647$	−24.1068	−0.947735	−0.473867	−	0.880596i	$-0.657142\pi$
$647$	−24.1068	−0.947735	−0.473867	+	0.880596i	$0.657142\pi$
$648$	−5.15444	−0.202485
$649$	−9.28151	−0.364331
$650$	−27.2088	−1.06722
$651$	−30.7798	−1.20635
$652$	−10.5998	−0.415118
$653$	22.4035	0.876717	0.438359	−	0.898800i	$-0.355560\pi$
$653$	22.4035	0.876717	0.438359	+	0.898800i	$0.355560\pi$
$654$	−62.3512	−2.43813
$655$	0.306840	0.0119892
$656$	−9.27284	−0.362044
$657$	1.44741	0.0564689
$658$	−32.9926	−1.28619
$659$	−30.7181	−1.19661	−0.598303	−	0.801270i	$-0.704158\pi$
$659$	−30.7181	−1.19661	−0.598303	+	0.801270i	$0.704158\pi$
$660$	−4.94409	−0.192449
$661$	−11.8599	−0.461295	−0.230648	−	0.973037i	$-0.574084\pi$
$661$	−11.8599	−0.461295	−0.230648	+	0.973037i	$0.574084\pi$
$662$	−27.5020	−1.06890
$663$	10.8843	0.422713
$664$	15.5105	0.601926
$665$	−25.4572	−0.987186
$666$	−64.7083	−2.50740
$667$	2.31545	0.0896544
$668$	7.60024	0.294062
$669$	−14.9302	−0.577235
$670$	−25.3922	−0.980987
$671$	14.6966	0.567357
$672$	68.9872	2.66124
$673$	1.93666	0.0746526	0.0373263	−	0.999303i	$-0.488116\pi$
$673$	1.93666	0.0746526	0.0373263	+	0.999303i	$0.488116\pi$
$674$	4.69808	0.180963
$675$	7.44580	0.286589
$676$	6.69461	0.257485
$677$	18.5223	0.711871	0.355935	−	0.934511i	$-0.384162\pi$
$677$	18.5223	0.711871	0.355935	+	0.934511i	$0.384162\pi$
$678$	−6.34419	−0.243647
$679$	−15.1579	−0.581705
$680$	−1.08591	−0.0416427
$681$	24.7818	0.949640
$682$	−5.95932	−0.228194
$683$	−30.3340	−1.16070	−0.580349	−	0.814368i	$-0.697084\pi$
$683$	−30.3340	−1.16070	−0.580349	+	0.814368i	$0.697084\pi$
$684$	−32.4567	−1.24101
$685$	−12.3579	−0.472171
$686$	−1.28890	−0.0492103
$687$	3.14701	0.120066
$688$	−4.71663	−0.179820
$689$	32.1447	1.22461
$690$	5.71450	0.217547
$691$	−36.4084	−1.38504	−0.692521	−	0.721398i	$-0.743500\pi$
$691$	−36.4084	−1.38504	−0.692521	+	0.721398i	$0.743500\pi$
$692$	32.1745	1.22309
$693$	14.2014	0.539465
$694$	−24.0415	−0.912602
$695$	24.9141	0.945047
$696$	5.64042	0.213800
$697$	1.96599	0.0744671
$698$	−19.2057	−0.726945
$699$	−15.2989	−0.578659
$700$	19.7846	0.747787
$701$	−32.4436	−1.22538	−0.612690	−	0.790324i	$-0.709912\pi$
$701$	−32.4436	−1.22538	−0.612690	+	0.790324i	$0.709912\pi$
$702$	−16.7894	−0.633676
$703$	−49.9562	−1.88414
$704$	3.92346	0.147871
$705$	−15.2384	−0.573910
$706$	−35.0437	−1.31889
$707$	15.5798	0.585937
$708$	−37.1437	−1.39595
$709$	4.86710	0.182788	0.0913940	−	0.995815i	$-0.470868\pi$
$709$	4.86710	0.182788	0.0913940	+	0.995815i	$0.470868\pi$
$710$	−3.62311	−0.135973
$711$	−18.1554	−0.680880
$712$	−1.11082	−0.0416299
$713$	2.98799	0.111901
$714$	−18.2444	−0.682779
$715$	−5.14880	−0.192554
$716$	−4.20365	−0.157098
$717$	34.9551	1.30542
$718$	22.8262	0.851864
$719$	−29.3259	−1.09367	−0.546837	−	0.837239i	$-0.684168\pi$
$719$	−29.3259	−1.09367	−0.546837	+	0.837239i	$0.684168\pi$
$720$	22.2638	0.829721
$721$	−25.2723	−0.941190
$722$	−22.0527	−0.820718
$723$	35.5167	1.32088
$724$	37.7512	1.40301
$725$	8.53522	0.316990
$726$	4.90847	0.182170
$727$	−22.2199	−0.824090	−0.412045	−	0.911163i	$-0.635185\pi$
$727$	−22.2199	−0.824090	−0.412045	+	0.911163i	$0.635185\pi$
$728$	13.6158	0.504635
$729$	−39.1996	−1.45184
$730$	0.879620	0.0325562
$731$	1.00000	0.0369863
$732$	58.8145	2.17385
$733$	−11.1130	−0.410469	−0.205235	−	0.978713i	$-0.565796\pi$
$733$	−11.1130	−0.410469	−0.205235	+	0.978713i	$0.565796\pi$
$734$	12.6143	0.465604
$735$	21.9904	0.811128
$736$	−6.69702	−0.246856
$737$	10.9358	0.402825
$738$	−14.1175	−0.519673
$739$	17.2563	0.634782	0.317391	−	0.948295i	$-0.397193\pi$
$739$	17.2563	0.634782	0.317391	+	0.948295i	$0.397193\pi$
$740$	−17.0590	−0.627102
$741$	−60.3405	−2.21666
$742$	−53.8811	−1.97804
$743$	38.8946	1.42690	0.713452	−	0.700704i	$-0.247131\pi$
$743$	38.8946	1.42690	0.713452	+	0.700704i	$0.247131\pi$
$744$	7.27873	0.266851
$745$	−28.0444	−1.02747
$746$	36.3392	1.33047
$747$	67.4218	2.46683
$748$	−1.53233	−0.0560274
$749$	−18.8869	−0.690112
$750$	51.3853	1.87633
$751$	−3.30930	−0.120758	−0.0603790	−	0.998176i	$-0.519231\pi$
$751$	−3.30930	−0.120758	−0.0603790	+	0.998176i	$0.519231\pi$
$752$	22.2759	0.812318
$753$	64.6371	2.35551
$754$	−19.2459	−0.700896
$755$	−19.0836	−0.694523
$756$	12.2082	0.444010
$757$	−50.4906	−1.83511	−0.917557	−	0.397605i	$-0.869841\pi$
$757$	−50.4906	−1.83511	−0.917557	+	0.397605i	$0.869841\pi$
$758$	10.3326	0.375298
$759$	−2.46109	−0.0893320
$760$	6.02005	0.218370
$761$	−30.6135	−1.10974	−0.554870	−	0.831937i	$-0.687232\pi$
$761$	−30.6135	−1.10974	−0.554870	+	0.831937i	$0.687232\pi$
$762$	−76.1986	−2.76039
$763$	47.2153	1.70931
$764$	19.4311	0.702992
$765$	−4.72027	−0.170662
$766$	36.2334	1.30917
$767$	−38.6816	−1.39671
$768$	−54.0649	−1.95090
$769$	−3.34187	−0.120511	−0.0602554	−	0.998183i	$-0.519192\pi$
$769$	−3.34187	−0.120511	−0.0602554	+	0.998183i	$0.519192\pi$
$770$	8.63045	0.311020
$771$	−51.8504	−1.86735
$772$	21.2164	0.763596
$773$	−44.1888	−1.58936	−0.794681	−	0.607028i	$-0.792362\pi$
$773$	−44.1888	−1.58936	−0.794681	+	0.607028i	$0.792362\pi$
$774$	−7.18088	−0.258111
$775$	11.0143	0.395647
$776$	3.58449	0.128676
$777$	87.4746	3.13813
$778$	23.7743	0.852352
$779$	−10.8990	−0.390498
$780$	−20.6050	−0.737778
$781$	1.56038	0.0558349
$782$	1.77110	0.0633344
$783$	5.26673	0.188218
$784$	−32.1462	−1.14808
$785$	−16.8025	−0.599708
$786$	−1.21910	−0.0434838
$787$	25.6510	0.914360	0.457180	−	0.889374i	$-0.348860\pi$
$787$	25.6510	0.914360	0.457180	+	0.889374i	$0.348860\pi$
$788$	21.2049	0.755392
$789$	72.4281	2.57851
$790$	−11.0334	−0.392550
$791$	4.80412	0.170815
$792$	−3.35831	−0.119332
$793$	61.2497	2.17504
$794$	−21.6003	−0.766567
$795$	−24.8862	−0.882621
$796$	10.8604	0.384935
$797$	9.65198	0.341891	0.170945	−	0.985280i	$-0.445318\pi$
$797$	9.65198	0.341891	0.170945	+	0.985280i	$0.445318\pi$
$798$	101.143	3.58043
$799$	−4.72284	−0.167082
$800$	−24.6866	−0.872804
$801$	−4.82857	−0.170609
$802$	−16.5953	−0.586000
$803$	−0.378831	−0.0133686
$804$	43.7640	1.54344
$805$	−4.32728	−0.152517
$806$	−24.8361	−0.874814
$807$	−19.9247	−0.701384
$808$	−3.68427	−0.129612
$809$	4.72435	0.166099	0.0830496	−	0.996545i	$-0.473534\pi$
$809$	4.72435	0.166099	0.0830496	+	0.996545i	$0.473534\pi$
$810$	−13.6163	−0.478427
$811$	−16.4749	−0.578511	−0.289256	−	0.957252i	$-0.593408\pi$
$811$	−16.4749	−0.578511	−0.289256	+	0.957252i	$0.593408\pi$
$812$	13.9945	0.491110
$813$	44.7132	1.56816
$814$	16.9361	0.593610
$815$	8.54603	0.299354
$816$	12.3182	0.431223
$817$	−5.54379	−0.193953
$818$	34.8935	1.22002
$819$	59.1857	2.06811
$820$	−3.72179	−0.129971
$821$	−13.7131	−0.478592	−0.239296	−	0.970947i	$-0.576917\pi$
$821$	−13.7131	−0.478592	−0.239296	+	0.970947i	$0.576917\pi$
$822$	49.0988	1.71252
$823$	34.3716	1.19812	0.599060	−	0.800704i	$-0.295541\pi$
$823$	34.3716	1.19812	0.599060	+	0.800704i	$0.295541\pi$
$824$	5.97634	0.208196
$825$	−9.07211	−0.315850
$826$	64.8384	2.25602
$827$	32.4075	1.12692	0.563460	−	0.826144i	$-0.309470\pi$
$827$	32.4075	1.12692	0.563460	+	0.826144i	$0.309470\pi$
$828$	−5.51710	−0.191732
$829$	−23.5988	−0.819620	−0.409810	−	0.912171i	$-0.634405\pi$
$829$	−23.5988	−0.819620	−0.409810	+	0.912171i	$0.634405\pi$
$830$	40.9735	1.42221
$831$	−4.04599	−0.140354
$832$	16.3514	0.566884
$833$	6.81550	0.236143
$834$	−98.9856	−3.42759
$835$	−6.12768	−0.212057
$836$	8.49490	0.293802
$837$	6.79650	0.234921
$838$	−73.8715	−2.55185
$839$	−42.2398	−1.45828	−0.729141	−	0.684364i	$-0.760080\pi$
$839$	−42.2398	−1.45828	−0.729141	+	0.684364i	$0.760080\pi$
$840$	−10.5413	−0.363708
$841$	−22.9627	−0.791816
$842$	−52.1630	−1.79765
$843$	18.2305	0.627890
$844$	35.9929	1.23893
$845$	−5.39751	−0.185680
$846$	33.9141	1.16599
$847$	−3.71692	−0.127715
$848$	36.3793	1.24927
$849$	−24.0597	−0.825728
$850$	6.52864	0.223931
$851$	−8.49171	−0.291092
$852$	6.24451	0.213933
$853$	35.3128	1.20909	0.604544	−	0.796572i	$-0.293355\pi$
$853$	35.3128	1.20909	0.604544	+	0.796572i	$0.293355\pi$
$854$	−102.667	−3.51320
$855$	26.1682	0.894933
$856$	4.46633	0.152656
$857$	−7.95792	−0.271837	−0.135919	−	0.990720i	$-0.543399\pi$
$857$	−7.95792	−0.271837	−0.135919	+	0.990720i	$0.543399\pi$
$858$	20.4565	0.698375
$859$	7.50464	0.256055	0.128028	−	0.991771i	$-0.459135\pi$
$859$	7.50464	0.256055	0.128028	+	0.991771i	$0.459135\pi$
$860$	−1.89309	−0.0645538
$861$	19.0845	0.650397
$862$	47.8077	1.62834
$863$	−36.8899	−1.25575	−0.627874	−	0.778315i	$-0.716075\pi$
$863$	−36.8899	−1.25575	−0.627874	+	0.778315i	$0.716075\pi$
$864$	−15.2331	−0.518241
$865$	−25.9406	−0.882008
$866$	−36.5245	−1.24115
$867$	−2.61165	−0.0886964
$868$	18.0593	0.612972
$869$	4.75181	0.161194
$870$	14.9001	0.505159
$871$	45.5761	1.54429
$872$	−11.1654	−0.378107
$873$	15.5812	0.527344
$874$	−9.81860	−0.332119
$875$	−38.9113	−1.31544
$876$	−1.51604	−0.0512224
$877$	−17.2652	−0.583006	−0.291503	−	0.956570i	$-0.594155\pi$
$877$	−17.2652	−0.583006	−0.291503	+	0.956570i	$0.594155\pi$
$878$	35.2380	1.18922
$879$	−77.6860	−2.62028
$880$	−5.82709	−0.196431
$881$	−15.6675	−0.527852	−0.263926	−	0.964543i	$-0.585017\pi$
$881$	−15.6675	−0.527852	−0.263926	+	0.964543i	$0.585017\pi$
$882$	−48.9412	−1.64794
$883$	34.5296	1.16201	0.581006	−	0.813899i	$-0.302659\pi$
$883$	34.5296	1.16201	0.581006	+	0.813899i	$0.302659\pi$
$884$	−6.38613	−0.214789
$885$	29.9470	1.00666
$886$	44.1370	1.48281
$887$	−18.1439	−0.609213	−0.304607	−	0.952478i	$-0.598525\pi$
$887$	−18.1439	−0.609213	−0.304607	+	0.952478i	$0.598525\pi$
$888$	−20.6858	−0.694170
$889$	57.7012	1.93523
$890$	−2.93442	−0.0983619
$891$	5.86419	0.196458
$892$	8.75995	0.293305
$893$	26.1824	0.876161
$894$	111.422	3.72652
$895$	3.38918	0.113288
$896$	25.4219	0.849286
$897$	−10.2569	−0.342467
$898$	−44.1801	−1.47431
$899$	7.79092	0.259842
$900$	−20.3372	−0.677906
$901$	−7.71299	−0.256957
$902$	3.69497	0.123029
$903$	9.70731	0.323039
$904$	−1.13607	−0.0377850
$905$	−30.4368	−1.01175
$906$	75.8204	2.51896
$907$	10.5620	0.350704	0.175352	−	0.984506i	$-0.443894\pi$
$907$	10.5620	0.350704	0.175352	+	0.984506i	$0.443894\pi$
$908$	−14.5401	−0.482531
$909$	−16.0149	−0.531181
$910$	35.9683	1.19234
$911$	−14.9802	−0.496317	−0.248159	−	0.968719i	$-0.579825\pi$
$911$	−14.9802	−0.496317	−0.248159	+	0.968719i	$0.579825\pi$
$912$	−68.2895	−2.26129
$913$	−17.6463	−0.584007
$914$	−62.6496	−2.07227
$915$	−47.4191	−1.56763
$916$	−1.84643	−0.0610078
$917$	0.923158	0.0304854
$918$	4.02855	0.132962
$919$	−5.55199	−0.183143	−0.0915717	−	0.995798i	$-0.529189\pi$
$919$	−5.55199	−0.183143	−0.0915717	+	0.995798i	$0.529189\pi$
$920$	1.02331	0.0337374
$921$	14.7881	0.487284
$922$	−24.6814	−0.812838
$923$	6.50306	0.214051
$924$	−14.8748	−0.489344
$925$	−31.3022	−1.02921
$926$	17.9575	0.590120
$927$	25.9782	0.853235
$928$	−17.4619	−0.573215
$929$	3.40408	0.111684	0.0558420	−	0.998440i	$-0.482216\pi$
$929$	3.40408	0.111684	0.0558420	+	0.998440i	$0.482216\pi$
$930$	19.2279	0.630508
$931$	−37.7837	−1.23831
$932$	8.97629	0.294028
$933$	−13.1930	−0.431918
$934$	58.4680	1.91313
$935$	1.23543	0.0404030
$936$	−13.9961	−0.457477
$937$	28.8206	0.941527	0.470763	−	0.882260i	$-0.343979\pi$
$937$	28.8206	0.941527	0.470763	+	0.882260i	$0.343979\pi$
$938$	−76.3948	−2.49438
$939$	−53.0700	−1.73187
$940$	8.94075	0.291615
$941$	9.36860	0.305408	0.152704	−	0.988272i	$-0.451202\pi$
$941$	9.36860	0.305408	0.152704	+	0.988272i	$0.451202\pi$
$942$	66.7576	2.17508
$943$	−1.85265	−0.0603306
$944$	−43.7774	−1.42483
$945$	−9.84287	−0.320189
$946$	1.87945	0.0611061
$947$	−17.8830	−0.581120	−0.290560	−	0.956857i	$-0.593842\pi$
$947$	−17.8830	−0.581120	−0.290560	+	0.956857i	$0.593842\pi$
$948$	19.0163	0.617620
$949$	−1.57882	−0.0512505
$950$	−36.1934	−1.17427
$951$	1.28110	0.0415424
$952$	−3.26706	−0.105886
$953$	−20.7617	−0.672538	−0.336269	−	0.941766i	$-0.609165\pi$
$953$	−20.7617	−0.672538	−0.336269	+	0.941766i	$0.609165\pi$
$954$	55.3860	1.79319
$955$	−15.6663	−0.506949
$956$	−20.5091	−0.663310
$957$	−6.41709	−0.207435
$958$	14.9767	0.483875
$959$	−37.1799	−1.20060
$960$	−12.6591	−0.408572
$961$	−20.9461	−0.675682
$962$	70.5829	2.27569
$963$	19.4144	0.625620
$964$	−20.8386	−0.671166
$965$	−17.1057	−0.550652
$966$	17.1926	0.553163
$967$	−7.39961	−0.237956	−0.118978	−	0.992897i	$-0.537962\pi$
$967$	−7.39961	−0.237956	−0.118978	+	0.992897i	$0.537962\pi$
$968$	0.878969	0.0282511
$969$	14.4785	0.465115
$970$	9.46901	0.304032
$971$	60.0242	1.92627	0.963134	−	0.269021i	$-0.0867000\pi$
$971$	60.0242	1.92627	0.963134	+	0.269021i	$0.0867000\pi$
$972$	33.3214	1.06879
$973$	74.9565	2.40299
$974$	1.11900	0.0358552
$975$	−37.8089	−1.21086
$976$	69.3185	2.21883
$977$	−18.6170	−0.595610	−0.297805	−	0.954627i	$-0.596255\pi$
$977$	−18.6170	−0.595610	−0.297805	+	0.954627i	$0.596255\pi$
$978$	−33.9540	−1.08573
$979$	1.26378	0.0403906
$980$	−12.9023	−0.412150
$981$	−48.5340	−1.54957
$982$	7.89300	0.251876
$983$	51.8626	1.65416	0.827080	−	0.562084i	$-0.190000\pi$
$983$	51.8626	1.65416	0.827080	+	0.562084i	$0.190000\pi$
$984$	−4.51305	−0.143871
$985$	−17.0964	−0.544736
$986$	4.61799	0.147067
$987$	−45.8460	−1.45930
$988$	35.4034	1.12633
$989$	−0.942351	−0.0299650
$990$	−8.87150	−0.281955
$991$	−51.4948	−1.63579	−0.817894	−	0.575370i	$-0.804858\pi$
$991$	−51.4948	−1.63579	−0.817894	+	0.575370i	$0.804858\pi$
$992$	−22.5339	−0.715451
$993$	−38.2164	−1.21276
$994$	−10.9005	−0.345742
$995$	−8.75613	−0.277588
$996$	−70.6188	−2.23764
$997$	28.9337	0.916340	0.458170	−	0.888865i	$-0.348505\pi$
$997$	28.9337	0.916340	0.458170	+	0.888865i	$0.348505\pi$
$998$	−5.94079	−0.188052
$999$	−19.3153	−0.611110

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8041.2.a.g.1.15	✓	69

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8041.2.a.g.1.15	✓	69	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:	\( N \)	\(=\)	\( 8041 = 11 \cdot 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8041.a (trivial)

\(f(q)\)	\(=\)	\(q-1.87945 q^{2} -2.61165 q^{3} +1.53233 q^{4} -1.23543 q^{5} +4.90847 q^{6} -3.71692 q^{7} +0.878969 q^{8} +3.82074 q^{9} +O(q^{10})\)	\(q-1.87945 q^{2} -2.61165 q^{3} +1.53233 q^{4} -1.23543 q^{5} +4.90847 q^{6} -3.71692 q^{7} +0.878969 q^{8} +3.82074 q^{9} +2.32194 q^{10} -1.00000 q^{11} -4.00191 q^{12} -4.16760 q^{13} +6.98576 q^{14} +3.22653 q^{15} -4.71663 q^{16} +1.00000 q^{17} -7.18088 q^{18} -5.54379 q^{19} -1.89309 q^{20} +9.70731 q^{21} +1.87945 q^{22} -0.942351 q^{23} -2.29556 q^{24} -3.47370 q^{25} +7.83280 q^{26} -2.14348 q^{27} -5.69553 q^{28} -2.45710 q^{29} -6.06409 q^{30} -3.17078 q^{31} +7.10672 q^{32} +2.61165 q^{33} -1.87945 q^{34} +4.59201 q^{35} +5.85461 q^{36} +9.01121 q^{37} +10.4193 q^{38} +10.8843 q^{39} -1.08591 q^{40} +1.96599 q^{41} -18.2444 q^{42} +1.00000 q^{43} -1.53233 q^{44} -4.72027 q^{45} +1.77110 q^{46} -4.72284 q^{47} +12.3182 q^{48} +6.81550 q^{49} +6.52864 q^{50} -2.61165 q^{51} -6.38613 q^{52} -7.71299 q^{53} +4.02855 q^{54} +1.23543 q^{55} -3.26706 q^{56} +14.4785 q^{57} +4.61799 q^{58} +9.28151 q^{59} +4.94409 q^{60} -14.6966 q^{61} +5.95932 q^{62} -14.2014 q^{63} -3.92346 q^{64} +5.14880 q^{65} -4.90847 q^{66} -10.9358 q^{67} +1.53233 q^{68} +2.46109 q^{69} -8.63045 q^{70} -1.56038 q^{71} +3.35831 q^{72} +0.378831 q^{73} -16.9361 q^{74} +9.07211 q^{75} -8.49490 q^{76} +3.71692 q^{77} -20.4565 q^{78} -4.75181 q^{79} +5.82709 q^{80} -5.86419 q^{81} -3.69497 q^{82} +17.6463 q^{83} +14.8748 q^{84} -1.23543 q^{85} -1.87945 q^{86} +6.41709 q^{87} -0.878969 q^{88} -1.26378 q^{89} +8.87150 q^{90} +15.4907 q^{91} -1.44399 q^{92} +8.28098 q^{93} +8.87633 q^{94} +6.84899 q^{95} -18.5603 q^{96} +4.07807 q^{97} -12.8094 q^{98} -3.82074 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 69 q - 11 q^{2} - 3 q^{3} + 65 q^{4} - 6 q^{5} - 10 q^{6} - 11 q^{7} - 33 q^{8} + 56 q^{9}+O(q^{10}) \) 69 * q - 11 * q^2 - 3 * q^3 + 65 * q^4 - 6 * q^5 - 10 * q^6 - 11 * q^7 - 33 * q^8 + 56 * q^9	\( 69 q - 11 q^{2} - 3 q^{3} + 65 q^{4} - 6 q^{5} - 10 q^{6} - 11 q^{7} - 33 q^{8} + 56 q^{9} - q^{10} - 69 q^{11} - 3 q^{12} - 28 q^{13} - 15 q^{14} - 45 q^{15} + 53 q^{16} + 69 q^{17} - 17 q^{18} - 32 q^{19} - 21 q^{20} - 38 q^{21} + 11 q^{22} - 41 q^{23} - 11 q^{24} + 67 q^{25} - 6 q^{26} - 3 q^{27} - 21 q^{28} - 22 q^{29} - 22 q^{30} - 27 q^{31} - 87 q^{32} + 3 q^{33} - 11 q^{34} - 44 q^{35} + 59 q^{36} + 24 q^{37} - 22 q^{38} - 59 q^{39} + q^{40} - 43 q^{41} - 15 q^{42} + 69 q^{43} - 65 q^{44} - 12 q^{45} - 21 q^{46} - 99 q^{47} + 2 q^{48} + 64 q^{49} - 78 q^{50} - 3 q^{51} - 57 q^{52} - 50 q^{53} + 20 q^{54} + 6 q^{55} - 59 q^{56} - 15 q^{57} + 22 q^{58} - 82 q^{59} - 86 q^{60} - 24 q^{61} + 15 q^{62} - 63 q^{63} + 63 q^{64} - 23 q^{65} + 10 q^{66} - 54 q^{67} + 65 q^{68} + 36 q^{69} + 9 q^{70} - 128 q^{71} - 69 q^{72} + 2 q^{73} - 58 q^{74} - 31 q^{75} - 76 q^{76} + 11 q^{77} - 19 q^{78} - 43 q^{79} - 19 q^{80} + 49 q^{81} - 2 q^{82} - 62 q^{83} - 82 q^{84} - 6 q^{85} - 11 q^{86} - 62 q^{87} + 33 q^{88} - 49 q^{89} - 37 q^{90} - 2 q^{91} - 96 q^{92} - 29 q^{93} - 75 q^{94} - 133 q^{95} - 86 q^{96} + 5 q^{97} - 72 q^{98} - 56 q^{99}+O(q^{100}) \) 69 * q - 11 * q^2 - 3 * q^3 + 65 * q^4 - 6 * q^5 - 10 * q^6 - 11 * q^7 - 33 * q^8 + 56 * q^9 - q^10 - 69 * q^11 - 3 * q^12 - 28 * q^13 - 15 * q^14 - 45 * q^15 + 53 * q^16 + 69 * q^17 - 17 * q^18 - 32 * q^19 - 21 * q^20 - 38 * q^21 + 11 * q^22 - 41 * q^23 - 11 * q^24 + 67 * q^25 - 6 * q^26 - 3 * q^27 - 21 * q^28 - 22 * q^29 - 22 * q^30 - 27 * q^31 - 87 * q^32 + 3 * q^33 - 11 * q^34 - 44 * q^35 + 59 * q^36 + 24 * q^37 - 22 * q^38 - 59 * q^39 + q^40 - 43 * q^41 - 15 * q^42 + 69 * q^43 - 65 * q^44 - 12 * q^45 - 21 * q^46 - 99 * q^47 + 2 * q^48 + 64 * q^49 - 78 * q^50 - 3 * q^51 - 57 * q^52 - 50 * q^53 + 20 * q^54 + 6 * q^55 - 59 * q^56 - 15 * q^57 + 22 * q^58 - 82 * q^59 - 86 * q^60 - 24 * q^61 + 15 * q^62 - 63 * q^63 + 63 * q^64 - 23 * q^65 + 10 * q^66 - 54 * q^67 + 65 * q^68 + 36 * q^69 + 9 * q^70 - 128 * q^71 - 69 * q^72 + 2 * q^73 - 58 * q^74 - 31 * q^75 - 76 * q^76 + 11 * q^77 - 19 * q^78 - 43 * q^79 - 19 * q^80 + 49 * q^81 - 2 * q^82 - 62 * q^83 - 82 * q^84 - 6 * q^85 - 11 * q^86 - 62 * q^87 + 33 * q^88 - 49 * q^89 - 37 * q^90 - 2 * q^91 - 96 * q^92 - 29 * q^93 - 75 * q^94 - 133 * q^95 - 86 * q^96 + 5 * q^97 - 72 * q^98 - 56 * q^99

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