LMFDB - Embedded newform 8043.2.a.n.1.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8043,2,Mod(1,8043)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8043, 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("8043.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8043 = 3 \cdot 7 \cdot 383 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8043.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:	$64.2236783457$
Analytic rank:	$1$
Dimension:	$40$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Character	$\chi$	$=$	8043.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.506041 q^{2} +1.00000 q^{3} -1.74392 q^{4} -0.0764488 q^{5} -0.506041 q^{6} +1.00000 q^{7} +1.89458 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.506041 q^{2} +1.00000 q^{3} -1.74392 q^{4} -0.0764488 q^{5} -0.506041 q^{6} +1.00000 q^{7} +1.89458 q^{8} +1.00000 q^{9} +0.0386862 q^{10} +2.12893 q^{11} -1.74392 q^{12} +3.26572 q^{13} -0.506041 q^{14} -0.0764488 q^{15} +2.52911 q^{16} +0.182019 q^{17} -0.506041 q^{18} -0.977071 q^{19} +0.133321 q^{20} +1.00000 q^{21} -1.07733 q^{22} -4.30362 q^{23} +1.89458 q^{24} -4.99416 q^{25} -1.65259 q^{26} +1.00000 q^{27} -1.74392 q^{28} -5.74330 q^{29} +0.0386862 q^{30} -7.92584 q^{31} -5.06899 q^{32} +2.12893 q^{33} -0.0921091 q^{34} -0.0764488 q^{35} -1.74392 q^{36} +4.63637 q^{37} +0.494438 q^{38} +3.26572 q^{39} -0.144838 q^{40} -3.13371 q^{41} -0.506041 q^{42} +1.11172 q^{43} -3.71269 q^{44} -0.0764488 q^{45} +2.17781 q^{46} +1.28598 q^{47} +2.52911 q^{48} +1.00000 q^{49} +2.52725 q^{50} +0.182019 q^{51} -5.69517 q^{52} -2.79987 q^{53} -0.506041 q^{54} -0.162754 q^{55} +1.89458 q^{56} -0.977071 q^{57} +2.90634 q^{58} -10.2924 q^{59} +0.133321 q^{60} -0.0886572 q^{61} +4.01080 q^{62} +1.00000 q^{63} -2.49311 q^{64} -0.249661 q^{65} -1.07733 q^{66} -5.90491 q^{67} -0.317427 q^{68} -4.30362 q^{69} +0.0386862 q^{70} +1.50622 q^{71} +1.89458 q^{72} +7.73426 q^{73} -2.34619 q^{74} -4.99416 q^{75} +1.70394 q^{76} +2.12893 q^{77} -1.65259 q^{78} -13.1518 q^{79} -0.193348 q^{80} +1.00000 q^{81} +1.58579 q^{82} +1.23270 q^{83} -1.74392 q^{84} -0.0139151 q^{85} -0.562574 q^{86} -5.74330 q^{87} +4.03343 q^{88} -6.43993 q^{89} +0.0386862 q^{90} +3.26572 q^{91} +7.50518 q^{92} -7.92584 q^{93} -0.650757 q^{94} +0.0746959 q^{95} -5.06899 q^{96} -0.769577 q^{97} -0.506041 q^{98} +2.12893 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 9 q^{2} + 40 q^{3} + 33 q^{4} - 27 q^{5} - 9 q^{6} + 40 q^{7} - 18 q^{8} + 40 q^{9}+O(q^{10}) $ 40 * q - 9 * q^2 + 40 * q^3 + 33 * q^4 - 27 * q^5 - 9 * q^6 + 40 * q^7 - 18 * q^8 + 40 * q^9	\( 40 q - 9 q^{2} + 40 q^{3} + 33 q^{4} - 27 q^{5} - 9 q^{6} + 40 q^{7} - 18 q^{8} + 40 q^{9} - 14 q^{10} - 29 q^{11} + 33 q^{12} - 40 q^{13} - 9 q^{14} - 27 q^{15} + 23 q^{16} - 46 q^{17} - 9 q^{18} + 17 q^{19} - 53 q^{20} + 40 q^{21} - 15 q^{22} - 62 q^{23} - 18 q^{24} + 35 q^{25} - 29 q^{26} + 40 q^{27} + 33 q^{28} - 47 q^{29} - 14 q^{30} + q^{31} - 45 q^{32} - 29 q^{33} + 9 q^{34} - 27 q^{35} + 33 q^{36} - 49 q^{37} - 52 q^{38} - 40 q^{39} - 12 q^{40} - 49 q^{41} - 9 q^{42} - 45 q^{43} - 68 q^{44} - 27 q^{45} - 36 q^{46} - 88 q^{47} + 23 q^{48} + 40 q^{49} - 32 q^{50} - 46 q^{51} - 34 q^{52} - 96 q^{53} - 9 q^{54} + 26 q^{55} - 18 q^{56} + 17 q^{57} + 12 q^{58} - 45 q^{59} - 53 q^{60} - 23 q^{61} - 15 q^{62} + 40 q^{63} + 50 q^{64} - 32 q^{65} - 15 q^{66} - 31 q^{67} - 76 q^{68} - 62 q^{69} - 14 q^{70} - 89 q^{71} - 18 q^{72} + 4 q^{73} + 10 q^{74} + 35 q^{75} + 51 q^{76} - 29 q^{77} - 29 q^{78} - 44 q^{79} - 53 q^{80} + 40 q^{81} - 15 q^{82} - 60 q^{83} + 33 q^{84} - 24 q^{85} - 9 q^{86} - 47 q^{87} - 64 q^{88} - 39 q^{89} - 14 q^{90} - 40 q^{91} - 80 q^{92} + q^{93} + 51 q^{94} - 71 q^{95} - 45 q^{96} - 21 q^{97} - 9 q^{98} - 29 q^{99}+O(q^{100}) \) 40 * q - 9 * q^2 + 40 * q^3 + 33 * q^4 - 27 * q^5 - 9 * q^6 + 40 * q^7 - 18 * q^8 + 40 * q^9 - 14 * q^10 - 29 * q^11 + 33 * q^12 - 40 * q^13 - 9 * q^14 - 27 * q^15 + 23 * q^16 - 46 * q^17 - 9 * q^18 + 17 * q^19 - 53 * q^20 + 40 * q^21 - 15 * q^22 - 62 * q^23 - 18 * q^24 + 35 * q^25 - 29 * q^26 + 40 * q^27 + 33 * q^28 - 47 * q^29 - 14 * q^30 + q^31 - 45 * q^32 - 29 * q^33 + 9 * q^34 - 27 * q^35 + 33 * q^36 - 49 * q^37 - 52 * q^38 - 40 * q^39 - 12 * q^40 - 49 * q^41 - 9 * q^42 - 45 * q^43 - 68 * q^44 - 27 * q^45 - 36 * q^46 - 88 * q^47 + 23 * q^48 + 40 * q^49 - 32 * q^50 - 46 * q^51 - 34 * q^52 - 96 * q^53 - 9 * q^54 + 26 * q^55 - 18 * q^56 + 17 * q^57 + 12 * q^58 - 45 * q^59 - 53 * q^60 - 23 * q^61 - 15 * q^62 + 40 * q^63 + 50 * q^64 - 32 * q^65 - 15 * q^66 - 31 * q^67 - 76 * q^68 - 62 * q^69 - 14 * q^70 - 89 * q^71 - 18 * q^72 + 4 * q^73 + 10 * q^74 + 35 * q^75 + 51 * q^76 - 29 * q^77 - 29 * q^78 - 44 * q^79 - 53 * q^80 + 40 * q^81 - 15 * q^82 - 60 * q^83 + 33 * q^84 - 24 * q^85 - 9 * q^86 - 47 * q^87 - 64 * q^88 - 39 * q^89 - 14 * q^90 - 40 * q^91 - 80 * q^92 + q^93 + 51 * q^94 - 71 * q^95 - 45 * q^96 - 21 * q^97 - 9 * q^98 - 29 * 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 $

$2$	−0.506041	−0.357825	−0.178912	−	0.983865i	$-0.557258\pi$
$2$	−0.506041	−0.357825	−0.178912	+	0.983865i	$0.557258\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.74392	−0.871961
$5$	−0.0764488	−0.0341890	−0.0170945	−	0.999854i	$-0.505442\pi$
$5$	−0.0764488	−0.0341890	−0.0170945	+	0.999854i	$0.505442\pi$
$6$	−0.506041	−0.206590
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.89458	0.669834
$9$	1.00000	0.333333
$10$	0.0386862	0.0122337
$11$	2.12893	0.641897	0.320948	−	0.947097i	$-0.395998\pi$
$11$	2.12893	0.641897	0.320948	+	0.947097i	$0.395998\pi$
$12$	−1.74392	−0.503427
$13$	3.26572	0.905748	0.452874	−	0.891574i	$-0.350399\pi$
$13$	3.26572	0.905748	0.452874	+	0.891574i	$0.350399\pi$
$14$	−0.506041	−0.135245
$15$	−0.0764488	−0.0197390
$16$	2.52911	0.632278
$17$	0.182019	0.0441461	0.0220730	−	0.999756i	$-0.492973\pi$
$17$	0.182019	0.0441461	0.0220730	+	0.999756i	$0.492973\pi$
$18$	−0.506041	−0.119275
$19$	−0.977071	−0.224156	−0.112078	−	0.993699i	$-0.535751\pi$
$19$	−0.977071	−0.224156	−0.112078	+	0.993699i	$0.535751\pi$
$20$	0.133321	0.0298114
$21$	1.00000	0.218218
$22$	−1.07733	−0.229687
$23$	−4.30362	−0.897366	−0.448683	−	0.893691i	$-0.648107\pi$
$23$	−4.30362	−0.897366	−0.448683	+	0.893691i	$0.648107\pi$
$24$	1.89458	0.386729
$25$	−4.99416	−0.998831
$26$	−1.65259	−0.324099
$27$	1.00000	0.192450
$28$	−1.74392	−0.329570
$29$	−5.74330	−1.06650	−0.533252	−	0.845956i	$-0.679030\pi$
$29$	−5.74330	−1.06650	−0.533252	+	0.845956i	$0.679030\pi$
$30$	0.0386862	0.00706311
$31$	−7.92584	−1.42352	−0.711762	−	0.702421i	$-0.752102\pi$
$31$	−7.92584	−1.42352	−0.711762	+	0.702421i	$0.752102\pi$
$32$	−5.06899	−0.896079
$33$	2.12893	0.370599
$34$	−0.0921091	−0.0157966
$35$	−0.0764488	−0.0129222
$36$	−1.74392	−0.290654
$37$	4.63637	0.762215	0.381107	−	0.924531i	$-0.375543\pi$
$37$	4.63637	0.762215	0.381107	+	0.924531i	$0.375543\pi$
$38$	0.494438	0.0802084
$39$	3.26572	0.522934
$40$	−0.144838	−0.0229009
$41$	−3.13371	−0.489404	−0.244702	−	0.969598i	$-0.578690\pi$
$41$	−3.13371	−0.489404	−0.244702	+	0.969598i	$0.578690\pi$
$42$	−0.506041	−0.0780838
$43$	1.11172	0.169535	0.0847676	−	0.996401i	$-0.472985\pi$
$43$	1.11172	0.169535	0.0847676	+	0.996401i	$0.472985\pi$
$44$	−3.71269	−0.559709
$45$	−0.0764488	−0.0113963
$46$	2.17781	0.321100
$47$	1.28598	0.187579	0.0937895	−	0.995592i	$-0.470102\pi$
$47$	1.28598	0.187579	0.0937895	+	0.995592i	$0.470102\pi$
$48$	2.52911	0.365046
$49$	1.00000	0.142857
$50$	2.52725	0.357407
$51$	0.182019	0.0254878
$52$	−5.69517	−0.789778
$53$	−2.79987	−0.384591	−0.192296	−	0.981337i	$-0.561593\pi$
$53$	−2.79987	−0.384591	−0.192296	+	0.981337i	$0.561593\pi$
$54$	−0.506041	−0.0688634
$55$	−0.162754	−0.0219458
$56$	1.89458	0.253174
$57$	−0.977071	−0.129416
$58$	2.90634	0.381622
$59$	−10.2924	−1.33995	−0.669976	−	0.742382i	$-0.733696\pi$
$59$	−10.2924	−1.33995	−0.669976	+	0.742382i	$0.733696\pi$
$60$	0.133321	0.0172116
$61$	−0.0886572	−0.0113514	−0.00567570	−	0.999984i	$-0.501807\pi$
$61$	−0.0886572	−0.0113514	−0.00567570	+	0.999984i	$0.501807\pi$
$62$	4.01080	0.509372
$63$	1.00000	0.125988
$64$	−2.49311	−0.311638
$65$	−0.249661	−0.0309666
$66$	−1.07733	−0.132610
$67$	−5.90491	−0.721400	−0.360700	−	0.932682i	$-0.617462\pi$
$67$	−5.90491	−0.721400	−0.360700	+	0.932682i	$0.617462\pi$
$68$	−0.317427	−0.0384937
$69$	−4.30362	−0.518095
$70$	0.0386862	0.00462389
$71$	1.50622	0.178756	0.0893778	−	0.995998i	$-0.471512\pi$
$71$	1.50622	0.178756	0.0893778	+	0.995998i	$0.471512\pi$
$72$	1.89458	0.223278
$73$	7.73426	0.905227	0.452613	−	0.891707i	$-0.350492\pi$
$73$	7.73426	0.905227	0.452613	+	0.891707i	$0.350492\pi$
$74$	−2.34619	−0.272739
$75$	−4.99416	−0.576675
$76$	1.70394	0.195455
$77$	2.12893	0.242614
$78$	−1.65259	−0.187119
$79$	−13.1518	−1.47969	−0.739845	−	0.672778i	$-0.765101\pi$
$79$	−13.1518	−1.47969	−0.739845	+	0.672778i	$0.765101\pi$
$80$	−0.193348	−0.0216169
$81$	1.00000	0.111111
$82$	1.58579	0.175121
$83$	1.23270	0.135307	0.0676535	−	0.997709i	$-0.478449\pi$
$83$	1.23270	0.135307	0.0676535	+	0.997709i	$0.478449\pi$
$84$	−1.74392	−0.190278
$85$	−0.0139151	−0.00150931
$86$	−0.562574	−0.0606639
$87$	−5.74330	−0.615746
$88$	4.03343	0.429965
$89$	−6.43993	−0.682632	−0.341316	−	0.939949i	$-0.610873\pi$
$89$	−6.43993	−0.682632	−0.341316	+	0.939949i	$0.610873\pi$
$90$	0.0386862	0.00407789
$91$	3.26572	0.342341
$92$	7.50518	0.782469
$93$	−7.92584	−0.821872
$94$	−0.650757	−0.0671204
$95$	0.0746959	0.00766364
$96$	−5.06899	−0.517352
$97$	−0.769577	−0.0781387	−0.0390693	−	0.999237i	$-0.512439\pi$
$97$	−0.769577	−0.0781387	−0.0390693	+	0.999237i	$0.512439\pi$
$98$	−0.506041	−0.0511178
$99$	2.12893	0.213966
$100$	8.70942	0.870942
$101$	−3.43117	−0.341414	−0.170707	−	0.985322i	$-0.554605\pi$
$101$	−3.43117	−0.341414	−0.170707	+	0.985322i	$0.554605\pi$
$102$	−0.0921091	−0.00912016
$103$	−18.6918	−1.84176	−0.920879	−	0.389847i	$-0.872528\pi$
$103$	−18.6918	−1.84176	−0.920879	+	0.389847i	$0.872528\pi$
$104$	6.18717	0.606702
$105$	−0.0764488	−0.00746064
$106$	1.41685	0.137616
$107$	6.63230	0.641169	0.320584	−	0.947220i	$-0.396121\pi$
$107$	6.63230	0.641169	0.320584	+	0.947220i	$0.396121\pi$
$108$	−1.74392	−0.167809
$109$	−10.6741	−1.02239	−0.511196	−	0.859464i	$-0.670798\pi$
$109$	−10.6741	−1.02239	−0.511196	+	0.859464i	$0.670798\pi$
$110$	0.0823603	0.00785275
$111$	4.63637	0.440065
$112$	2.52911	0.238979
$113$	−14.0969	−1.32612	−0.663062	−	0.748564i	$-0.730744\pi$
$113$	−14.0969	−1.32612	−0.663062	+	0.748564i	$0.730744\pi$
$114$	0.494438	0.0463084
$115$	0.329006	0.0306800
$116$	10.0159	0.929950
$117$	3.26572	0.301916
$118$	5.20836	0.479469
$119$	0.182019	0.0166857
$120$	−0.144838	−0.0132219
$121$	−6.46765	−0.587969
$122$	0.0448642	0.00406181
$123$	−3.13371	−0.282557
$124$	13.8221	1.24126
$125$	0.764041	0.0683379
$126$	−0.506041	−0.0450817
$127$	19.8131	1.75813	0.879064	−	0.476704i	$-0.158169\pi$
$127$	19.8131	1.75813	0.879064	+	0.476704i	$0.158169\pi$
$128$	11.3996	1.00759
$129$	1.11172	0.0978812
$130$	0.126338	0.0110806
$131$	7.88735	0.689121	0.344560	−	0.938764i	$-0.388028\pi$
$131$	7.88735	0.689121	0.344560	+	0.938764i	$0.388028\pi$
$132$	−3.71269	−0.323148
$133$	−0.977071	−0.0847228
$134$	2.98813	0.258135
$135$	−0.0764488	−0.00657967
$136$	0.344849	0.0295706
$137$	21.5324	1.83964	0.919819	−	0.392343i	$-0.128335\pi$
$137$	21.5324	1.83964	0.919819	+	0.392343i	$0.128335\pi$
$138$	2.17781	0.185387
$139$	14.2266	1.20668	0.603341	−	0.797483i	$-0.293836\pi$
$139$	14.2266	1.20668	0.603341	+	0.797483i	$0.293836\pi$
$140$	0.133321	0.0112677
$141$	1.28598	0.108299
$142$	−0.762210	−0.0639632
$143$	6.95250	0.581397
$144$	2.52911	0.210759
$145$	0.439069	0.0364627
$146$	−3.91385	−0.323913
$147$	1.00000	0.0824786
$148$	−8.08547	−0.664622
$149$	−15.5329	−1.27251	−0.636254	−	0.771480i	$-0.719517\pi$
$149$	−15.5329	−1.27251	−0.636254	+	0.771480i	$0.719517\pi$
$150$	2.52725	0.206349
$151$	3.53847	0.287957	0.143978	−	0.989581i	$-0.454010\pi$
$151$	3.53847	0.287957	0.143978	+	0.989581i	$0.454010\pi$
$152$	−1.85114	−0.150147
$153$	0.182019	0.0147154
$154$	−1.07733	−0.0868134
$155$	0.605921	0.0486688
$156$	−5.69517	−0.455978
$157$	0.838130	0.0668901	0.0334450	−	0.999441i	$-0.489352\pi$
$157$	0.838130	0.0668901	0.0334450	+	0.999441i	$0.489352\pi$
$158$	6.65533	0.529470
$159$	−2.79987	−0.222044
$160$	0.387518	0.0306360
$161$	−4.30362	−0.339173
$162$	−0.506041	−0.0397583
$163$	21.2112	1.66139	0.830695	−	0.556728i	$-0.187944\pi$
$163$	21.2112	1.66139	0.830695	+	0.556728i	$0.187944\pi$
$164$	5.46495	0.426741
$165$	−0.162754	−0.0126704
$166$	−0.623799	−0.0484162
$167$	−11.0598	−0.855833	−0.427916	−	0.903818i	$-0.640752\pi$
$167$	−11.0598	−0.855833	−0.427916	+	0.903818i	$0.640752\pi$
$168$	1.89458	0.146170
$169$	−2.33506	−0.179620
$170$	0.00704163	0.000540068 0
$171$	−0.977071	−0.0747185
$172$	−1.93875	−0.147828
$173$	12.5137	0.951400	0.475700	−	0.879608i	$-0.342195\pi$
$173$	12.5137	0.951400	0.475700	+	0.879608i	$0.342195\pi$
$174$	2.90634	0.220329
$175$	−4.99416	−0.377523
$176$	5.38430	0.405857
$177$	−10.2924	−0.773622
$178$	3.25887	0.244263
$179$	11.0731	0.827642	0.413821	−	0.910358i	$-0.364194\pi$
$179$	11.0731	0.827642	0.413821	+	0.910358i	$0.364194\pi$
$180$	0.133321	0.00993715
$181$	−19.5818	−1.45550	−0.727751	−	0.685841i	$-0.759434\pi$
$181$	−19.5818	−1.45550	−0.727751	+	0.685841i	$0.759434\pi$
$182$	−1.65259	−0.122498
$183$	−0.0886572	−0.00655373
$184$	−8.15354	−0.601087
$185$	−0.354445	−0.0260593
$186$	4.01080	0.294086
$187$	0.387506	0.0283372
$188$	−2.24264	−0.163562
$189$	1.00000	0.0727393
$190$	−0.0377992	−0.00274224
$191$	−0.884335	−0.0639883	−0.0319941	−	0.999488i	$-0.510186\pi$
$191$	−0.884335	−0.0639883	−0.0319941	+	0.999488i	$0.510186\pi$
$192$	−2.49311	−0.179925
$193$	−4.01830	−0.289244	−0.144622	−	0.989487i	$-0.546197\pi$
$193$	−4.01830	−0.289244	−0.144622	+	0.989487i	$0.546197\pi$
$194$	0.389437	0.0279600
$195$	−0.249661	−0.0178786
$196$	−1.74392	−0.124566
$197$	−10.2792	−0.732361	−0.366180	−	0.930544i	$-0.619335\pi$
$197$	−10.2792	−0.732361	−0.366180	+	0.930544i	$0.619335\pi$
$198$	−1.07733	−0.0765622
$199$	−3.37900	−0.239531	−0.119765	−	0.992802i	$-0.538214\pi$
$199$	−3.37900	−0.239531	−0.119765	+	0.992802i	$0.538214\pi$
$200$	−9.46182	−0.669051
$201$	−5.90491	−0.416500
$202$	1.73631	0.122166
$203$	−5.74330	−0.403101
$204$	−0.317427	−0.0222243
$205$	0.239569	0.0167322
$206$	9.45882	0.659027
$207$	−4.30362	−0.299122
$208$	8.25938	0.572685
$209$	−2.08012	−0.143885
$210$	0.0386862	0.00266960
$211$	17.5703	1.20959	0.604795	−	0.796381i	$-0.293255\pi$
$211$	17.5703	1.20959	0.604795	+	0.796381i	$0.293255\pi$
$212$	4.88275	0.335349
$213$	1.50622	0.103205
$214$	−3.35621	−0.229426
$215$	−0.0849894	−0.00579623
$216$	1.89458	0.128910
$217$	−7.92584	−0.538041
$218$	5.40153	0.365838
$219$	7.73426	0.522633
$220$	0.283831	0.0191359
$221$	0.594424	0.0399853
$222$	−2.34619	−0.157466
$223$	11.9559	0.800626	0.400313	−	0.916378i	$-0.368901\pi$
$223$	11.9559	0.800626	0.400313	+	0.916378i	$0.368901\pi$
$224$	−5.06899	−0.338686
$225$	−4.99416	−0.332944
$226$	7.13361	0.474520
$227$	−21.3219	−1.41518	−0.707592	−	0.706621i	$-0.750218\pi$
$227$	−21.3219	−1.41518	−0.707592	+	0.706621i	$0.750218\pi$
$228$	1.70394	0.112846
$229$	25.9207	1.71289	0.856445	−	0.516238i	$-0.172668\pi$
$229$	25.9207	1.71289	0.856445	+	0.516238i	$0.172668\pi$
$230$	−0.166491	−0.0109781
$231$	2.12893	0.140073
$232$	−10.8811	−0.714381
$233$	24.3752	1.59687	0.798436	−	0.602079i	$-0.205661\pi$
$233$	24.3752	1.59687	0.798436	+	0.602079i	$0.205661\pi$
$234$	−1.65259	−0.108033
$235$	−0.0983114	−0.00641313
$236$	17.9491	1.16839
$237$	−13.1518	−0.854299
$238$	−0.0921091	−0.00597054
$239$	−13.4994	−0.873202	−0.436601	−	0.899655i	$-0.643818\pi$
$239$	−13.4994	−0.873202	−0.436601	+	0.899655i	$0.643818\pi$
$240$	−0.193348	−0.0124805
$241$	−29.7260	−1.91482	−0.957411	−	0.288727i	$-0.906768\pi$
$241$	−29.7260	−1.91482	−0.957411	+	0.288727i	$0.906768\pi$
$242$	3.27290	0.210390
$243$	1.00000	0.0641500
$244$	0.154611	0.00989798
$245$	−0.0764488	−0.00488414
$246$	1.58579	0.101106
$247$	−3.19084	−0.203029
$248$	−15.0161	−0.953525
$249$	1.23270	0.0781195
$250$	−0.386636	−0.0244530
$251$	−0.0773386	−0.00488157	−0.00244078	−	0.999997i	$-0.500777\pi$
$251$	−0.0773386	−0.00488157	−0.00244078	+	0.999997i	$0.500777\pi$
$252$	−1.74392	−0.109857
$253$	−9.16210	−0.576016
$254$	−10.0262	−0.629102
$255$	−0.0139151	−0.000871400 0
$256$	−0.782446	−0.0489029
$257$	15.9600	0.995557	0.497778	−	0.867304i	$-0.334149\pi$
$257$	15.9600	0.995557	0.497778	+	0.867304i	$0.334149\pi$
$258$	−0.562574	−0.0350243
$259$	4.63637	0.288090
$260$	0.435389	0.0270017
$261$	−5.74330	−0.355501
$262$	−3.99132	−0.246585
$263$	−28.8253	−1.77744	−0.888722	−	0.458447i	$-0.848406\pi$
$263$	−28.8253	−1.77744	−0.888722	+	0.458447i	$0.848406\pi$
$264$	4.03343	0.248240
$265$	0.214047	0.0131488
$266$	0.494438	0.0303159
$267$	−6.43993	−0.394118
$268$	10.2977	0.629033
$269$	−22.2831	−1.35862	−0.679312	−	0.733849i	$-0.737722\pi$
$269$	−22.2831	−1.35862	−0.679312	+	0.733849i	$0.737722\pi$
$270$	0.0386862	0.00235437
$271$	−5.72737	−0.347913	−0.173956	−	0.984753i	$-0.555655\pi$
$271$	−5.72737	−0.347913	−0.173956	+	0.984753i	$0.555655\pi$
$272$	0.460346	0.0279126
$273$	3.26572	0.197651
$274$	−10.8963	−0.658268
$275$	−10.6322	−0.641146
$276$	7.50518	0.451758
$277$	−12.0448	−0.723705	−0.361852	−	0.932235i	$-0.617856\pi$
$277$	−12.0448	−0.723705	−0.361852	+	0.932235i	$0.617856\pi$
$278$	−7.19923	−0.431781
$279$	−7.92584	−0.474508
$280$	−0.144838	−0.00865574
$281$	−15.4010	−0.918749	−0.459374	−	0.888243i	$-0.651926\pi$
$281$	−15.4010	−0.918749	−0.459374	+	0.888243i	$0.651926\pi$
$282$	−0.650757	−0.0387520
$283$	−1.99595	−0.118647	−0.0593236	−	0.998239i	$-0.518894\pi$
$283$	−1.99595	−0.118647	−0.0593236	+	0.998239i	$0.518894\pi$
$284$	−2.62673	−0.155868
$285$	0.0746959	0.00442461
$286$	−3.51825	−0.208038
$287$	−3.13371	−0.184977
$288$	−5.06899	−0.298693
$289$	−16.9669	−0.998051
$290$	−0.222187	−0.0130472
$291$	−0.769577	−0.0451134
$292$	−13.4880	−0.789323
$293$	7.86395	0.459417	0.229709	−	0.973259i	$-0.426223\pi$
$293$	7.86395	0.459417	0.229709	+	0.973259i	$0.426223\pi$
$294$	−0.506041	−0.0295129
$295$	0.786840	0.0458116
$296$	8.78397	0.510558
$297$	2.12893	0.123533
$298$	7.86030	0.455335
$299$	−14.0544	−0.812788
$300$	8.70942	0.502839
$301$	1.11172	0.0640783
$302$	−1.79061	−0.103038
$303$	−3.43117	−0.197115
$304$	−2.47112	−0.141729
$305$	0.00677774	0.000388092 0
$306$	−0.0921091	−0.00526552
$307$	11.3913	0.650138	0.325069	−	0.945690i	$-0.394612\pi$
$307$	11.3913	0.650138	0.325069	+	0.945690i	$0.394612\pi$
$308$	−3.71269	−0.211550
$309$	−18.6918	−1.06334
$310$	−0.306621	−0.0174149
$311$	−20.5285	−1.16407	−0.582033	−	0.813165i	$-0.697742\pi$
$311$	−20.5285	−1.16407	−0.582033	+	0.813165i	$0.697742\pi$
$312$	6.18717	0.350279
$313$	−1.03747	−0.0586415	−0.0293207	−	0.999570i	$-0.509334\pi$
$313$	−1.03747	−0.0586415	−0.0293207	+	0.999570i	$0.509334\pi$
$314$	−0.424128	−0.0239349
$315$	−0.0764488	−0.00430740
$316$	22.9357	1.29023
$317$	14.6073	0.820429	0.410214	−	0.911989i	$-0.365454\pi$
$317$	14.6073	0.820429	0.410214	+	0.911989i	$0.365454\pi$
$318$	1.41685	0.0794528
$319$	−12.2271	−0.684585
$320$	0.190595	0.0106546
$321$	6.63230	0.370179
$322$	2.17781	0.121364
$323$	−0.177846	−0.00989559
$324$	−1.74392	−0.0968846
$325$	−16.3095	−0.904690
$326$	−10.7337	−0.594486
$327$	−10.6741	−0.590279
$328$	−5.93706	−0.327819
$329$	1.28598	0.0708982
$330$	0.0823603	0.00453379
$331$	−9.41115	−0.517284	−0.258642	−	0.965973i	$-0.583275\pi$
$331$	−9.41115	−0.517284	−0.258642	+	0.965973i	$0.583275\pi$
$332$	−2.14974	−0.117982
$333$	4.63637	0.254072
$334$	5.59671	0.306238
$335$	0.451424	0.0246639
$336$	2.52911	0.137974
$337$	14.6978	0.800638	0.400319	−	0.916376i	$-0.368899\pi$
$337$	14.6978	0.800638	0.400319	+	0.916376i	$0.368899\pi$
$338$	1.18163	0.0642724
$339$	−14.0969	−0.765638
$340$	0.0242669	0.00131606
$341$	−16.8736	−0.913755
$342$	0.494438	0.0267361
$343$	1.00000	0.0539949
$344$	2.10623	0.113560
$345$	0.329006	0.0177131
$346$	−6.33245	−0.340435
$347$	4.16923	0.223816	0.111908	−	0.993719i	$-0.464304\pi$
$347$	4.16923	0.223816	0.111908	+	0.993719i	$0.464304\pi$
$348$	10.0159	0.536907
$349$	−5.67983	−0.304035	−0.152017	−	0.988378i	$-0.548577\pi$
$349$	−5.67983	−0.304035	−0.152017	+	0.988378i	$0.548577\pi$
$350$	2.52725	0.135087
$351$	3.26572	0.174311
$352$	−10.7915	−0.575190
$353$	2.86856	0.152678	0.0763390	−	0.997082i	$-0.475677\pi$
$353$	2.86856	0.152678	0.0763390	+	0.997082i	$0.475677\pi$
$354$	5.20836	0.276821
$355$	−0.115149	−0.00611147
$356$	11.2307	0.595228
$357$	0.182019	0.00963347
$358$	−5.60344	−0.296151
$359$	−26.9608	−1.42294	−0.711468	−	0.702718i	$-0.751970\pi$
$359$	−26.9608	−1.42294	−0.711468	+	0.702718i	$0.751970\pi$
$360$	−0.144838	−0.00763365
$361$	−18.0453	−0.949754
$362$	9.90918	0.520815
$363$	−6.46765	−0.339464
$364$	−5.69517	−0.298508
$365$	−0.591275	−0.0309488
$366$	0.0448642	0.00234509
$367$	23.6372	1.23385	0.616925	−	0.787022i	$-0.288378\pi$
$367$	23.6372	1.23385	0.616925	+	0.787022i	$0.288378\pi$
$368$	−10.8843	−0.567385
$369$	−3.13371	−0.163135
$370$	0.179364	0.00932468
$371$	−2.79987	−0.145362
$372$	13.8221	0.716640
$373$	−24.3675	−1.26170	−0.630850	−	0.775905i	$-0.717293\pi$
$373$	−24.3675	−1.26170	−0.630850	+	0.775905i	$0.717293\pi$
$374$	−0.196094	−0.0101398
$375$	0.764041	0.0394549
$376$	2.43638	0.125647
$377$	−18.7560	−0.965984
$378$	−0.506041	−0.0260279
$379$	−4.03061	−0.207039	−0.103519	−	0.994627i	$-0.533010\pi$
$379$	−4.03061	−0.207039	−0.103519	+	0.994627i	$0.533010\pi$
$380$	−0.130264	−0.00668240
$381$	19.8131	1.01506
$382$	0.447510	0.0228966
$383$	−1.00000	−0.0510976
$383$	−1.00000	−0.0510976
$384$	11.3996	0.581733
$385$	−0.162754	−0.00829472
$386$	2.03342	0.103499
$387$	1.11172	0.0565117
$388$	1.34208	0.0681339
$389$	12.5253	0.635059	0.317529	−	0.948248i	$-0.397147\pi$
$389$	12.5253	0.635059	0.317529	+	0.948248i	$0.397147\pi$
$390$	0.126338	0.00639740
$391$	−0.783340	−0.0396152
$392$	1.89458	0.0956906
$393$	7.88735	0.397864
$394$	5.20168	0.262057
$395$	1.00544	0.0505890
$396$	−3.71269	−0.186570
$397$	27.6064	1.38552	0.692761	−	0.721167i	$-0.256394\pi$
$397$	27.6064	1.38552	0.692761	+	0.721167i	$0.256394\pi$
$398$	1.70991	0.0857101
$399$	−0.977071	−0.0489147
$400$	−12.6308	−0.631539
$401$	2.88755	0.144197	0.0720986	−	0.997398i	$-0.477030\pi$
$401$	2.88755	0.144197	0.0720986	+	0.997398i	$0.477030\pi$
$402$	2.98813	0.149034
$403$	−25.8836	−1.28935
$404$	5.98369	0.297700
$405$	−0.0764488	−0.00379877
$406$	2.90634	0.144239
$407$	9.87051	0.489263
$408$	0.344849	0.0170726
$409$	−10.5139	−0.519879	−0.259940	−	0.965625i	$-0.583703\pi$
$409$	−10.5139	−0.519879	−0.259940	+	0.965625i	$0.583703\pi$
$410$	−0.121232	−0.00598720
$411$	21.5324	1.06212
$412$	32.5971	1.60594
$413$	−10.2924	−0.506455
$414$	2.17781	0.107033
$415$	−0.0942388	−0.00462600
$416$	−16.5539	−0.811622
$417$	14.2266	0.696678
$418$	1.05262	0.0514855
$419$	−21.5439	−1.05249	−0.526245	−	0.850333i	$-0.676401\pi$
$419$	−21.5439	−1.05249	−0.526245	+	0.850333i	$0.676401\pi$
$420$	0.133321	0.00650539
$421$	9.11116	0.444051	0.222025	−	0.975041i	$-0.428733\pi$
$421$	9.11116	0.444051	0.222025	+	0.975041i	$0.428733\pi$
$422$	−8.89129	−0.432821
$423$	1.28598	0.0625263
$424$	−5.30457	−0.257612
$425$	−0.909031	−0.0440945
$426$	−0.762210	−0.0369292
$427$	−0.0886572	−0.00429042
$428$	−11.5662	−0.559074
$429$	6.95250	0.335670
$430$	0.0430081	0.00207404
$431$	31.7345	1.52860	0.764298	−	0.644864i	$-0.223086\pi$
$431$	31.7345	1.52860	0.764298	+	0.644864i	$0.223086\pi$
$432$	2.52911	0.121682
$433$	−35.7895	−1.71993	−0.859966	−	0.510351i	$-0.829515\pi$
$433$	−35.7895	−1.71993	−0.859966	+	0.510351i	$0.829515\pi$
$434$	4.01080	0.192525
$435$	0.439069	0.0210517
$436$	18.6148	0.891487
$437$	4.20494	0.201150
$438$	−3.91385	−0.187011
$439$	5.32725	0.254256	0.127128	−	0.991886i	$-0.459424\pi$
$439$	5.32725	0.254256	0.127128	+	0.991886i	$0.459424\pi$
$440$	−0.308351	−0.0147000
$441$	1.00000	0.0476190
$442$	−0.300803	−0.0143077
$443$	−27.1888	−1.29178	−0.645889	−	0.763431i	$-0.723513\pi$
$443$	−27.1888	−1.29178	−0.645889	+	0.763431i	$0.723513\pi$
$444$	−8.08547	−0.383720
$445$	0.492325	0.0233385
$446$	−6.05017	−0.286484
$447$	−15.5329	−0.734683
$448$	−2.49311	−0.117788
$449$	−8.16557	−0.385357	−0.192678	−	0.981262i	$-0.561717\pi$
$449$	−8.16557	−0.385357	−0.192678	+	0.981262i	$0.561717\pi$
$450$	2.52725	0.119136
$451$	−6.67146	−0.314147
$452$	24.5839	1.15633
$453$	3.53847	0.166252
$454$	10.7898	0.506388
$455$	−0.249661	−0.0117043
$456$	−1.85114	−0.0866875
$457$	−3.23866	−0.151498	−0.0757490	−	0.997127i	$-0.524135\pi$
$457$	−3.23866	−0.151498	−0.0757490	+	0.997127i	$0.524135\pi$
$458$	−13.1169	−0.612915
$459$	0.182019	0.00849592
$460$	−0.573762	−0.0267518
$461$	−10.1381	−0.472181	−0.236090	−	0.971731i	$-0.575866\pi$
$461$	−10.1381	−0.472181	−0.236090	+	0.971731i	$0.575866\pi$
$462$	−1.07733	−0.0501217
$463$	7.99157	0.371400	0.185700	−	0.982607i	$-0.440545\pi$
$463$	7.99157	0.371400	0.185700	+	0.982607i	$0.440545\pi$
$464$	−14.5254	−0.674327
$465$	0.605921	0.0280989
$466$	−12.3348	−0.571401
$467$	−0.259080	−0.0119888	−0.00599439	−	0.999982i	$-0.501908\pi$
$467$	−0.259080	−0.0119888	−0.00599439	+	0.999982i	$0.501908\pi$
$468$	−5.69517	−0.263259
$469$	−5.90491	−0.272663
$470$	0.0497496	0.00229478
$471$	0.838130	0.0386190
$472$	−19.4997	−0.897547
$473$	2.36677	0.108824
$474$	6.65533	0.305690
$475$	4.87965	0.223893
$476$	−0.317427	−0.0145492
$477$	−2.79987	−0.128197
$478$	6.83124	0.312454
$479$	−22.3544	−1.02140	−0.510699	−	0.859760i	$-0.670613\pi$
$479$	−22.3544	−1.02140	−0.510699	+	0.859760i	$0.670613\pi$
$480$	0.387518	0.0176877
$481$	15.1411	0.690375
$482$	15.0426	0.685171
$483$	−4.30362	−0.195821
$484$	11.2791	0.512686
$485$	0.0588332	0.00267148
$486$	−0.506041	−0.0229545
$487$	−9.92356	−0.449679	−0.224840	−	0.974396i	$-0.572186\pi$
$487$	−9.92356	−0.449679	−0.224840	+	0.974396i	$0.572186\pi$
$488$	−0.167968	−0.00760356
$489$	21.2112	0.959203
$490$	0.0386862	0.00174767
$491$	−24.6763	−1.11363	−0.556813	−	0.830638i	$-0.687976\pi$
$491$	−24.6763	−1.11363	−0.556813	+	0.830638i	$0.687976\pi$
$492$	5.46495	0.246379
$493$	−1.04539	−0.0470820
$494$	1.61470	0.0726487
$495$	−0.162754	−0.00731526
$496$	−20.0453	−0.900062
$497$	1.50622	0.0675633
$498$	−0.623799	−0.0279531
$499$	−17.2924	−0.774114	−0.387057	−	0.922056i	$-0.626508\pi$
$499$	−17.2924	−0.774114	−0.387057	+	0.922056i	$0.626508\pi$
$500$	−1.33243	−0.0595880
$501$	−11.0598	−0.494115
$502$	0.0391365	0.00174675
$503$	−31.7449	−1.41544	−0.707719	−	0.706495i	$-0.750276\pi$
$503$	−31.7449	−1.41544	−0.707719	+	0.706495i	$0.750276\pi$
$504$	1.89458	0.0843912
$505$	0.262309	0.0116726
$506$	4.63640	0.206113
$507$	−2.33506	−0.103703
$508$	−34.5525	−1.53302
$509$	43.2689	1.91786	0.958931	−	0.283639i	$-0.0915418\pi$
$509$	43.2689	1.91786	0.958931	+	0.283639i	$0.0915418\pi$
$510$	0.00704163	0.000311809 0
$511$	7.73426	0.342144
$512$	−22.4032	−0.990093
$513$	−0.977071	−0.0431387
$514$	−8.07641	−0.356235
$515$	1.42897	0.0629678
$516$	−1.93875	−0.0853486
$517$	2.73776	0.120406
$518$	−2.34619	−0.103086
$519$	12.5137	0.549291
$520$	−0.473002	−0.0207425
$521$	35.2537	1.54449	0.772247	−	0.635323i	$-0.219133\pi$
$521$	35.2537	1.54449	0.772247	+	0.635323i	$0.219133\pi$
$522$	2.90634	0.127207
$523$	21.8641	0.956048	0.478024	−	0.878347i	$-0.341353\pi$
$523$	21.8641	0.956048	0.478024	+	0.878347i	$0.341353\pi$
$524$	−13.7549	−0.600887
$525$	−4.99416	−0.217963
$526$	14.5868	0.636014
$527$	−1.44265	−0.0628430
$528$	5.38430	0.234322
$529$	−4.47888	−0.194734
$530$	−0.108316	−0.00470496
$531$	−10.2924	−0.446651
$532$	1.70394	0.0738750
$533$	−10.2338	−0.443277
$534$	3.25887	0.141025
$535$	−0.507031	−0.0219209
$536$	−11.1873	−0.483218
$537$	11.0731	0.477839
$538$	11.2762	0.486150
$539$	2.12893	0.0916995
$540$	0.133321	0.00573722
$541$	14.8089	0.636684	0.318342	−	0.947976i	$-0.396874\pi$
$541$	14.8089	0.636684	0.318342	+	0.947976i	$0.396874\pi$
$542$	2.89828	0.124492
$543$	−19.5818	−0.840335
$544$	−0.922652	−0.0395584
$545$	0.816022	0.0349545
$546$	−1.65259	−0.0707243
$547$	10.9199	0.466901	0.233450	−	0.972369i	$-0.424998\pi$
$547$	10.9199	0.466901	0.233450	+	0.972369i	$0.424998\pi$
$548$	−37.5509	−1.60409
$549$	−0.0886572	−0.00378380
$550$	5.38033	0.229418
$551$	5.61161	0.239063
$552$	−8.15354	−0.347038
$553$	−13.1518	−0.559270
$554$	6.09518	0.258960
$555$	−0.354445	−0.0150454
$556$	−24.8100	−1.05218
$557$	6.09910	0.258427	0.129214	−	0.991617i	$-0.458755\pi$
$557$	6.09910	0.258427	0.129214	+	0.991617i	$0.458755\pi$
$558$	4.01080	0.169791
$559$	3.63056	0.153556
$560$	−0.193348	−0.00817043
$561$	0.387506	0.0163605
$562$	7.79355	0.328751
$563$	1.55041	0.0653420	0.0326710	−	0.999466i	$-0.489599\pi$
$563$	1.55041	0.0653420	0.0326710	+	0.999466i	$0.489599\pi$
$564$	−2.24264	−0.0944323
$565$	1.07769	0.0453388
$566$	1.01003	0.0424549
$567$	1.00000	0.0419961
$568$	2.85365	0.119737
$569$	−36.3129	−1.52231	−0.761157	−	0.648568i	$-0.775368\pi$
$569$	−36.3129	−1.52231	−0.761157	+	0.648568i	$0.775368\pi$
$570$	−0.0377992	−0.00158323
$571$	22.9642	0.961023	0.480512	−	0.876988i	$-0.340451\pi$
$571$	22.9642	0.961023	0.480512	+	0.876988i	$0.340451\pi$
$572$	−12.1246	−0.506956
$573$	−0.884335	−0.0369436
$574$	1.58579	0.0661895
$575$	21.4929	0.896317
$576$	−2.49311	−0.103879
$577$	29.0479	1.20928	0.604640	−	0.796499i	$-0.293317\pi$
$577$	29.0479	1.20928	0.604640	+	0.796499i	$0.293317\pi$
$578$	8.58593	0.357128
$579$	−4.01830	−0.166995
$580$	−0.765702	−0.0317940
$581$	1.23270	0.0511412
$582$	0.389437	0.0161427
$583$	−5.96072	−0.246868
$584$	14.6532	0.606352
$585$	−0.249661	−0.0103222
$586$	−3.97948	−0.164391
$587$	−37.7597	−1.55851	−0.779254	−	0.626708i	$-0.784402\pi$
$587$	−37.7597	−1.55851	−0.779254	+	0.626708i	$0.784402\pi$
$588$	−1.74392	−0.0719182
$589$	7.74411	0.319091
$590$	−0.398173	−0.0163925
$591$	−10.2792	−0.422829
$592$	11.7259	0.481931
$593$	−25.8625	−1.06204	−0.531022	−	0.847358i	$-0.678192\pi$
$593$	−25.8625	−1.06204	−0.531022	+	0.847358i	$0.678192\pi$
$594$	−1.07733	−0.0442032
$595$	−0.0139151	−0.000570465 0
$596$	27.0883	1.10958
$597$	−3.37900	−0.138293
$598$	7.11211	0.290836
$599$	22.6343	0.924811	0.462405	−	0.886669i	$-0.346986\pi$
$599$	22.6343	0.924811	0.462405	+	0.886669i	$0.346986\pi$
$600$	−9.46182	−0.386277
$601$	8.60449	0.350985	0.175492	−	0.984481i	$-0.443848\pi$
$601$	8.60449	0.350985	0.175492	+	0.984481i	$0.443848\pi$
$602$	−0.562574	−0.0229288
$603$	−5.90491	−0.240467
$604$	−6.17082	−0.251087
$605$	0.494445	0.0201020
$606$	1.73631	0.0705328
$607$	−45.8881	−1.86254	−0.931270	−	0.364330i	$-0.881298\pi$
$607$	−45.8881	−1.86254	−0.931270	+	0.364330i	$0.881298\pi$
$608$	4.95276	0.200861
$609$	−5.74330	−0.232730
$610$	−0.00342981	−0.000138869 0
$611$	4.19964	0.169899
$612$	−0.317427	−0.0128312
$613$	−34.9127	−1.41011	−0.705054	−	0.709153i	$-0.749077\pi$
$613$	−34.9127	−1.41011	−0.705054	+	0.709153i	$0.749077\pi$
$614$	−5.76449	−0.232636
$615$	0.239569	0.00966034
$616$	4.03343	0.162511
$617$	−33.3926	−1.34434	−0.672168	−	0.740399i	$-0.734637\pi$
$617$	−33.3926	−1.34434	−0.672168	+	0.740399i	$0.734637\pi$
$618$	9.45882	0.380490
$619$	−25.5126	−1.02544	−0.512718	−	0.858557i	$-0.671361\pi$
$619$	−25.5126	−1.02544	−0.512718	+	0.858557i	$0.671361\pi$
$620$	−1.05668	−0.0424373
$621$	−4.30362	−0.172698
$622$	10.3883	0.416532
$623$	−6.43993	−0.258011
$624$	8.25938	0.330640
$625$	24.9124	0.996495
$626$	0.525004	0.0209834
$627$	−2.08012	−0.0830719
$628$	−1.46163	−0.0583256
$629$	0.843908	0.0336488
$630$	0.0386862	0.00154130
$631$	35.5887	1.41676	0.708382	−	0.705830i	$-0.249426\pi$
$631$	35.5887	1.41676	0.708382	+	0.705830i	$0.249426\pi$
$632$	−24.9171	−0.991147
$633$	17.5703	0.698357
$634$	−7.39190	−0.293570
$635$	−1.51469	−0.0601085
$636$	4.88275	0.193614
$637$	3.26572	0.129393
$638$	6.18741	0.244962
$639$	1.50622	0.0595852
$640$	−0.871485	−0.0344485
$641$	−40.6386	−1.60513	−0.802564	−	0.596567i	$-0.796531\pi$
$641$	−40.6386	−1.60513	−0.802564	+	0.596567i	$0.796531\pi$
$642$	−3.35621	−0.132459
$643$	−14.4015	−0.567940	−0.283970	−	0.958833i	$-0.591652\pi$
$643$	−14.4015	−0.567940	−0.283970	+	0.958833i	$0.591652\pi$
$644$	7.50518	0.295745
$645$	−0.0849894	−0.00334645
$646$	0.0899971	0.00354089
$647$	22.7207	0.893243	0.446622	−	0.894723i	$-0.352627\pi$
$647$	22.7207	0.893243	0.446622	+	0.894723i	$0.352627\pi$
$648$	1.89458	0.0744260
$649$	−21.9118	−0.860111
$650$	8.25329	0.323721
$651$	−7.92584	−0.310638
$652$	−36.9907	−1.44867
$653$	−20.6170	−0.806804	−0.403402	−	0.915023i	$-0.632172\pi$
$653$	−20.6170	−0.806804	−0.403402	+	0.915023i	$0.632172\pi$
$654$	5.40153	0.211216
$655$	−0.602979	−0.0235603
$656$	−7.92551	−0.309439
$657$	7.73426	0.301742
$658$	−0.650757	−0.0253691
$659$	−26.2446	−1.02235	−0.511173	−	0.859478i	$-0.670789\pi$
$659$	−26.2446	−1.02235	−0.511173	+	0.859478i	$0.670789\pi$
$660$	0.283831	0.0110481
$661$	−30.5464	−1.18812	−0.594058	−	0.804422i	$-0.702475\pi$
$661$	−30.5464	−1.18812	−0.594058	+	0.804422i	$0.702475\pi$
$662$	4.76243	0.185097
$663$	0.594424	0.0230855
$664$	2.33546	0.0906332
$665$	0.0746959	0.00289658
$666$	−2.34619	−0.0909131
$667$	24.7170	0.957045
$668$	19.2874	0.746253
$669$	11.9559	0.462242
$670$	−0.228439	−0.00882536
$671$	−0.188745	−0.00728642
$672$	−5.06899	−0.195541
$673$	−7.97023	−0.307230	−0.153615	−	0.988131i	$-0.549092\pi$
$673$	−7.97023	−0.307230	−0.153615	+	0.988131i	$0.549092\pi$
$674$	−7.43767	−0.286488
$675$	−4.99416	−0.192225
$676$	4.07216	0.156621
$677$	10.4197	0.400463	0.200231	−	0.979749i	$-0.435831\pi$
$677$	10.4197	0.400463	0.200231	+	0.979749i	$0.435831\pi$
$678$	7.13361	0.273964
$679$	−0.769577	−0.0295336
$680$	−0.0263633	−0.00101099
$681$	−21.3219	−0.817057
$682$	8.53872	0.326964
$683$	−12.5373	−0.479726	−0.239863	−	0.970807i	$-0.577103\pi$
$683$	−12.5373	−0.479726	−0.239863	+	0.970807i	$0.577103\pi$
$684$	1.70394	0.0651516
$685$	−1.64613	−0.0628953
$686$	−0.506041	−0.0193207
$687$	25.9207	0.988938
$688$	2.81165	0.107193
$689$	−9.14359	−0.348343
$690$	−0.166491	−0.00633819
$691$	7.14712	0.271889	0.135945	−	0.990716i	$-0.456593\pi$
$691$	7.14712	0.271889	0.135945	+	0.990716i	$0.456593\pi$
$692$	−21.8230	−0.829584
$693$	2.12893	0.0808714
$694$	−2.10980	−0.0800870
$695$	−1.08760	−0.0412552
$696$	−10.8811	−0.412448
$697$	−0.570395	−0.0216053
$698$	2.87423	0.108791
$699$	24.3752	0.921955
$700$	8.70942	0.329185
$701$	−11.1296	−0.420358	−0.210179	−	0.977663i	$-0.567405\pi$
$701$	−11.1296	−0.420358	−0.210179	+	0.977663i	$0.567405\pi$
$702$	−1.65259	−0.0623730
$703$	−4.53006	−0.170855
$704$	−5.30765	−0.200040
$705$	−0.0983114	−0.00370262
$706$	−1.45161	−0.0546320
$707$	−3.43117	−0.129042
$708$	17.9491	0.674569
$709$	22.3435	0.839126	0.419563	−	0.907726i	$-0.362183\pi$
$709$	22.3435	0.839126	0.419563	+	0.907726i	$0.362183\pi$
$710$	0.0582700	0.00218684
$711$	−13.1518	−0.493230
$712$	−12.2010	−0.457250
$713$	34.1098	1.27742
$714$	−0.0921091	−0.00344709
$715$	−0.531510	−0.0198774
$716$	−19.3106	−0.721671
$717$	−13.4994	−0.504144
$718$	13.6433	0.509162
$719$	45.8924	1.71150	0.855748	−	0.517392i	$-0.173097\pi$
$719$	45.8924	1.71150	0.855748	+	0.517392i	$0.173097\pi$
$720$	−0.193348	−0.00720564
$721$	−18.6918	−0.696119
$722$	9.13168	0.339846
$723$	−29.7260	−1.10552
$724$	34.1491	1.26914
$725$	28.6829	1.06526
$726$	3.27290	0.121469
$727$	−4.24065	−0.157277	−0.0786385	−	0.996903i	$-0.525057\pi$
$727$	−4.24065	−0.157277	−0.0786385	+	0.996903i	$0.525057\pi$
$728$	6.18717	0.229312
$729$	1.00000	0.0370370
$730$	0.299209	0.0110742
$731$	0.202354	0.00748431
$732$	0.154611	0.00571460
$733$	23.3046	0.860773	0.430387	−	0.902645i	$-0.358377\pi$
$733$	23.3046	0.860773	0.430387	+	0.902645i	$0.358377\pi$
$734$	−11.9614	−0.441502
$735$	−0.0764488	−0.00281986
$736$	21.8150	0.804111
$737$	−12.5711	−0.463064
$738$	1.58579	0.0583736
$739$	33.7993	1.24333	0.621664	−	0.783284i	$-0.286457\pi$
$739$	33.7993	1.24333	0.621664	+	0.783284i	$0.286457\pi$
$740$	0.618125	0.0227227
$741$	−3.19084	−0.117219
$742$	1.41685	0.0520141
$743$	35.8416	1.31490	0.657451	−	0.753497i	$-0.271635\pi$
$743$	35.8416	1.31490	0.657451	+	0.753497i	$0.271635\pi$
$744$	−15.0161	−0.550518
$745$	1.18748	0.0435057
$746$	12.3309	0.451467
$747$	1.23270	0.0451023
$748$	−0.675780	−0.0247090
$749$	6.63230	0.242339
$750$	−0.386636	−0.0141180
$751$	−7.36701	−0.268826	−0.134413	−	0.990925i	$-0.542915\pi$
$751$	−7.36701	−0.268826	−0.134413	+	0.990925i	$0.542915\pi$
$752$	3.25238	0.118602
$753$	−0.0773386	−0.00281837
$754$	9.49131	0.345653
$755$	−0.270512	−0.00984494
$756$	−1.74392	−0.0634259
$757$	50.4405	1.83329	0.916646	−	0.399700i	$-0.130886\pi$
$757$	50.4405	1.83329	0.916646	+	0.399700i	$0.130886\pi$
$758$	2.03966	0.0740836
$759$	−9.16210	−0.332563
$760$	0.141517	0.00513337
$761$	−8.95723	−0.324699	−0.162350	−	0.986733i	$-0.551907\pi$
$761$	−8.95723	−0.324699	−0.162350	+	0.986733i	$0.551907\pi$
$762$	−10.0262	−0.363212
$763$	−10.6741	−0.386428
$764$	1.54221	0.0557953
$765$	−0.0139151	−0.000503103 0
$766$	0.506041	0.0182840
$767$	−33.6120	−1.21366
$768$	−0.782446	−0.0282341
$769$	6.54661	0.236077	0.118038	−	0.993009i	$-0.462339\pi$
$769$	6.54661	0.236077	0.118038	+	0.993009i	$0.462339\pi$
$770$	0.0823603	0.00296806
$771$	15.9600	0.574785
$772$	7.00761	0.252209
$773$	−39.5074	−1.42098	−0.710490	−	0.703707i	$-0.751527\pi$
$773$	−39.5074	−1.42098	−0.710490	+	0.703707i	$0.751527\pi$
$774$	−0.562574	−0.0202213
$775$	39.5829	1.42186
$776$	−1.45802	−0.0523400
$777$	4.63637	0.166329
$778$	−6.33832	−0.227240
$779$	3.06186	0.109703
$780$	0.435389	0.0155894
$781$	3.20664	0.114743
$782$	0.396402	0.0141753
$783$	−5.74330	−0.205249
$784$	2.52911	0.0903254
$785$	−0.0640741	−0.00228690
$786$	−3.99132	−0.142366
$787$	22.8204	0.813460	0.406730	−	0.913548i	$-0.366669\pi$
$787$	22.8204	0.813460	0.406730	+	0.913548i	$0.366669\pi$
$788$	17.9261	0.638590
$789$	−28.8253	−1.02621
$790$	−0.508792	−0.0181020
$791$	−14.0969	−0.501228
$792$	4.03343	0.143322
$793$	−0.289530	−0.0102815
$794$	−13.9699	−0.495775
$795$	0.214047	0.00759145
$796$	5.89271	0.208862
$797$	−2.99244	−0.105998	−0.0529988	−	0.998595i	$-0.516878\pi$
$797$	−2.99244	−0.105998	−0.0529988	+	0.998595i	$0.516878\pi$
$798$	0.494438	0.0175029
$799$	0.234072	0.00828088
$800$	25.3153	0.895032
$801$	−6.43993	−0.227544
$802$	−1.46122	−0.0515973
$803$	16.4657	0.581062
$804$	10.2977	0.363172
$805$	0.329006	0.0115960
$806$	13.0982	0.461363
$807$	−22.2831	−0.784402
$808$	−6.50061	−0.228691
$809$	−2.35965	−0.0829608	−0.0414804	−	0.999139i	$-0.513207\pi$
$809$	−2.35965	−0.0829608	−0.0414804	+	0.999139i	$0.513207\pi$
$810$	0.0386862	0.00135930
$811$	34.2361	1.20219	0.601096	−	0.799177i	$-0.294731\pi$
$811$	34.2361	1.20219	0.601096	+	0.799177i	$0.294731\pi$
$812$	10.0159	0.351488
$813$	−5.72737	−0.200868
$814$	−4.99488	−0.175071
$815$	−1.62157	−0.0568012
$816$	0.460346	0.0161153
$817$	−1.08623	−0.0380022
$818$	5.32047	0.186026
$819$	3.26572	0.114114
$820$	−0.417789	−0.0145898
$821$	−23.3540	−0.815059	−0.407529	−	0.913192i	$-0.633610\pi$
$821$	−23.3540	−0.815059	−0.407529	+	0.913192i	$0.633610\pi$
$822$	−10.8963	−0.380051
$823$	45.7099	1.59335	0.796674	−	0.604410i	$-0.206591\pi$
$823$	45.7099	1.59335	0.796674	+	0.604410i	$0.206591\pi$
$824$	−35.4131	−1.23367
$825$	−10.6322	−0.370166
$826$	5.20836	0.181222
$827$	37.5563	1.30596	0.652980	−	0.757375i	$-0.273519\pi$
$827$	37.5563	1.30596	0.652980	+	0.757375i	$0.273519\pi$
$828$	7.50518	0.260823
$829$	12.9023	0.448117	0.224058	−	0.974576i	$-0.428069\pi$
$829$	12.9023	0.448117	0.224058	+	0.974576i	$0.428069\pi$
$830$	0.0476887	0.00165530
$831$	−12.0448	−0.417831
$832$	−8.14180	−0.282266
$833$	0.182019	0.00630658
$834$	−7.19923	−0.249289
$835$	0.845509	0.0292600
$836$	3.62756	0.125462
$837$	−7.92584	−0.273957
$838$	10.9021	0.376607
$839$	−41.5854	−1.43569	−0.717844	−	0.696204i	$-0.754871\pi$
$839$	−41.5854	−1.43569	−0.717844	+	0.696204i	$0.754871\pi$
$840$	−0.144838	−0.00499739
$841$	3.98549	0.137431
$842$	−4.61062	−0.158893
$843$	−15.4010	−0.530440
$844$	−30.6413	−1.05472
$845$	0.178512	0.00614101
$846$	−0.650757	−0.0223735
$847$	−6.46765	−0.222231
$848$	−7.08118	−0.243169
$849$	−1.99595	−0.0685009
$850$	0.460007	0.0157781
$851$	−19.9532	−0.683986
$852$	−2.62673	−0.0899904
$853$	32.0453	1.09721	0.548605	−	0.836082i	$-0.315159\pi$
$853$	32.0453	1.09721	0.548605	+	0.836082i	$0.315159\pi$
$854$	0.0448642	0.00153522
$855$	0.0746959	0.00255455
$856$	12.5654	0.429477
$857$	−19.8001	−0.676360	−0.338180	−	0.941081i	$-0.609811\pi$
$857$	−19.8001	−0.676360	−0.338180	+	0.941081i	$0.609811\pi$
$858$	−3.51825	−0.120111
$859$	26.0520	0.888884	0.444442	−	0.895808i	$-0.353402\pi$
$859$	26.0520	0.888884	0.444442	+	0.895808i	$0.353402\pi$
$860$	0.148215	0.00505409
$861$	−3.13371	−0.106797
$862$	−16.0589	−0.546969
$863$	−3.83076	−0.130401	−0.0652003	−	0.997872i	$-0.520769\pi$
$863$	−3.83076	−0.130401	−0.0652003	+	0.997872i	$0.520769\pi$
$864$	−5.06899	−0.172451
$865$	−0.956659	−0.0325274
$866$	18.1109	0.615435
$867$	−16.9669	−0.576225
$868$	13.8221	0.469151
$869$	−27.9992	−0.949808
$870$	−0.222187	−0.00753283
$871$	−19.2838	−0.653407
$872$	−20.2229	−0.684834
$873$	−0.769577	−0.0260462
$874$	−2.12787	−0.0719763
$875$	0.764041	0.0258293
$876$	−13.4880	−0.455716
$877$	33.0386	1.11563	0.557817	−	0.829964i	$-0.311639\pi$
$877$	33.0386	1.11563	0.557817	+	0.829964i	$0.311639\pi$
$878$	−2.69581	−0.0909791
$879$	7.86395	0.265245
$880$	−0.411624	−0.0138758
$881$	50.4715	1.70043	0.850213	−	0.526438i	$-0.176473\pi$
$881$	50.4715	1.70043	0.850213	+	0.526438i	$0.176473\pi$
$882$	−0.506041	−0.0170393
$883$	32.8223	1.10456	0.552279	−	0.833659i	$-0.313758\pi$
$883$	32.8223	1.10456	0.552279	+	0.833659i	$0.313758\pi$
$884$	−1.03663	−0.0348656
$885$	0.786840	0.0264493
$886$	13.7586	0.462231
$887$	−8.96987	−0.301179	−0.150589	−	0.988596i	$-0.548117\pi$
$887$	−8.96987	−0.301179	−0.150589	+	0.988596i	$0.548117\pi$
$888$	8.78397	0.294771
$889$	19.8131	0.664510
$890$	−0.249137	−0.00835108
$891$	2.12893	0.0713219
$892$	−20.8502	−0.698115
$893$	−1.25649	−0.0420469
$894$	7.86030	0.262888
$895$	−0.846525	−0.0282962
$896$	11.3996	0.380834
$897$	−14.0544	−0.469263
$898$	4.13211	0.137890
$899$	45.5205	1.51819
$900$	8.70942	0.290314
$901$	−0.509629	−0.0169782
$902$	3.37603	0.112409
$903$	1.11172	0.0369956
$904$	−26.7077	−0.888284
$905$	1.49700	0.0497621
$906$	−1.79061	−0.0594890
$907$	9.68899	0.321718	0.160859	−	0.986977i	$-0.448574\pi$
$907$	9.68899	0.321718	0.160859	+	0.986977i	$0.448574\pi$
$908$	37.1837	1.23399
$909$	−3.43117	−0.113805
$910$	0.126338	0.00418808
$911$	9.38182	0.310834	0.155417	−	0.987849i	$-0.450328\pi$
$911$	9.38182	0.310834	0.155417	+	0.987849i	$0.450328\pi$
$912$	−2.47112	−0.0818270
$913$	2.62434	0.0868531
$914$	1.63889	0.0542097
$915$	0.00677774	0.000224065 0
$916$	−45.2038	−1.49357
$917$	7.88735	0.260463
$918$	−0.0921091	−0.00304005
$919$	19.7199	0.650498	0.325249	−	0.945628i	$-0.394552\pi$
$919$	19.7199	0.650498	0.325249	+	0.945628i	$0.394552\pi$
$920$	0.623328	0.0205505
$921$	11.3913	0.375358
$922$	5.13032	0.168958
$923$	4.91890	0.161908
$924$	−3.71269	−0.122139
$925$	−23.1548	−0.761324
$926$	−4.04406	−0.132896
$927$	−18.6918	−0.613920
$928$	29.1127	0.955672
$929$	−17.6046	−0.577587	−0.288794	−	0.957391i	$-0.593254\pi$
$929$	−17.6046	−0.577587	−0.288794	+	0.957391i	$0.593254\pi$
$930$	−0.306621	−0.0100545
$931$	−0.977071	−0.0320222
$932$	−42.5085	−1.39241
$933$	−20.5285	−0.672073
$934$	0.131105	0.00428989
$935$	−0.0296244	−0.000968820 0
$936$	6.18717	0.202234
$937$	32.3246	1.05600	0.528000	−	0.849245i	$-0.322942\pi$
$937$	32.3246	1.05600	0.528000	+	0.849245i	$0.322942\pi$
$938$	2.98813	0.0975658
$939$	−1.03747	−0.0338567
$940$	0.171447	0.00559200
$941$	−40.9031	−1.33340	−0.666701	−	0.745325i	$-0.732294\pi$
$941$	−40.9031	−1.33340	−0.666701	+	0.745325i	$0.732294\pi$
$942$	−0.424128	−0.0138188
$943$	13.4863	0.439174
$944$	−26.0306	−0.847223
$945$	−0.0764488	−0.00248688
$946$	−1.19768	−0.0389400
$947$	11.2410	0.365283	0.182641	−	0.983180i	$-0.441535\pi$
$947$	11.2410	0.365283	0.182641	+	0.983180i	$0.441535\pi$
$948$	22.9357	0.744916
$949$	25.2579	0.819908
$950$	−2.46930	−0.0801147
$951$	14.6073	0.473675
$952$	0.344849	0.0111766
$953$	21.2449	0.688190	0.344095	−	0.938935i	$-0.388186\pi$
$953$	21.2449	0.688190	0.344095	+	0.938935i	$0.388186\pi$
$954$	1.41685	0.0458721
$955$	0.0676064	0.00218769
$956$	23.5419	0.761399
$957$	−12.2271	−0.395246
$958$	11.3122	0.365482
$959$	21.5324	0.695318
$960$	0.190595	0.00615143
$961$	31.8190	1.02642
$962$	−7.66202	−0.247033
$963$	6.63230	0.213723
$964$	51.8399	1.66965
$965$	0.307194	0.00988894
$966$	2.17781	0.0700698
$967$	25.3265	0.814445	0.407222	−	0.913329i	$-0.366497\pi$
$967$	25.3265	0.814445	0.407222	+	0.913329i	$0.366497\pi$
$968$	−12.2535	−0.393842
$969$	−0.177846	−0.00571322
$970$	−0.0297720	−0.000955922 0
$971$	22.9939	0.737908	0.368954	−	0.929448i	$-0.379716\pi$
$971$	22.9939	0.737908	0.368954	+	0.929448i	$0.379716\pi$
$972$	−1.74392	−0.0559363
$973$	14.2266	0.456083
$974$	5.02173	0.160906
$975$	−16.3095	−0.522323
$976$	−0.224224	−0.00717724
$977$	−10.4927	−0.335690	−0.167845	−	0.985813i	$-0.553681\pi$
$977$	−10.4927	−0.335690	−0.167845	+	0.985813i	$0.553681\pi$
$978$	−10.7337	−0.343227
$979$	−13.7102	−0.438179
$980$	0.133321	0.00425878
$981$	−10.6741	−0.340798
$982$	12.4872	0.398483
$983$	−21.3950	−0.682396	−0.341198	−	0.939992i	$-0.610833\pi$
$983$	−21.3950	−0.682396	−0.341198	+	0.939992i	$0.610833\pi$
$984$	−5.93706	−0.189267
$985$	0.785831	0.0250387
$986$	0.529010	0.0168471
$987$	1.28598	0.0409331
$988$	5.56458	0.177033
$989$	−4.78440	−0.152135
$990$	0.0823603	0.00261758
$991$	−32.3747	−1.02842	−0.514208	−	0.857665i	$-0.671914\pi$
$991$	−32.3747	−1.02842	−0.514208	+	0.857665i	$0.671914\pi$
$992$	40.1760	1.27559
$993$	−9.41115	−0.298654
$994$	−0.762210	−0.0241758
$995$	0.258320	0.00818931
$996$	−2.14974	−0.0681172
$997$	0.873801	0.0276736	0.0138368	−	0.999904i	$-0.495595\pi$
$997$	0.873801	0.0276736	0.0138368	+	0.999904i	$0.495595\pi$
$998$	8.75066	0.276997
$999$	4.63637	0.146688

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8043.2.a.n.1.20	✓	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8043.2.a.n.1.20	✓	40	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:	\( N \)	\(=\)	\( 8043 = 3 \cdot 7 \cdot 383 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8043.a (trivial)

\(f(q)\)	\(=\)	\(q-0.506041 q^{2} +1.00000 q^{3} -1.74392 q^{4} -0.0764488 q^{5} -0.506041 q^{6} +1.00000 q^{7} +1.89458 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.506041 q^{2} +1.00000 q^{3} -1.74392 q^{4} -0.0764488 q^{5} -0.506041 q^{6} +1.00000 q^{7} +1.89458 q^{8} +1.00000 q^{9} +0.0386862 q^{10} +2.12893 q^{11} -1.74392 q^{12} +3.26572 q^{13} -0.506041 q^{14} -0.0764488 q^{15} +2.52911 q^{16} +0.182019 q^{17} -0.506041 q^{18} -0.977071 q^{19} +0.133321 q^{20} +1.00000 q^{21} -1.07733 q^{22} -4.30362 q^{23} +1.89458 q^{24} -4.99416 q^{25} -1.65259 q^{26} +1.00000 q^{27} -1.74392 q^{28} -5.74330 q^{29} +0.0386862 q^{30} -7.92584 q^{31} -5.06899 q^{32} +2.12893 q^{33} -0.0921091 q^{34} -0.0764488 q^{35} -1.74392 q^{36} +4.63637 q^{37} +0.494438 q^{38} +3.26572 q^{39} -0.144838 q^{40} -3.13371 q^{41} -0.506041 q^{42} +1.11172 q^{43} -3.71269 q^{44} -0.0764488 q^{45} +2.17781 q^{46} +1.28598 q^{47} +2.52911 q^{48} +1.00000 q^{49} +2.52725 q^{50} +0.182019 q^{51} -5.69517 q^{52} -2.79987 q^{53} -0.506041 q^{54} -0.162754 q^{55} +1.89458 q^{56} -0.977071 q^{57} +2.90634 q^{58} -10.2924 q^{59} +0.133321 q^{60} -0.0886572 q^{61} +4.01080 q^{62} +1.00000 q^{63} -2.49311 q^{64} -0.249661 q^{65} -1.07733 q^{66} -5.90491 q^{67} -0.317427 q^{68} -4.30362 q^{69} +0.0386862 q^{70} +1.50622 q^{71} +1.89458 q^{72} +7.73426 q^{73} -2.34619 q^{74} -4.99416 q^{75} +1.70394 q^{76} +2.12893 q^{77} -1.65259 q^{78} -13.1518 q^{79} -0.193348 q^{80} +1.00000 q^{81} +1.58579 q^{82} +1.23270 q^{83} -1.74392 q^{84} -0.0139151 q^{85} -0.562574 q^{86} -5.74330 q^{87} +4.03343 q^{88} -6.43993 q^{89} +0.0386862 q^{90} +3.26572 q^{91} +7.50518 q^{92} -7.92584 q^{93} -0.650757 q^{94} +0.0746959 q^{95} -5.06899 q^{96} -0.769577 q^{97} -0.506041 q^{98} +2.12893 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 9 q^{2} + 40 q^{3} + 33 q^{4} - 27 q^{5} - 9 q^{6} + 40 q^{7} - 18 q^{8} + 40 q^{9}+O(q^{10}) \) 40 * q - 9 * q^2 + 40 * q^3 + 33 * q^4 - 27 * q^5 - 9 * q^6 + 40 * q^7 - 18 * q^8 + 40 * q^9	\( 40 q - 9 q^{2} + 40 q^{3} + 33 q^{4} - 27 q^{5} - 9 q^{6} + 40 q^{7} - 18 q^{8} + 40 q^{9} - 14 q^{10} - 29 q^{11} + 33 q^{12} - 40 q^{13} - 9 q^{14} - 27 q^{15} + 23 q^{16} - 46 q^{17} - 9 q^{18} + 17 q^{19} - 53 q^{20} + 40 q^{21} - 15 q^{22} - 62 q^{23} - 18 q^{24} + 35 q^{25} - 29 q^{26} + 40 q^{27} + 33 q^{28} - 47 q^{29} - 14 q^{30} + q^{31} - 45 q^{32} - 29 q^{33} + 9 q^{34} - 27 q^{35} + 33 q^{36} - 49 q^{37} - 52 q^{38} - 40 q^{39} - 12 q^{40} - 49 q^{41} - 9 q^{42} - 45 q^{43} - 68 q^{44} - 27 q^{45} - 36 q^{46} - 88 q^{47} + 23 q^{48} + 40 q^{49} - 32 q^{50} - 46 q^{51} - 34 q^{52} - 96 q^{53} - 9 q^{54} + 26 q^{55} - 18 q^{56} + 17 q^{57} + 12 q^{58} - 45 q^{59} - 53 q^{60} - 23 q^{61} - 15 q^{62} + 40 q^{63} + 50 q^{64} - 32 q^{65} - 15 q^{66} - 31 q^{67} - 76 q^{68} - 62 q^{69} - 14 q^{70} - 89 q^{71} - 18 q^{72} + 4 q^{73} + 10 q^{74} + 35 q^{75} + 51 q^{76} - 29 q^{77} - 29 q^{78} - 44 q^{79} - 53 q^{80} + 40 q^{81} - 15 q^{82} - 60 q^{83} + 33 q^{84} - 24 q^{85} - 9 q^{86} - 47 q^{87} - 64 q^{88} - 39 q^{89} - 14 q^{90} - 40 q^{91} - 80 q^{92} + q^{93} + 51 q^{94} - 71 q^{95} - 45 q^{96} - 21 q^{97} - 9 q^{98} - 29 q^{99}+O(q^{100}) \) 40 * q - 9 * q^2 + 40 * q^3 + 33 * q^4 - 27 * q^5 - 9 * q^6 + 40 * q^7 - 18 * q^8 + 40 * q^9 - 14 * q^10 - 29 * q^11 + 33 * q^12 - 40 * q^13 - 9 * q^14 - 27 * q^15 + 23 * q^16 - 46 * q^17 - 9 * q^18 + 17 * q^19 - 53 * q^20 + 40 * q^21 - 15 * q^22 - 62 * q^23 - 18 * q^24 + 35 * q^25 - 29 * q^26 + 40 * q^27 + 33 * q^28 - 47 * q^29 - 14 * q^30 + q^31 - 45 * q^32 - 29 * q^33 + 9 * q^34 - 27 * q^35 + 33 * q^36 - 49 * q^37 - 52 * q^38 - 40 * q^39 - 12 * q^40 - 49 * q^41 - 9 * q^42 - 45 * q^43 - 68 * q^44 - 27 * q^45 - 36 * q^46 - 88 * q^47 + 23 * q^48 + 40 * q^49 - 32 * q^50 - 46 * q^51 - 34 * q^52 - 96 * q^53 - 9 * q^54 + 26 * q^55 - 18 * q^56 + 17 * q^57 + 12 * q^58 - 45 * q^59 - 53 * q^60 - 23 * q^61 - 15 * q^62 + 40 * q^63 + 50 * q^64 - 32 * q^65 - 15 * q^66 - 31 * q^67 - 76 * q^68 - 62 * q^69 - 14 * q^70 - 89 * q^71 - 18 * q^72 + 4 * q^73 + 10 * q^74 + 35 * q^75 + 51 * q^76 - 29 * q^77 - 29 * q^78 - 44 * q^79 - 53 * q^80 + 40 * q^81 - 15 * q^82 - 60 * q^83 + 33 * q^84 - 24 * q^85 - 9 * q^86 - 47 * q^87 - 64 * q^88 - 39 * q^89 - 14 * q^90 - 40 * q^91 - 80 * q^92 + q^93 + 51 * q^94 - 71 * q^95 - 45 * q^96 - 21 * q^97 - 9 * q^98 - 29 * q^99

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