LMFDB - Embedded newform 6046.2.a.f.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6046,2,Mod(1,6046)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6046, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6046.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6046 = 2 \cdot 3023 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6046.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$48.2775530621$
Analytic rank:	$0$
Dimension:	$67$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Character	$\chi$	$=$	6046.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -3.02893 q^{3} +1.00000 q^{4} -3.61003 q^{5} -3.02893 q^{6} +0.772328 q^{7} +1.00000 q^{8} +6.17443 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -3.02893 q^{3} +1.00000 q^{4} -3.61003 q^{5} -3.02893 q^{6} +0.772328 q^{7} +1.00000 q^{8} +6.17443 q^{9} -3.61003 q^{10} +4.11696 q^{11} -3.02893 q^{12} -3.35641 q^{13} +0.772328 q^{14} +10.9345 q^{15} +1.00000 q^{16} +1.78903 q^{17} +6.17443 q^{18} +7.40068 q^{19} -3.61003 q^{20} -2.33933 q^{21} +4.11696 q^{22} -4.71134 q^{23} -3.02893 q^{24} +8.03235 q^{25} -3.35641 q^{26} -9.61513 q^{27} +0.772328 q^{28} +5.79004 q^{29} +10.9345 q^{30} +1.09806 q^{31} +1.00000 q^{32} -12.4700 q^{33} +1.78903 q^{34} -2.78813 q^{35} +6.17443 q^{36} -6.54221 q^{37} +7.40068 q^{38} +10.1663 q^{39} -3.61003 q^{40} -3.22100 q^{41} -2.33933 q^{42} -3.63904 q^{43} +4.11696 q^{44} -22.2899 q^{45} -4.71134 q^{46} -7.11603 q^{47} -3.02893 q^{48} -6.40351 q^{49} +8.03235 q^{50} -5.41883 q^{51} -3.35641 q^{52} +11.1004 q^{53} -9.61513 q^{54} -14.8624 q^{55} +0.772328 q^{56} -22.4161 q^{57} +5.79004 q^{58} -5.83020 q^{59} +10.9345 q^{60} +12.1417 q^{61} +1.09806 q^{62} +4.76868 q^{63} +1.00000 q^{64} +12.1168 q^{65} -12.4700 q^{66} -7.87060 q^{67} +1.78903 q^{68} +14.2703 q^{69} -2.78813 q^{70} +12.8834 q^{71} +6.17443 q^{72} +11.1783 q^{73} -6.54221 q^{74} -24.3294 q^{75} +7.40068 q^{76} +3.17964 q^{77} +10.1663 q^{78} -16.3236 q^{79} -3.61003 q^{80} +10.6003 q^{81} -3.22100 q^{82} -11.9000 q^{83} -2.33933 q^{84} -6.45844 q^{85} -3.63904 q^{86} -17.5376 q^{87} +4.11696 q^{88} -14.3610 q^{89} -22.2899 q^{90} -2.59225 q^{91} -4.71134 q^{92} -3.32596 q^{93} -7.11603 q^{94} -26.7167 q^{95} -3.02893 q^{96} -8.03623 q^{97} -6.40351 q^{98} +25.4199 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 67 q + 67 q^{2} + 21 q^{3} + 67 q^{4} + 21 q^{5} + 21 q^{6} + 38 q^{7} + 67 q^{8} + 90 q^{9}+O(q^{10}) $ 67 * q + 67 * q^2 + 21 * q^3 + 67 * q^4 + 21 * q^5 + 21 * q^6 + 38 * q^7 + 67 * q^8 + 90 * q^9	\( 67 q + 67 q^{2} + 21 q^{3} + 67 q^{4} + 21 q^{5} + 21 q^{6} + 38 q^{7} + 67 q^{8} + 90 q^{9} + 21 q^{10} + 56 q^{11} + 21 q^{12} + 33 q^{13} + 38 q^{14} + 25 q^{15} + 67 q^{16} + 30 q^{17} + 90 q^{18} + 36 q^{19} + 21 q^{20} + 20 q^{21} + 56 q^{22} + 65 q^{23} + 21 q^{24} + 72 q^{25} + 33 q^{26} + 57 q^{27} + 38 q^{28} + 84 q^{29} + 25 q^{30} + 52 q^{31} + 67 q^{32} - 9 q^{33} + 30 q^{34} + 30 q^{35} + 90 q^{36} + 52 q^{37} + 36 q^{38} + 41 q^{39} + 21 q^{40} + 46 q^{41} + 20 q^{42} + 61 q^{43} + 56 q^{44} + 23 q^{45} + 65 q^{46} + 51 q^{47} + 21 q^{48} + 81 q^{49} + 72 q^{50} + 33 q^{51} + 33 q^{52} + 72 q^{53} + 57 q^{54} + 14 q^{55} + 38 q^{56} - 26 q^{57} + 84 q^{58} + 71 q^{59} + 25 q^{60} + 42 q^{61} + 52 q^{62} + 63 q^{63} + 67 q^{64} - 2 q^{65} - 9 q^{66} + 70 q^{67} + 30 q^{68} + 21 q^{69} + 30 q^{70} + 104 q^{71} + 90 q^{72} - 31 q^{73} + 52 q^{74} + 69 q^{75} + 36 q^{76} + 48 q^{77} + 41 q^{78} + 79 q^{79} + 21 q^{80} + 123 q^{81} + 46 q^{82} + 41 q^{83} + 20 q^{84} + 6 q^{85} + 61 q^{86} + 19 q^{87} + 56 q^{88} + 58 q^{89} + 23 q^{90} + 31 q^{91} + 65 q^{92} + 13 q^{93} + 51 q^{94} + 77 q^{95} + 21 q^{96} - 8 q^{97} + 81 q^{98} + 129 q^{99}+O(q^{100}) \) 67 * q + 67 * q^2 + 21 * q^3 + 67 * q^4 + 21 * q^5 + 21 * q^6 + 38 * q^7 + 67 * q^8 + 90 * q^9 + 21 * q^10 + 56 * q^11 + 21 * q^12 + 33 * q^13 + 38 * q^14 + 25 * q^15 + 67 * q^16 + 30 * q^17 + 90 * q^18 + 36 * q^19 + 21 * q^20 + 20 * q^21 + 56 * q^22 + 65 * q^23 + 21 * q^24 + 72 * q^25 + 33 * q^26 + 57 * q^27 + 38 * q^28 + 84 * q^29 + 25 * q^30 + 52 * q^31 + 67 * q^32 - 9 * q^33 + 30 * q^34 + 30 * q^35 + 90 * q^36 + 52 * q^37 + 36 * q^38 + 41 * q^39 + 21 * q^40 + 46 * q^41 + 20 * q^42 + 61 * q^43 + 56 * q^44 + 23 * q^45 + 65 * q^46 + 51 * q^47 + 21 * q^48 + 81 * q^49 + 72 * q^50 + 33 * q^51 + 33 * q^52 + 72 * q^53 + 57 * q^54 + 14 * q^55 + 38 * q^56 - 26 * q^57 + 84 * q^58 + 71 * q^59 + 25 * q^60 + 42 * q^61 + 52 * q^62 + 63 * q^63 + 67 * q^64 - 2 * q^65 - 9 * q^66 + 70 * q^67 + 30 * q^68 + 21 * q^69 + 30 * q^70 + 104 * q^71 + 90 * q^72 - 31 * q^73 + 52 * q^74 + 69 * q^75 + 36 * q^76 + 48 * q^77 + 41 * q^78 + 79 * q^79 + 21 * q^80 + 123 * q^81 + 46 * q^82 + 41 * q^83 + 20 * q^84 + 6 * q^85 + 61 * q^86 + 19 * q^87 + 56 * q^88 + 58 * q^89 + 23 * q^90 + 31 * q^91 + 65 * q^92 + 13 * q^93 + 51 * q^94 + 77 * q^95 + 21 * q^96 - 8 * q^97 + 81 * q^98 + 129 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−3.02893	−1.74875	−0.874377	−	0.485247i	$-0.838730\pi$
$3$	−3.02893	−1.74875	−0.874377	+	0.485247i	$0.838730\pi$
$4$	1.00000	0.500000
$5$	−3.61003	−1.61446	−0.807228	−	0.590239i	$-0.799033\pi$
$5$	−3.61003	−1.61446	−0.807228	+	0.590239i	$0.799033\pi$
$6$	−3.02893	−1.23656
$7$	0.772328	0.291913	0.145956	−	0.989291i	$-0.453374\pi$
$7$	0.772328	0.291913	0.145956	+	0.989291i	$0.453374\pi$
$8$	1.00000	0.353553
$9$	6.17443	2.05814
$10$	−3.61003	−1.14159
$11$	4.11696	1.24131	0.620655	−	0.784083i	$-0.286867\pi$
$11$	4.11696	1.24131	0.620655	+	0.784083i	$0.286867\pi$
$12$	−3.02893	−0.874377
$13$	−3.35641	−0.930900	−0.465450	−	0.885074i	$-0.654108\pi$
$13$	−3.35641	−0.930900	−0.465450	+	0.885074i	$0.654108\pi$
$14$	0.772328	0.206413
$15$	10.9345	2.82329
$16$	1.00000	0.250000
$17$	1.78903	0.433902	0.216951	−	0.976182i	$-0.430389\pi$
$17$	1.78903	0.433902	0.216951	+	0.976182i	$0.430389\pi$
$18$	6.17443	1.45533
$19$	7.40068	1.69783	0.848916	−	0.528528i	$-0.177256\pi$
$19$	7.40068	1.69783	0.848916	+	0.528528i	$0.177256\pi$
$20$	−3.61003	−0.807228
$21$	−2.33933	−0.510483
$22$	4.11696	0.877739
$23$	−4.71134	−0.982382	−0.491191	−	0.871052i	$-0.663438\pi$
$23$	−4.71134	−0.982382	−0.491191	+	0.871052i	$0.663438\pi$
$24$	−3.02893	−0.618278
$25$	8.03235	1.60647
$26$	−3.35641	−0.658246
$27$	−9.61513	−1.85043
$28$	0.772328	0.145956
$29$	5.79004	1.07518	0.537592	−	0.843205i	$-0.319334\pi$
$29$	5.79004	1.07518	0.537592	+	0.843205i	$0.319334\pi$
$30$	10.9345	1.99637
$31$	1.09806	0.197218	0.0986090	−	0.995126i	$-0.468561\pi$
$31$	1.09806	0.197218	0.0986090	+	0.995126i	$0.468561\pi$
$32$	1.00000	0.176777
$33$	−12.4700	−2.17075
$34$	1.78903	0.306815
$35$	−2.78813	−0.471280
$36$	6.17443	1.02907
$37$	−6.54221	−1.07553	−0.537767	−	0.843094i	$-0.680732\pi$
$37$	−6.54221	−1.07553	−0.537767	+	0.843094i	$0.680732\pi$
$38$	7.40068	1.20055
$39$	10.1663	1.62792
$40$	−3.61003	−0.570797
$41$	−3.22100	−0.503036	−0.251518	−	0.967853i	$-0.580930\pi$
$41$	−3.22100	−0.503036	−0.251518	+	0.967853i	$0.580930\pi$
$42$	−2.33933	−0.360966
$43$	−3.63904	−0.554948	−0.277474	−	0.960733i	$-0.589497\pi$
$43$	−3.63904	−0.554948	−0.277474	+	0.960733i	$0.589497\pi$
$44$	4.11696	0.620655
$45$	−22.2899	−3.32278
$46$	−4.71134	−0.694649
$47$	−7.11603	−1.03798	−0.518990	−	0.854780i	$-0.673692\pi$
$47$	−7.11603	−1.03798	−0.518990	+	0.854780i	$0.673692\pi$
$48$	−3.02893	−0.437189
$49$	−6.40351	−0.914787
$50$	8.03235	1.13595
$51$	−5.41883	−0.758789
$52$	−3.35641	−0.465450
$53$	11.1004	1.52476	0.762380	−	0.647129i	$-0.224031\pi$
$53$	11.1004	1.52476	0.762380	+	0.647129i	$0.224031\pi$
$54$	−9.61513	−1.30845
$55$	−14.8624	−2.00404
$56$	0.772328	0.103207
$57$	−22.4161	−2.96909
$58$	5.79004	0.760269
$59$	−5.83020	−0.759027	−0.379514	−	0.925186i	$-0.623909\pi$
$59$	−5.83020	−0.759027	−0.379514	+	0.925186i	$0.623909\pi$
$60$	10.9345	1.41164
$61$	12.1417	1.55459	0.777293	−	0.629139i	$-0.216592\pi$
$61$	12.1417	1.55459	0.777293	+	0.629139i	$0.216592\pi$
$62$	1.09806	0.139454
$63$	4.76868	0.600798
$64$	1.00000	0.125000
$65$	12.1168	1.50290
$66$	−12.4700	−1.53495
$67$	−7.87060	−0.961547	−0.480774	−	0.876845i	$-0.659644\pi$
$67$	−7.87060	−0.961547	−0.480774	+	0.876845i	$0.659644\pi$
$68$	1.78903	0.216951
$69$	14.2703	1.71795
$70$	−2.78813	−0.333245
$71$	12.8834	1.52898	0.764491	−	0.644634i	$-0.222990\pi$
$71$	12.8834	1.52898	0.764491	+	0.644634i	$0.222990\pi$
$72$	6.17443	0.727663
$73$	11.1783	1.30833	0.654163	−	0.756354i	$-0.273021\pi$
$73$	11.1783	1.30833	0.654163	+	0.756354i	$0.273021\pi$
$74$	−6.54221	−0.760517
$75$	−24.3294	−2.80932
$76$	7.40068	0.848916
$77$	3.17964	0.362354
$78$	10.1663	1.15111
$79$	−16.3236	−1.83654	−0.918272	−	0.395951i	$-0.870415\pi$
$79$	−16.3236	−1.83654	−0.918272	+	0.395951i	$0.870415\pi$
$80$	−3.61003	−0.403614
$81$	10.6003	1.17781
$82$	−3.22100	−0.355700
$83$	−11.9000	−1.30619	−0.653095	−	0.757276i	$-0.726530\pi$
$83$	−11.9000	−1.30619	−0.653095	+	0.757276i	$0.726530\pi$
$84$	−2.33933	−0.255242
$85$	−6.45844	−0.700516
$86$	−3.63904	−0.392407
$87$	−17.5376	−1.88023
$88$	4.11696	0.438870
$89$	−14.3610	−1.52226	−0.761132	−	0.648597i	$-0.775356\pi$
$89$	−14.3610	−1.52226	−0.761132	+	0.648597i	$0.775356\pi$
$90$	−22.2899	−2.34956
$91$	−2.59225	−0.271741
$92$	−4.71134	−0.491191
$93$	−3.32596	−0.344886
$94$	−7.11603	−0.733962
$95$	−26.7167	−2.74108
$96$	−3.02893	−0.309139
$97$	−8.03623	−0.815956	−0.407978	−	0.912992i	$-0.633766\pi$
$97$	−8.03623	−0.815956	−0.407978	+	0.912992i	$0.633766\pi$
$98$	−6.40351	−0.646852
$99$	25.4199	2.55479
$100$	8.03235	0.803235
$101$	−0.535057	−0.0532401	−0.0266201	−	0.999646i	$-0.508474\pi$
$101$	−0.535057	−0.0532401	−0.0266201	+	0.999646i	$0.508474\pi$
$102$	−5.41883	−0.536545
$103$	9.08396	0.895069	0.447534	−	0.894267i	$-0.352302\pi$
$103$	9.08396	0.895069	0.447534	+	0.894267i	$0.352302\pi$
$104$	−3.35641	−0.329123
$105$	8.44506	0.824153
$106$	11.1004	1.07817
$107$	18.0535	1.74530	0.872649	−	0.488348i	$-0.162400\pi$
$107$	18.0535	1.74530	0.872649	+	0.488348i	$0.162400\pi$
$108$	−9.61513	−0.925216
$109$	20.6088	1.97396	0.986981	−	0.160837i	$-0.0514193\pi$
$109$	20.6088	1.97396	0.986981	+	0.160837i	$0.0514193\pi$
$110$	−14.8624	−1.41707
$111$	19.8159	1.88084
$112$	0.772328	0.0729781
$113$	−0.635364	−0.0597700	−0.0298850	−	0.999553i	$-0.509514\pi$
$113$	−0.635364	−0.0597700	−0.0298850	+	0.999553i	$0.509514\pi$
$114$	−22.4161	−2.09946
$115$	17.0081	1.58601
$116$	5.79004	0.537592
$117$	−20.7239	−1.91593
$118$	−5.83020	−0.536713
$119$	1.38171	0.126662
$120$	10.9345	0.998183
$121$	5.94938	0.540852
$122$	12.1417	1.09926
$123$	9.75620	0.879687
$124$	1.09806	0.0986090
$125$	−10.9469	−0.979119
$126$	4.76868	0.424828
$127$	−2.09599	−0.185989	−0.0929946	−	0.995667i	$-0.529644\pi$
$127$	−2.09599	−0.185989	−0.0929946	+	0.995667i	$0.529644\pi$
$128$	1.00000	0.0883883
$129$	11.0224	0.970467
$130$	12.1168	1.06271
$131$	1.85297	0.161895	0.0809473	−	0.996718i	$-0.474205\pi$
$131$	1.85297	0.161895	0.0809473	+	0.996718i	$0.474205\pi$
$132$	−12.4700	−1.08537
$133$	5.71575	0.495618
$134$	−7.87060	−0.679917
$135$	34.7109	2.98744
$136$	1.78903	0.153408
$137$	3.23610	0.276479	0.138240	−	0.990399i	$-0.455856\pi$
$137$	3.23610	0.276479	0.138240	+	0.990399i	$0.455856\pi$
$138$	14.2703	1.21477
$139$	−18.6236	−1.57963	−0.789817	−	0.613342i	$-0.789825\pi$
$139$	−18.6236	−1.57963	−0.789817	+	0.613342i	$0.789825\pi$
$140$	−2.78813	−0.235640
$141$	21.5540	1.81517
$142$	12.8834	1.08115
$143$	−13.8182	−1.15554
$144$	6.17443	0.514536
$145$	−20.9022	−1.73584
$146$	11.1783	0.925126
$147$	19.3958	1.59974
$148$	−6.54221	−0.537767
$149$	13.4584	1.10255	0.551277	−	0.834322i	$-0.314141\pi$
$149$	13.4584	1.10255	0.551277	+	0.834322i	$0.314141\pi$
$150$	−24.3294	−1.98649
$151$	8.38784	0.682592	0.341296	−	0.939956i	$-0.389134\pi$
$151$	8.38784	0.682592	0.341296	+	0.939956i	$0.389134\pi$
$152$	7.40068	0.600274
$153$	11.0462	0.893033
$154$	3.17964	0.256223
$155$	−3.96405	−0.318400
$156$	10.1663	0.813958
$157$	6.12890	0.489140	0.244570	−	0.969632i	$-0.421353\pi$
$157$	6.12890	0.489140	0.244570	+	0.969632i	$0.421353\pi$
$158$	−16.3236	−1.29863
$159$	−33.6224	−2.66643
$160$	−3.61003	−0.285398
$161$	−3.63870	−0.286770
$162$	10.6003	0.832836
$163$	9.71610	0.761024	0.380512	−	0.924776i	$-0.375748\pi$
$163$	9.71610	0.761024	0.380512	+	0.924776i	$0.375748\pi$
$164$	−3.22100	−0.251518
$165$	45.0171	3.50458
$166$	−11.9000	−0.923616
$167$	8.05320	0.623176	0.311588	−	0.950217i	$-0.399139\pi$
$167$	8.05320	0.623176	0.311588	+	0.950217i	$0.399139\pi$
$168$	−2.33933	−0.180483
$169$	−1.73452	−0.133425
$170$	−6.45844	−0.495340
$171$	45.6949	3.49438
$172$	−3.63904	−0.277474
$173$	−1.53021	−0.116340	−0.0581698	−	0.998307i	$-0.518526\pi$
$173$	−1.53021	−0.116340	−0.0581698	+	0.998307i	$0.518526\pi$
$174$	−17.5376	−1.32952
$175$	6.20361	0.468949
$176$	4.11696	0.310328
$177$	17.6593	1.32735
$178$	−14.3610	−1.07640
$179$	19.5884	1.46410	0.732051	−	0.681250i	$-0.238563\pi$
$179$	19.5884	1.46410	0.732051	+	0.681250i	$0.238563\pi$
$180$	−22.2899	−1.66139
$181$	4.41090	0.327859	0.163930	−	0.986472i	$-0.447583\pi$
$181$	4.41090	0.327859	0.163930	+	0.986472i	$0.447583\pi$
$182$	−2.59225	−0.192150
$183$	−36.7764	−2.71859
$184$	−4.71134	−0.347325
$185$	23.6176	1.73640
$186$	−3.32596	−0.243871
$187$	7.36535	0.538608
$188$	−7.11603	−0.518990
$189$	−7.42603	−0.540164
$190$	−26.7167	−1.93823
$191$	16.0385	1.16051	0.580253	−	0.814437i	$-0.302954\pi$
$191$	16.0385	1.16051	0.580253	+	0.814437i	$0.302954\pi$
$192$	−3.02893	−0.218594
$193$	−18.7416	−1.34905	−0.674527	−	0.738250i	$-0.735652\pi$
$193$	−18.7416	−1.34905	−0.674527	+	0.738250i	$0.735652\pi$
$194$	−8.03623	−0.576968
$195$	−36.7008	−2.62820
$196$	−6.40351	−0.457394
$197$	−1.80509	−0.128608	−0.0643038	−	0.997930i	$-0.520483\pi$
$197$	−1.80509	−0.128608	−0.0643038	+	0.997930i	$0.520483\pi$
$198$	25.4199	1.80651
$199$	−13.3082	−0.943394	−0.471697	−	0.881761i	$-0.656358\pi$
$199$	−13.3082	−0.943394	−0.471697	+	0.881761i	$0.656358\pi$
$200$	8.03235	0.567973
$201$	23.8395	1.68151
$202$	−0.535057	−0.0376464
$203$	4.47181	0.313859
$204$	−5.41883	−0.379394
$205$	11.6279	0.812130
$206$	9.08396	0.632909
$207$	−29.0898	−2.02188
$208$	−3.35641	−0.232725
$209$	30.4683	2.10754
$210$	8.44506	0.582764
$211$	5.51899	0.379943	0.189972	−	0.981790i	$-0.439160\pi$
$211$	5.51899	0.379943	0.189972	+	0.981790i	$0.439160\pi$
$212$	11.1004	0.762380
$213$	−39.0231	−2.67382
$214$	18.0535	1.23411
$215$	13.1370	0.895939
$216$	−9.61513	−0.654226
$217$	0.848065	0.0575704
$218$	20.6088	1.39580
$219$	−33.8584	−2.28794
$220$	−14.8624	−1.00202
$221$	−6.00470	−0.403920
$222$	19.8159	1.32996
$223$	−9.18970	−0.615388	−0.307694	−	0.951485i	$-0.599557\pi$
$223$	−9.18970	−0.615388	−0.307694	+	0.951485i	$0.599557\pi$
$224$	0.772328	0.0516033
$225$	49.5952	3.30634
$226$	−0.635364	−0.0422638
$227$	12.7855	0.848603	0.424301	−	0.905521i	$-0.360520\pi$
$227$	12.7855	0.848603	0.424301	+	0.905521i	$0.360520\pi$
$228$	−22.4161	−1.48455
$229$	8.61768	0.569472	0.284736	−	0.958606i	$-0.408094\pi$
$229$	8.61768	0.569472	0.284736	+	0.958606i	$0.408094\pi$
$230$	17.0081	1.12148
$231$	−9.63093	−0.633668
$232$	5.79004	0.380135
$233$	−7.55167	−0.494727	−0.247363	−	0.968923i	$-0.579564\pi$
$233$	−7.55167	−0.494727	−0.247363	+	0.968923i	$0.579564\pi$
$234$	−20.7239	−1.35476
$235$	25.6891	1.67577
$236$	−5.83020	−0.379514
$237$	49.4429	3.21166
$238$	1.38171	0.0895632
$239$	13.1414	0.850047	0.425024	−	0.905182i	$-0.360266\pi$
$239$	13.1414	0.850047	0.425024	+	0.905182i	$0.360266\pi$
$240$	10.9345	0.705822
$241$	5.68263	0.366051	0.183025	−	0.983108i	$-0.441411\pi$
$241$	5.68263	0.366051	0.183025	+	0.983108i	$0.441411\pi$
$242$	5.94938	0.382440
$243$	−3.26214	−0.209266
$244$	12.1417	0.777293
$245$	23.1169	1.47688
$246$	9.75620	0.622033
$247$	−24.8397	−1.58051
$248$	1.09806	0.0697271
$249$	36.0442	2.28421
$250$	−10.9469	−0.692341
$251$	−8.80361	−0.555679	−0.277839	−	0.960628i	$-0.589618\pi$
$251$	−8.80361	−0.555679	−0.277839	+	0.960628i	$0.589618\pi$
$252$	4.76868	0.300399
$253$	−19.3964	−1.21944
$254$	−2.09599	−0.131514
$255$	19.5622	1.22503
$256$	1.00000	0.0625000
$257$	−12.6343	−0.788106	−0.394053	−	0.919088i	$-0.628927\pi$
$257$	−12.6343	−0.788106	−0.394053	+	0.919088i	$0.628927\pi$
$258$	11.0224	0.686224
$259$	−5.05273	−0.313962
$260$	12.1168	0.751449
$261$	35.7502	2.21288
$262$	1.85297	0.114477
$263$	28.0483	1.72953	0.864765	−	0.502176i	$-0.167467\pi$
$263$	28.0483	1.72953	0.864765	+	0.502176i	$0.167467\pi$
$264$	−12.4700	−0.767475
$265$	−40.0729	−2.46166
$266$	5.71575	0.350455
$267$	43.4985	2.66207
$268$	−7.87060	−0.480774
$269$	9.11937	0.556018	0.278009	−	0.960578i	$-0.410326\pi$
$269$	9.11937	0.556018	0.278009	+	0.960578i	$0.410326\pi$
$270$	34.7109	2.11244
$271$	12.4816	0.758202	0.379101	−	0.925355i	$-0.376233\pi$
$271$	12.4816	0.758202	0.379101	+	0.925355i	$0.376233\pi$
$272$	1.78903	0.108476
$273$	7.85174	0.475209
$274$	3.23610	0.195500
$275$	33.0689	1.99413
$276$	14.2703	0.858973
$277$	−12.7735	−0.767485	−0.383743	−	0.923440i	$-0.625365\pi$
$277$	−12.7735	−0.767485	−0.383743	+	0.923440i	$0.625365\pi$
$278$	−18.6236	−1.11697
$279$	6.77991	0.405903
$280$	−2.78813	−0.166623
$281$	−3.81431	−0.227543	−0.113771	−	0.993507i	$-0.536293\pi$
$281$	−3.81431	−0.227543	−0.113771	+	0.993507i	$0.536293\pi$
$282$	21.5540	1.28352
$283$	0.741675	0.0440880	0.0220440	−	0.999757i	$-0.492983\pi$
$283$	0.741675	0.0440880	0.0220440	+	0.999757i	$0.492983\pi$
$284$	12.8834	0.764491
$285$	80.9231	4.79347
$286$	−13.8182	−0.817088
$287$	−2.48767	−0.146843
$288$	6.17443	0.363832
$289$	−13.7994	−0.811729
$290$	−20.9022	−1.22742
$291$	24.3412	1.42691
$292$	11.1783	0.654163
$293$	−16.6692	−0.973826	−0.486913	−	0.873451i	$-0.661877\pi$
$293$	−16.6692	−0.973826	−0.486913	+	0.873451i	$0.661877\pi$
$294$	19.3958	1.13119
$295$	21.0472	1.22542
$296$	−6.54221	−0.380258
$297$	−39.5851	−2.29696
$298$	13.4584	0.779623
$299$	15.8132	0.914500
$300$	−24.3294	−1.40466
$301$	−2.81053	−0.161996
$302$	8.38784	0.482666
$303$	1.62065	0.0931039
$304$	7.40068	0.424458
$305$	−43.8320	−2.50981
$306$	11.0462	0.631470
$307$	24.0996	1.37544	0.687718	−	0.725978i	$-0.258612\pi$
$307$	24.0996	1.37544	0.687718	+	0.725978i	$0.258612\pi$
$308$	3.17964	0.181177
$309$	−27.5147	−1.56526
$310$	−3.96405	−0.225143
$311$	−10.1640	−0.576348	−0.288174	−	0.957578i	$-0.593048\pi$
$311$	−10.1640	−0.576348	−0.288174	+	0.957578i	$0.593048\pi$
$312$	10.1663	0.575555
$313$	14.4141	0.814732	0.407366	−	0.913265i	$-0.366447\pi$
$313$	14.4141	0.814732	0.407366	+	0.913265i	$0.366447\pi$
$314$	6.12890	0.345874
$315$	−17.2151	−0.969962
$316$	−16.3236	−0.918272
$317$	−1.58165	−0.0888345	−0.0444172	−	0.999013i	$-0.514143\pi$
$317$	−1.58165	−0.0888345	−0.0444172	+	0.999013i	$0.514143\pi$
$318$	−33.6224	−1.88545
$319$	23.8374	1.33464
$320$	−3.61003	−0.201807
$321$	−54.6828	−3.05210
$322$	−3.63870	−0.202777
$323$	13.2400	0.736693
$324$	10.6003	0.588904
$325$	−26.9598	−1.49546
$326$	9.71610	0.538125
$327$	−62.4226	−3.45198
$328$	−3.22100	−0.177850
$329$	−5.49591	−0.302999
$330$	45.0171	2.47811
$331$	−1.58444	−0.0870888	−0.0435444	−	0.999051i	$-0.513865\pi$
$331$	−1.58444	−0.0870888	−0.0435444	+	0.999051i	$0.513865\pi$
$332$	−11.9000	−0.653095
$333$	−40.3944	−2.21360
$334$	8.05320	0.440652
$335$	28.4131	1.55238
$336$	−2.33933	−0.127621
$337$	−15.0854	−0.821752	−0.410876	−	0.911691i	$-0.634777\pi$
$337$	−15.0854	−0.821752	−0.410876	+	0.911691i	$0.634777\pi$
$338$	−1.73452	−0.0943455
$339$	1.92447	0.104523
$340$	−6.45844	−0.350258
$341$	4.52069	0.244809
$342$	45.6949	2.47090
$343$	−10.3519	−0.558950
$344$	−3.63904	−0.196204
$345$	−51.5164	−2.77355
$346$	−1.53021	−0.0822645
$347$	21.5861	1.15880	0.579401	−	0.815043i	$-0.303287\pi$
$347$	21.5861	1.15880	0.579401	+	0.815043i	$0.303287\pi$
$348$	−17.5376	−0.940116
$349$	−19.4566	−1.04149	−0.520745	−	0.853712i	$-0.674346\pi$
$349$	−19.4566	−1.04149	−0.520745	+	0.853712i	$0.674346\pi$
$350$	6.20361	0.331597
$351$	32.2723	1.72257
$352$	4.11696	0.219435
$353$	−27.2505	−1.45040	−0.725198	−	0.688541i	$-0.758252\pi$
$353$	−27.2505	−1.45040	−0.725198	+	0.688541i	$0.758252\pi$
$354$	17.6593	0.938580
$355$	−46.5097	−2.46848
$356$	−14.3610	−0.761132
$357$	−4.18512	−0.221500
$358$	19.5884	1.03528
$359$	8.51153	0.449221	0.224611	−	0.974449i	$-0.427889\pi$
$359$	8.51153	0.449221	0.224611	+	0.974449i	$0.427889\pi$
$360$	−22.2899	−1.17478
$361$	35.7700	1.88263
$362$	4.41090	0.231832
$363$	−18.0203	−0.945818
$364$	−2.59225	−0.135871
$365$	−40.3542	−2.11224
$366$	−36.7764	−1.92233
$367$	33.9406	1.77168	0.885841	−	0.463988i	$-0.153582\pi$
$367$	33.9406	1.77168	0.885841	+	0.463988i	$0.153582\pi$
$368$	−4.71134	−0.245596
$369$	−19.8879	−1.03532
$370$	23.6176	1.22782
$371$	8.57317	0.445097
$372$	−3.32596	−0.172443
$373$	−8.31065	−0.430309	−0.215154	−	0.976580i	$-0.569026\pi$
$373$	−8.31065	−0.430309	−0.215154	+	0.976580i	$0.569026\pi$
$374$	7.36535	0.380853
$375$	33.1574	1.71224
$376$	−7.11603	−0.366981
$377$	−19.4337	−1.00089
$378$	−7.42603	−0.381954
$379$	36.3581	1.86759	0.933795	−	0.357808i	$-0.116476\pi$
$379$	36.3581	1.86759	0.933795	+	0.357808i	$0.116476\pi$
$380$	−26.7167	−1.37054
$381$	6.34862	0.325249
$382$	16.0385	0.820601
$383$	26.3657	1.34722	0.673611	−	0.739086i	$-0.264742\pi$
$383$	26.3657	1.34722	0.673611	+	0.739086i	$0.264742\pi$
$384$	−3.02893	−0.154570
$385$	−11.4786	−0.585005
$386$	−18.7416	−0.953925
$387$	−22.4690	−1.14216
$388$	−8.03623	−0.407978
$389$	36.5001	1.85063	0.925315	−	0.379199i	$-0.123801\pi$
$389$	36.5001	1.85063	0.925315	+	0.379199i	$0.123801\pi$
$390$	−36.7008	−1.85842
$391$	−8.42870	−0.426258
$392$	−6.40351	−0.323426
$393$	−5.61252	−0.283114
$394$	−1.80509	−0.0909393
$395$	58.9286	2.96502
$396$	25.4199	1.27740
$397$	33.5994	1.68631	0.843154	−	0.537672i	$-0.180696\pi$
$397$	33.5994	1.68631	0.843154	+	0.537672i	$0.180696\pi$
$398$	−13.3082	−0.667080
$399$	−17.3126	−0.866715
$400$	8.03235	0.401617
$401$	34.8436	1.74001	0.870004	−	0.493044i	$-0.164116\pi$
$401$	34.8436	1.74001	0.870004	+	0.493044i	$0.164116\pi$
$402$	23.8395	1.18901
$403$	−3.68555	−0.183590
$404$	−0.535057	−0.0266201
$405$	−38.2674	−1.90152
$406$	4.47181	0.221932
$407$	−26.9340	−1.33507
$408$	−5.41883	−0.268272
$409$	38.8971	1.92334	0.961668	−	0.274217i	$-0.0884187\pi$
$409$	38.8971	1.92334	0.961668	+	0.274217i	$0.0884187\pi$
$410$	11.6279	0.574263
$411$	−9.80194	−0.483494
$412$	9.08396	0.447534
$413$	−4.50283	−0.221570
$414$	−29.0898	−1.42969
$415$	42.9593	2.10879
$416$	−3.35641	−0.164561
$417$	56.4097	2.76239
$418$	30.4683	1.49025
$419$	15.0420	0.734848	0.367424	−	0.930053i	$-0.380240\pi$
$419$	15.0420	0.734848	0.367424	+	0.930053i	$0.380240\pi$
$420$	8.44506	0.412077
$421$	−6.75253	−0.329098	−0.164549	−	0.986369i	$-0.552617\pi$
$421$	−6.75253	−0.329098	−0.164549	+	0.986369i	$0.552617\pi$
$422$	5.51899	0.268660
$423$	−43.9374	−2.13631
$424$	11.1004	0.539084
$425$	14.3701	0.697051
$426$	−39.0231	−1.89067
$427$	9.37737	0.453803
$428$	18.0535	0.872649
$429$	41.8544	2.02075
$430$	13.1370	0.633525
$431$	−34.8373	−1.67805	−0.839026	−	0.544092i	$-0.816874\pi$
$431$	−34.8373	−1.67805	−0.839026	+	0.544092i	$0.816874\pi$
$432$	−9.61513	−0.462608
$433$	5.77818	0.277681	0.138841	−	0.990315i	$-0.455662\pi$
$433$	5.77818	0.277681	0.138841	+	0.990315i	$0.455662\pi$
$434$	0.848065	0.0407084
$435$	63.3114	3.03555
$436$	20.6088	0.986981
$437$	−34.8671	−1.66792
$438$	−33.8584	−1.61782
$439$	−13.7709	−0.657247	−0.328623	−	0.944461i	$-0.606585\pi$
$439$	−13.7709	−0.657247	−0.328623	+	0.944461i	$0.606585\pi$
$440$	−14.8624	−0.708536
$441$	−39.5380	−1.88276
$442$	−6.00470	−0.285614
$443$	−22.6228	−1.07484	−0.537422	−	0.843314i	$-0.680602\pi$
$443$	−22.6228	−1.07484	−0.537422	+	0.843314i	$0.680602\pi$
$444$	19.8159	0.940422
$445$	51.8437	2.45763
$446$	−9.18970	−0.435145
$447$	−40.7645	−1.92810
$448$	0.772328	0.0364891
$449$	−8.52906	−0.402511	−0.201256	−	0.979539i	$-0.564502\pi$
$449$	−8.52906	−0.402511	−0.201256	+	0.979539i	$0.564502\pi$
$450$	49.5952	2.33794
$451$	−13.2607	−0.624424
$452$	−0.635364	−0.0298850
$453$	−25.4062	−1.19369
$454$	12.7855	0.600053
$455$	9.35810	0.438715
$456$	−22.4161	−1.04973
$457$	−30.1252	−1.40920	−0.704599	−	0.709605i	$-0.748873\pi$
$457$	−30.1252	−1.40920	−0.704599	+	0.709605i	$0.748873\pi$
$458$	8.61768	0.402678
$459$	−17.2017	−0.802907
$460$	17.0081	0.793007
$461$	11.8267	0.550824	0.275412	−	0.961326i	$-0.411186\pi$
$461$	11.8267	0.550824	0.275412	+	0.961326i	$0.411186\pi$
$462$	−9.63093	−0.448071
$463$	27.2504	1.26644	0.633218	−	0.773974i	$-0.281734\pi$
$463$	27.2504	1.26644	0.633218	+	0.773974i	$0.281734\pi$
$464$	5.79004	0.268796
$465$	12.0068	0.556803
$466$	−7.55167	−0.349825
$467$	25.8445	1.19594	0.597971	−	0.801518i	$-0.295974\pi$
$467$	25.8445	1.19594	0.597971	+	0.801518i	$0.295974\pi$
$468$	−20.7239	−0.957963
$469$	−6.07869	−0.280688
$470$	25.6891	1.18495
$471$	−18.5640	−0.855385
$472$	−5.83020	−0.268357
$473$	−14.9818	−0.688863
$474$	49.4429	2.27099
$475$	59.4448	2.72751
$476$	1.38171	0.0633308
$477$	68.5388	3.13817
$478$	13.1414	0.601074
$479$	13.6620	0.624234	0.312117	−	0.950044i	$-0.398962\pi$
$479$	13.6620	0.624234	0.312117	+	0.950044i	$0.398962\pi$
$480$	10.9345	0.499092
$481$	21.9583	1.00121
$482$	5.68263	0.258837
$483$	11.0214	0.501490
$484$	5.94938	0.270426
$485$	29.0111	1.31733
$486$	−3.26214	−0.147974
$487$	−20.7001	−0.938012	−0.469006	−	0.883195i	$-0.655388\pi$
$487$	−20.7001	−0.938012	−0.469006	+	0.883195i	$0.655388\pi$
$488$	12.1417	0.549629
$489$	−29.4294	−1.33084
$490$	23.1169	1.04431
$491$	6.73929	0.304140	0.152070	−	0.988370i	$-0.451406\pi$
$491$	6.73929	0.304140	0.152070	+	0.988370i	$0.451406\pi$
$492$	9.75620	0.439843
$493$	10.3585	0.466524
$494$	−24.8397	−1.11759
$495$	−91.7667	−4.12460
$496$	1.09806	0.0493045
$497$	9.95024	0.446329
$498$	36.0442	1.61518
$499$	−35.4419	−1.58660	−0.793299	−	0.608832i	$-0.791638\pi$
$499$	−35.4419	−1.58660	−0.793299	+	0.608832i	$0.791638\pi$
$500$	−10.9469	−0.489559
$501$	−24.3926	−1.08978
$502$	−8.80361	−0.392924
$503$	1.31732	0.0587362	0.0293681	−	0.999569i	$-0.490650\pi$
$503$	1.31732	0.0587362	0.0293681	+	0.999569i	$0.490650\pi$
$504$	4.76868	0.212414
$505$	1.93157	0.0859539
$506$	−19.3964	−0.862275
$507$	5.25375	0.233327
$508$	−2.09599	−0.0929946
$509$	29.7300	1.31776	0.658879	−	0.752249i	$-0.271031\pi$
$509$	29.7300	1.31776	0.658879	+	0.752249i	$0.271031\pi$
$510$	19.5622	0.866228
$511$	8.63335	0.381917
$512$	1.00000	0.0441942
$513$	−71.1584	−3.14172
$514$	−12.6343	−0.557275
$515$	−32.7934	−1.44505
$516$	11.0224	0.485234
$517$	−29.2964	−1.28846
$518$	−5.05273	−0.222004
$519$	4.63490	0.203449
$520$	12.1168	0.531355
$521$	−2.60587	−0.114165	−0.0570827	−	0.998369i	$-0.518180\pi$
$521$	−2.60587	−0.114165	−0.0570827	+	0.998369i	$0.518180\pi$
$522$	35.7502	1.56474
$523$	26.3948	1.15417	0.577083	−	0.816685i	$-0.304191\pi$
$523$	26.3948	1.15417	0.577083	+	0.816685i	$0.304191\pi$
$524$	1.85297	0.0809473
$525$	−18.7903	−0.820076
$526$	28.0483	1.22296
$527$	1.96446	0.0855734
$528$	−12.4700	−0.542687
$529$	−0.803286	−0.0349255
$530$	−40.0729	−1.74066
$531$	−35.9982	−1.56219
$532$	5.71575	0.247809
$533$	10.8110	0.468277
$534$	43.4985	1.88236
$535$	−65.1738	−2.81771
$536$	−7.87060	−0.339958
$537$	−59.3318	−2.56036
$538$	9.11937	0.393164
$539$	−26.3630	−1.13554
$540$	34.7109	1.49372
$541$	−12.6003	−0.541729	−0.270865	−	0.962617i	$-0.587310\pi$
$541$	−12.6003	−0.541729	−0.270865	+	0.962617i	$0.587310\pi$
$542$	12.4816	0.536130
$543$	−13.3603	−0.573345
$544$	1.78903	0.0767038
$545$	−74.3984	−3.18688
$546$	7.85174	0.336024
$547$	21.3958	0.914819	0.457410	−	0.889256i	$-0.348777\pi$
$547$	21.3958	0.914819	0.457410	+	0.889256i	$0.348777\pi$
$548$	3.23610	0.138240
$549$	74.9681	3.19956
$550$	33.0689	1.41006
$551$	42.8502	1.82548
$552$	14.2703	0.607385
$553$	−12.6071	−0.536110
$554$	−12.7735	−0.542694
$555$	−71.5362	−3.03654
$556$	−18.6236	−0.789817
$557$	22.1047	0.936606	0.468303	−	0.883568i	$-0.344866\pi$
$557$	22.1047	0.936606	0.468303	+	0.883568i	$0.344866\pi$
$558$	6.77991	0.287017
$559$	12.2141	0.516601
$560$	−2.78813	−0.117820
$561$	−22.3091	−0.941893
$562$	−3.81431	−0.160897
$563$	−25.1298	−1.05910	−0.529548	−	0.848280i	$-0.677638\pi$
$563$	−25.1298	−1.05910	−0.529548	+	0.848280i	$0.677638\pi$
$564$	21.5540	0.907586
$565$	2.29369	0.0964961
$566$	0.741675	0.0311749
$567$	8.18689	0.343817
$568$	12.8834	0.540577
$569$	−33.7371	−1.41433	−0.707166	−	0.707048i	$-0.750027\pi$
$569$	−33.7371	−1.41433	−0.707166	+	0.707048i	$0.750027\pi$
$570$	80.9231	3.38949
$571$	6.17861	0.258567	0.129283	−	0.991608i	$-0.458732\pi$
$571$	6.17861	0.258567	0.129283	+	0.991608i	$0.458732\pi$
$572$	−13.8182	−0.577768
$573$	−48.5795	−2.02944
$574$	−2.48767	−0.103833
$575$	−37.8431	−1.57817
$576$	6.17443	0.257268
$577$	−18.8520	−0.784817	−0.392409	−	0.919791i	$-0.628358\pi$
$577$	−18.8520	−0.784817	−0.392409	+	0.919791i	$0.628358\pi$
$578$	−13.7994	−0.573979
$579$	56.7672	2.35916
$580$	−20.9022	−0.867918
$581$	−9.19067	−0.381293
$582$	24.3412	1.00898
$583$	45.7000	1.89270
$584$	11.1783	0.462563
$585$	74.8140	3.09318
$586$	−16.6692	−0.688599
$587$	−25.4716	−1.05132	−0.525662	−	0.850693i	$-0.676182\pi$
$587$	−25.4716	−1.05132	−0.525662	+	0.850693i	$0.676182\pi$
$588$	19.3958	0.799869
$589$	8.12641	0.334843
$590$	21.0472	0.866500
$591$	5.46750	0.224903
$592$	−6.54221	−0.268883
$593$	3.04901	0.125208	0.0626039	−	0.998038i	$-0.480060\pi$
$593$	3.04901	0.125208	0.0626039	+	0.998038i	$0.480060\pi$
$594$	−39.5851	−1.62420
$595$	−4.98804	−0.204490
$596$	13.4584	0.551277
$597$	40.3097	1.64976
$598$	15.8132	0.646649
$599$	13.1187	0.536014	0.268007	−	0.963417i	$-0.413635\pi$
$599$	13.1187	0.536014	0.268007	+	0.963417i	$0.413635\pi$
$600$	−24.3294	−0.993245
$601$	−8.64733	−0.352732	−0.176366	−	0.984325i	$-0.556434\pi$
$601$	−8.64733	−0.352732	−0.176366	+	0.984325i	$0.556434\pi$
$602$	−2.81053	−0.114549
$603$	−48.5965	−1.97900
$604$	8.38784	0.341296
$605$	−21.4775	−0.873183
$606$	1.62065	0.0658344
$607$	24.1545	0.980403	0.490201	−	0.871609i	$-0.336923\pi$
$607$	24.1545	0.980403	0.490201	+	0.871609i	$0.336923\pi$
$608$	7.40068	0.300137
$609$	−13.5448	−0.548863
$610$	−43.8320	−1.77470
$611$	23.8843	0.966255
$612$	11.0462	0.446516
$613$	−27.4325	−1.10799	−0.553995	−	0.832520i	$-0.686897\pi$
$613$	−27.4325	−1.10799	−0.553995	+	0.832520i	$0.686897\pi$
$614$	24.0996	0.972581
$615$	−35.2202	−1.42022
$616$	3.17964	0.128112
$617$	39.4984	1.59015	0.795073	−	0.606514i	$-0.207432\pi$
$617$	39.4984	1.59015	0.795073	+	0.606514i	$0.207432\pi$
$618$	−27.5147	−1.10680
$619$	−9.66937	−0.388645	−0.194322	−	0.980938i	$-0.562251\pi$
$619$	−9.66937	−0.388645	−0.194322	+	0.980938i	$0.562251\pi$
$620$	−3.96405	−0.159200
$621$	45.3001	1.81783
$622$	−10.1640	−0.407540
$623$	−11.0914	−0.444368
$624$	10.1663	0.406979
$625$	−0.643126	−0.0257251
$626$	14.4141	0.576102
$627$	−92.2864	−3.68556
$628$	6.12890	0.244570
$629$	−11.7042	−0.466676
$630$	−17.2151	−0.685866
$631$	27.7637	1.10525	0.552627	−	0.833429i	$-0.313626\pi$
$631$	27.7637	1.10525	0.552627	+	0.833429i	$0.313626\pi$
$632$	−16.3236	−0.649316
$633$	−16.7167	−0.664427
$634$	−1.58165	−0.0628155
$635$	7.56660	0.300271
$636$	−33.6224	−1.33322
$637$	21.4928	0.851576
$638$	23.8374	0.943730
$639$	79.5479	3.14686
$640$	−3.61003	−0.142699
$641$	14.5878	0.576184	0.288092	−	0.957603i	$-0.406979\pi$
$641$	14.5878	0.576184	0.288092	+	0.957603i	$0.406979\pi$
$642$	−54.6828	−2.15816
$643$	−19.2829	−0.760445	−0.380222	−	0.924895i	$-0.624153\pi$
$643$	−19.2829	−0.760445	−0.380222	+	0.924895i	$0.624153\pi$
$644$	−3.63870	−0.143385
$645$	−39.7912	−1.56678
$646$	13.2400	0.520921
$647$	19.0143	0.747529	0.373764	−	0.927524i	$-0.378067\pi$
$647$	19.0143	0.747529	0.373764	+	0.927524i	$0.378067\pi$
$648$	10.6003	0.416418
$649$	−24.0027	−0.942189
$650$	−26.9598	−1.05745
$651$	−2.56873	−0.100677
$652$	9.71610	0.380512
$653$	26.4335	1.03442	0.517211	−	0.855858i	$-0.326970\pi$
$653$	26.4335	1.03442	0.517211	+	0.855858i	$0.326970\pi$
$654$	−62.4226	−2.44092
$655$	−6.68928	−0.261372
$656$	−3.22100	−0.125759
$657$	69.0199	2.69272
$658$	−5.49591	−0.214253
$659$	−30.9379	−1.20517	−0.602585	−	0.798055i	$-0.705862\pi$
$659$	−30.9379	−1.20517	−0.602585	+	0.798055i	$0.705862\pi$
$660$	45.0171	1.75229
$661$	6.65789	0.258962	0.129481	−	0.991582i	$-0.458669\pi$
$661$	6.65789	0.258962	0.129481	+	0.991582i	$0.458669\pi$
$662$	−1.58444	−0.0615811
$663$	18.1878	0.706357
$664$	−11.9000	−0.461808
$665$	−20.6341	−0.800154
$666$	−40.3944	−1.56525
$667$	−27.2788	−1.05624
$668$	8.05320	0.311588
$669$	27.8350	1.07616
$670$	28.4131	1.09770
$671$	49.9869	1.92972
$672$	−2.33933	−0.0902416
$673$	−23.9147	−0.921845	−0.460922	−	0.887441i	$-0.652481\pi$
$673$	−23.9147	−0.921845	−0.460922	+	0.887441i	$0.652481\pi$
$674$	−15.0854	−0.581066
$675$	−77.2320	−2.97266
$676$	−1.73452	−0.0667123
$677$	2.37965	0.0914575	0.0457288	−	0.998954i	$-0.485439\pi$
$677$	2.37965	0.0914575	0.0457288	+	0.998954i	$0.485439\pi$
$678$	1.92447	0.0739090
$679$	−6.20661	−0.238188
$680$	−6.45844	−0.247670
$681$	−38.7264	−1.48400
$682$	4.52069	0.173106
$683$	−48.6653	−1.86212	−0.931062	−	0.364860i	$-0.881117\pi$
$683$	−48.6653	−1.86212	−0.931062	+	0.364860i	$0.881117\pi$
$684$	45.6949	1.74719
$685$	−11.6824	−0.446363
$686$	−10.3519	−0.395238
$687$	−26.1024	−0.995867
$688$	−3.63904	−0.138737
$689$	−37.2576	−1.41940
$690$	−51.5164	−1.96119
$691$	10.1610	0.386542	0.193271	−	0.981145i	$-0.438090\pi$
$691$	10.1610	0.386542	0.193271	+	0.981145i	$0.438090\pi$
$692$	−1.53021	−0.0581698
$693$	19.6325	0.745777
$694$	21.5861	0.819397
$695$	67.2319	2.55025
$696$	−17.5376	−0.664762
$697$	−5.76246	−0.218269
$698$	−19.4566	−0.736445
$699$	22.8735	0.865156
$700$	6.20361	0.234474
$701$	−37.2971	−1.40869	−0.704346	−	0.709857i	$-0.748760\pi$
$701$	−37.2971	−1.40869	−0.704346	+	0.709857i	$0.748760\pi$
$702$	32.2723	1.21804
$703$	−48.4168	−1.82607
$704$	4.11696	0.155164
$705$	−77.8106	−2.93052
$706$	−27.2505	−1.02558
$707$	−0.413239	−0.0155415
$708$	17.6593	0.663676
$709$	−28.2972	−1.06272	−0.531362	−	0.847145i	$-0.678319\pi$
$709$	−28.2972	−1.06272	−0.531362	+	0.847145i	$0.678319\pi$
$710$	−46.5097	−1.74548
$711$	−100.789	−3.77987
$712$	−14.3610	−0.538201
$713$	−5.17335	−0.193743
$714$	−4.18512	−0.156624
$715$	49.8842	1.86556
$716$	19.5884	0.732051
$717$	−39.8044	−1.48652
$718$	8.51153	0.317647
$719$	4.04821	0.150973	0.0754863	−	0.997147i	$-0.475949\pi$
$719$	4.04821	0.150973	0.0754863	+	0.997147i	$0.475949\pi$
$720$	−22.2899	−0.830695
$721$	7.01579	0.261282
$722$	35.7700	1.33122
$723$	−17.2123	−0.640133
$724$	4.41090	0.163930
$725$	46.5076	1.72725
$726$	−18.0203	−0.668794
$727$	37.8207	1.40269	0.701345	−	0.712822i	$-0.252583\pi$
$727$	37.8207	1.40269	0.701345	+	0.712822i	$0.252583\pi$
$728$	−2.59225	−0.0960751
$729$	−21.9200	−0.811853
$730$	−40.3542	−1.49358
$731$	−6.51033	−0.240793
$732$	−36.7764	−1.35929
$733$	−33.0604	−1.22111	−0.610557	−	0.791972i	$-0.709054\pi$
$733$	−33.0604	−1.22111	−0.610557	+	0.791972i	$0.709054\pi$
$734$	33.9406	1.25277
$735$	−70.0195	−2.58271
$736$	−4.71134	−0.173662
$737$	−32.4030	−1.19358
$738$	−19.8879	−0.732082
$739$	−34.4290	−1.26649	−0.633245	−	0.773951i	$-0.718278\pi$
$739$	−34.4290	−1.26649	−0.633245	+	0.773951i	$0.718278\pi$
$740$	23.6176	0.868201
$741$	75.2377	2.76393
$742$	8.57317	0.314731
$743$	−2.51262	−0.0921792	−0.0460896	−	0.998937i	$-0.514676\pi$
$743$	−2.51262	−0.0921792	−0.0460896	+	0.998937i	$0.514676\pi$
$744$	−3.32596	−0.121936
$745$	−48.5852	−1.78002
$746$	−8.31065	−0.304274
$747$	−73.4754	−2.68833
$748$	7.36535	0.269304
$749$	13.9432	0.509474
$750$	33.1574	1.21074
$751$	−14.3616	−0.524062	−0.262031	−	0.965059i	$-0.584392\pi$
$751$	−14.3616	−0.524062	−0.262031	+	0.965059i	$0.584392\pi$
$752$	−7.11603	−0.259495
$753$	26.6655	0.971746
$754$	−19.4337	−0.707735
$755$	−30.2804	−1.10202
$756$	−7.42603	−0.270082
$757$	12.1632	0.442079	0.221039	−	0.975265i	$-0.429055\pi$
$757$	12.1632	0.442079	0.221039	+	0.975265i	$0.429055\pi$
$758$	36.3581	1.32059
$759$	58.7504	2.13250
$760$	−26.7167	−0.969116
$761$	−6.73107	−0.244001	−0.122001	−	0.992530i	$-0.538931\pi$
$761$	−6.73107	−0.244001	−0.122001	+	0.992530i	$0.538931\pi$
$762$	6.34862	0.229986
$763$	15.9167	0.576224
$764$	16.0385	0.580253
$765$	−39.8772	−1.44176
$766$	26.3657	0.952630
$767$	19.5685	0.706579
$768$	−3.02893	−0.109297
$769$	43.6186	1.57293	0.786463	−	0.617637i	$-0.211910\pi$
$769$	43.6186	1.57293	0.786463	+	0.617637i	$0.211910\pi$
$770$	−11.4786	−0.413661
$771$	38.2684	1.37820
$772$	−18.7416	−0.674527
$773$	9.10377	0.327440	0.163720	−	0.986507i	$-0.447651\pi$
$773$	9.10377	0.327440	0.163720	+	0.986507i	$0.447651\pi$
$774$	−22.4690	−0.807630
$775$	8.82003	0.316825
$776$	−8.03623	−0.288484
$777$	15.3044	0.549042
$778$	36.5001	1.30859
$779$	−23.8376	−0.854071
$780$	−36.7008	−1.31410
$781$	53.0406	1.89794
$782$	−8.42870	−0.301410
$783$	−55.6719	−1.98955
$784$	−6.40351	−0.228697
$785$	−22.1255	−0.789695
$786$	−5.61252	−0.200192
$787$	6.54386	0.233263	0.116632	−	0.993175i	$-0.462790\pi$
$787$	6.54386	0.233263	0.116632	+	0.993175i	$0.462790\pi$
$788$	−1.80509	−0.0643038
$789$	−84.9563	−3.02452
$790$	58.9286	2.09659
$791$	−0.490709	−0.0174476
$792$	25.4199	0.903256
$793$	−40.7525	−1.44716
$794$	33.5994	1.19240
$795$	121.378	4.30484
$796$	−13.3082	−0.471697
$797$	16.8575	0.597123	0.298561	−	0.954390i	$-0.403493\pi$
$797$	16.8575	0.597123	0.298561	+	0.954390i	$0.403493\pi$
$798$	−17.3126	−0.612860
$799$	−12.7308	−0.450382
$800$	8.03235	0.283986
$801$	−88.6710	−3.13304
$802$	34.8436	1.23037
$803$	46.0208	1.62404
$804$	23.8395	0.840755
$805$	13.1358	0.462977
$806$	−3.68555	−0.129818
$807$	−27.6220	−0.972339
$808$	−0.535057	−0.0188232
$809$	46.5662	1.63718	0.818590	−	0.574379i	$-0.194756\pi$
$809$	46.5662	1.63718	0.818590	+	0.574379i	$0.194756\pi$
$810$	−38.2674	−1.34458
$811$	20.5759	0.722518	0.361259	−	0.932465i	$-0.382347\pi$
$811$	20.5759	0.722518	0.361259	+	0.932465i	$0.382347\pi$
$812$	4.47181	0.156930
$813$	−37.8059	−1.32591
$814$	−26.9340	−0.944038
$815$	−35.0754	−1.22864
$816$	−5.41883	−0.189697
$817$	−26.9313	−0.942208
$818$	38.8971	1.36000
$819$	−16.0056	−0.559283
$820$	11.6279	0.406065
$821$	20.2159	0.705540	0.352770	−	0.935710i	$-0.385240\pi$
$821$	20.2159	0.705540	0.352770	+	0.935710i	$0.385240\pi$
$822$	−9.80194	−0.341882
$823$	1.31358	0.0457884	0.0228942	−	0.999738i	$-0.492712\pi$
$823$	1.31358	0.0457884	0.0228942	+	0.999738i	$0.492712\pi$
$824$	9.08396	0.316455
$825$	−100.163	−3.48724
$826$	−4.50283	−0.156673
$827$	49.0403	1.70530	0.852648	−	0.522485i	$-0.174995\pi$
$827$	49.0403	1.70530	0.852648	+	0.522485i	$0.174995\pi$
$828$	−29.0898	−1.01094
$829$	−38.4798	−1.33646	−0.668230	−	0.743955i	$-0.732948\pi$
$829$	−38.4798	−1.33646	−0.668230	+	0.743955i	$0.732948\pi$
$830$	42.9593	1.49114
$831$	38.6901	1.34214
$832$	−3.35641	−0.116363
$833$	−11.4560	−0.396928
$834$	56.4097	1.95331
$835$	−29.0723	−1.00609
$836$	30.4683	1.05377
$837$	−10.5580	−0.364938
$838$	15.0420	0.519616
$839$	53.3513	1.84189	0.920946	−	0.389689i	$-0.127418\pi$
$839$	53.3513	1.84189	0.920946	+	0.389689i	$0.127418\pi$
$840$	8.44506	0.291382
$841$	4.52454	0.156019
$842$	−6.75253	−0.232707
$843$	11.5533	0.397916
$844$	5.51899	0.189972
$845$	6.26168	0.215408
$846$	−43.9374	−1.51060
$847$	4.59487	0.157882
$848$	11.1004	0.381190
$849$	−2.24648	−0.0770991
$850$	14.3701	0.492889
$851$	30.8226	1.05658
$852$	−39.0231	−1.33691
$853$	10.1590	0.347837	0.173918	−	0.984760i	$-0.444357\pi$
$853$	10.1590	0.347837	0.173918	+	0.984760i	$0.444357\pi$
$854$	9.37737	0.320887
$855$	−164.960	−5.64152
$856$	18.0535	0.617056
$857$	25.8326	0.882426	0.441213	−	0.897402i	$-0.354548\pi$
$857$	25.8326	0.882426	0.441213	+	0.897402i	$0.354548\pi$
$858$	41.8544	1.42889
$859$	−22.2232	−0.758247	−0.379123	−	0.925346i	$-0.623774\pi$
$859$	−22.2232	−0.758247	−0.379123	+	0.925346i	$0.623774\pi$
$860$	13.1370	0.447970
$861$	7.53498	0.256792
$862$	−34.8373	−1.18656
$863$	−17.1815	−0.584864	−0.292432	−	0.956286i	$-0.594465\pi$
$863$	−17.1815	−0.584864	−0.292432	+	0.956286i	$0.594465\pi$
$864$	−9.61513	−0.327113
$865$	5.52410	0.187825
$866$	5.77818	0.196350
$867$	41.7974	1.41951
$868$	0.848065	0.0287852
$869$	−67.2035	−2.27972
$870$	63.3114	2.14646
$871$	26.4170	0.895105
$872$	20.6088	0.697901
$873$	−49.6191	−1.67935
$874$	−34.8671	−1.17940
$875$	−8.45458	−0.285817
$876$	−33.8584	−1.14397
$877$	40.4601	1.36624	0.683119	−	0.730307i	$-0.260623\pi$
$877$	40.4601	1.36624	0.683119	+	0.730307i	$0.260623\pi$
$878$	−13.7709	−0.464744
$879$	50.4899	1.70298
$880$	−14.8624	−0.501011
$881$	−32.5453	−1.09648	−0.548239	−	0.836322i	$-0.684702\pi$
$881$	−32.5453	−1.09648	−0.548239	+	0.836322i	$0.684702\pi$
$882$	−39.5380	−1.33131
$883$	39.6077	1.33291	0.666453	−	0.745547i	$-0.267812\pi$
$883$	39.6077	1.33291	0.666453	+	0.745547i	$0.267812\pi$
$884$	−6.00470	−0.201960
$885$	−63.7506	−2.14295
$886$	−22.6228	−0.760029
$887$	−21.3205	−0.715874	−0.357937	−	0.933746i	$-0.616520\pi$
$887$	−21.3205	−0.715874	−0.357937	+	0.933746i	$0.616520\pi$
$888$	19.8159	0.664979
$889$	−1.61879	−0.0542926
$890$	51.8437	1.73781
$891$	43.6409	1.46203
$892$	−9.18970	−0.307694
$893$	−52.6634	−1.76231
$894$	−40.7645	−1.36337
$895$	−70.7146	−2.36373
$896$	0.772328	0.0258017
$897$	−47.8970	−1.59924
$898$	−8.52906	−0.284618
$899$	6.35783	0.212045
$900$	49.5952	1.65317
$901$	19.8589	0.661597
$902$	−13.2607	−0.441535
$903$	8.51290	0.283292
$904$	−0.635364	−0.0211319
$905$	−15.9235	−0.529315
$906$	−25.4062	−0.844064
$907$	−17.7411	−0.589084	−0.294542	−	0.955639i	$-0.595167\pi$
$907$	−17.7411	−0.589084	−0.294542	+	0.955639i	$0.595167\pi$
$908$	12.7855	0.424301
$909$	−3.30367	−0.109576
$910$	9.35810	0.310218
$911$	29.1506	0.965801	0.482901	−	0.875675i	$-0.339583\pi$
$911$	29.1506	0.965801	0.482901	+	0.875675i	$0.339583\pi$
$912$	−22.4161	−0.742273
$913$	−48.9917	−1.62139
$914$	−30.1252	−0.996454
$915$	132.764	4.38904
$916$	8.61768	0.284736
$917$	1.43110	0.0472591
$918$	−17.2017	−0.567741
$919$	−30.9616	−1.02133	−0.510665	−	0.859780i	$-0.670601\pi$
$919$	−30.9616	−1.02133	−0.510665	+	0.859780i	$0.670601\pi$
$920$	17.0081	0.560740
$921$	−72.9960	−2.40530
$922$	11.8267	0.389491
$923$	−43.2421	−1.42333
$924$	−9.63093	−0.316834
$925$	−52.5493	−1.72781
$926$	27.2504	0.895505
$927$	56.0882	1.84218
$928$	5.79004	0.190067
$929$	−20.4251	−0.670126	−0.335063	−	0.942196i	$-0.608758\pi$
$929$	−20.4251	−0.670126	−0.335063	+	0.942196i	$0.608758\pi$
$930$	12.0068	0.393719
$931$	−47.3903	−1.55315
$932$	−7.55167	−0.247363
$933$	30.7861	1.00789
$934$	25.8445	0.845659
$935$	−26.5892	−0.869559
$936$	−20.7239	−0.677382
$937$	−16.2941	−0.532304	−0.266152	−	0.963931i	$-0.585752\pi$
$937$	−16.2941	−0.532304	−0.266152	+	0.963931i	$0.585752\pi$
$938$	−6.07869	−0.198476
$939$	−43.6593	−1.42477
$940$	25.6891	0.837886
$941$	0.591462	0.0192811	0.00964055	−	0.999954i	$-0.496931\pi$
$941$	0.591462	0.0192811	0.00964055	+	0.999954i	$0.496931\pi$
$942$	−18.5640	−0.604849
$943$	15.1752	0.494174
$944$	−5.83020	−0.189757
$945$	26.8082	0.872072
$946$	−14.9818	−0.487099
$947$	24.1883	0.786015	0.393007	−	0.919535i	$-0.371435\pi$
$947$	24.1883	0.786015	0.393007	+	0.919535i	$0.371435\pi$
$948$	49.4429	1.60583
$949$	−37.5191	−1.21792
$950$	59.4448	1.92864
$951$	4.79072	0.155350
$952$	1.38171	0.0447816
$953$	43.5555	1.41090	0.705450	−	0.708760i	$-0.250745\pi$
$953$	43.5555	1.41090	0.705450	+	0.708760i	$0.250745\pi$
$954$	68.5388	2.21902
$955$	−57.8995	−1.87358
$956$	13.1414	0.425024
$957$	−72.2018	−2.33395
$958$	13.6620	0.441400
$959$	2.49933	0.0807077
$960$	10.9345	0.352911
$961$	−29.7943	−0.961105
$962$	21.9583	0.707965
$963$	111.470	3.59207
$964$	5.68263	0.183025
$965$	67.6580	2.17799
$966$	11.0214	0.354607
$967$	26.3388	0.846999	0.423500	−	0.905896i	$-0.360801\pi$
$967$	26.3388	0.846999	0.423500	+	0.905896i	$0.360801\pi$
$968$	5.94938	0.191220
$969$	−40.1030	−1.28830
$970$	29.0111	0.931490
$971$	11.0735	0.355366	0.177683	−	0.984088i	$-0.443140\pi$
$971$	11.0735	0.355366	0.177683	+	0.984088i	$0.443140\pi$
$972$	−3.26214	−0.104633
$973$	−14.3835	−0.461115
$974$	−20.7001	−0.663274
$975$	81.6595	2.61520
$976$	12.1417	0.388646
$977$	−17.9305	−0.573648	−0.286824	−	0.957983i	$-0.592600\pi$
$977$	−17.9305	−0.573648	−0.286824	+	0.957983i	$0.592600\pi$
$978$	−29.4294	−0.941048
$979$	−59.1237	−1.88960
$980$	23.1169	0.738442
$981$	127.247	4.06270
$982$	6.73929	0.215059
$983$	0.955723	0.0304828	0.0152414	−	0.999884i	$-0.495148\pi$
$983$	0.955723	0.0304828	0.0152414	+	0.999884i	$0.495148\pi$
$984$	9.75620	0.311016
$985$	6.51645	0.207631
$986$	10.3585	0.329883
$987$	16.6467	0.529871
$988$	−24.8397	−0.790256
$989$	17.1447	0.545171
$990$	−91.7667	−2.91654
$991$	31.1488	0.989474	0.494737	−	0.869043i	$-0.335264\pi$
$991$	31.1488	0.989474	0.494737	+	0.869043i	$0.335264\pi$
$992$	1.09806	0.0348635
$993$	4.79917	0.152297
$994$	9.95024	0.315602
$995$	48.0431	1.52307
$996$	36.0442	1.14210
$997$	28.9764	0.917691	0.458845	−	0.888516i	$-0.348263\pi$
$997$	28.9764	0.917691	0.458845	+	0.888516i	$0.348263\pi$
$998$	−35.4419	−1.12189
$999$	62.9042	1.99020

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6046.2.a.f.1.3	✓	67

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6046.2.a.f.1.3	✓	67	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:	\( N \)	\(=\)	\( 6046 = 2 \cdot 3023 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6046.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -3.02893 q^{3} +1.00000 q^{4} -3.61003 q^{5} -3.02893 q^{6} +0.772328 q^{7} +1.00000 q^{8} +6.17443 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -3.02893 q^{3} +1.00000 q^{4} -3.61003 q^{5} -3.02893 q^{6} +0.772328 q^{7} +1.00000 q^{8} +6.17443 q^{9} -3.61003 q^{10} +4.11696 q^{11} -3.02893 q^{12} -3.35641 q^{13} +0.772328 q^{14} +10.9345 q^{15} +1.00000 q^{16} +1.78903 q^{17} +6.17443 q^{18} +7.40068 q^{19} -3.61003 q^{20} -2.33933 q^{21} +4.11696 q^{22} -4.71134 q^{23} -3.02893 q^{24} +8.03235 q^{25} -3.35641 q^{26} -9.61513 q^{27} +0.772328 q^{28} +5.79004 q^{29} +10.9345 q^{30} +1.09806 q^{31} +1.00000 q^{32} -12.4700 q^{33} +1.78903 q^{34} -2.78813 q^{35} +6.17443 q^{36} -6.54221 q^{37} +7.40068 q^{38} +10.1663 q^{39} -3.61003 q^{40} -3.22100 q^{41} -2.33933 q^{42} -3.63904 q^{43} +4.11696 q^{44} -22.2899 q^{45} -4.71134 q^{46} -7.11603 q^{47} -3.02893 q^{48} -6.40351 q^{49} +8.03235 q^{50} -5.41883 q^{51} -3.35641 q^{52} +11.1004 q^{53} -9.61513 q^{54} -14.8624 q^{55} +0.772328 q^{56} -22.4161 q^{57} +5.79004 q^{58} -5.83020 q^{59} +10.9345 q^{60} +12.1417 q^{61} +1.09806 q^{62} +4.76868 q^{63} +1.00000 q^{64} +12.1168 q^{65} -12.4700 q^{66} -7.87060 q^{67} +1.78903 q^{68} +14.2703 q^{69} -2.78813 q^{70} +12.8834 q^{71} +6.17443 q^{72} +11.1783 q^{73} -6.54221 q^{74} -24.3294 q^{75} +7.40068 q^{76} +3.17964 q^{77} +10.1663 q^{78} -16.3236 q^{79} -3.61003 q^{80} +10.6003 q^{81} -3.22100 q^{82} -11.9000 q^{83} -2.33933 q^{84} -6.45844 q^{85} -3.63904 q^{86} -17.5376 q^{87} +4.11696 q^{88} -14.3610 q^{89} -22.2899 q^{90} -2.59225 q^{91} -4.71134 q^{92} -3.32596 q^{93} -7.11603 q^{94} -26.7167 q^{95} -3.02893 q^{96} -8.03623 q^{97} -6.40351 q^{98} +25.4199 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 67 q + 67 q^{2} + 21 q^{3} + 67 q^{4} + 21 q^{5} + 21 q^{6} + 38 q^{7} + 67 q^{8} + 90 q^{9}+O(q^{10}) \) 67 * q + 67 * q^2 + 21 * q^3 + 67 * q^4 + 21 * q^5 + 21 * q^6 + 38 * q^7 + 67 * q^8 + 90 * q^9	\( 67 q + 67 q^{2} + 21 q^{3} + 67 q^{4} + 21 q^{5} + 21 q^{6} + 38 q^{7} + 67 q^{8} + 90 q^{9} + 21 q^{10} + 56 q^{11} + 21 q^{12} + 33 q^{13} + 38 q^{14} + 25 q^{15} + 67 q^{16} + 30 q^{17} + 90 q^{18} + 36 q^{19} + 21 q^{20} + 20 q^{21} + 56 q^{22} + 65 q^{23} + 21 q^{24} + 72 q^{25} + 33 q^{26} + 57 q^{27} + 38 q^{28} + 84 q^{29} + 25 q^{30} + 52 q^{31} + 67 q^{32} - 9 q^{33} + 30 q^{34} + 30 q^{35} + 90 q^{36} + 52 q^{37} + 36 q^{38} + 41 q^{39} + 21 q^{40} + 46 q^{41} + 20 q^{42} + 61 q^{43} + 56 q^{44} + 23 q^{45} + 65 q^{46} + 51 q^{47} + 21 q^{48} + 81 q^{49} + 72 q^{50} + 33 q^{51} + 33 q^{52} + 72 q^{53} + 57 q^{54} + 14 q^{55} + 38 q^{56} - 26 q^{57} + 84 q^{58} + 71 q^{59} + 25 q^{60} + 42 q^{61} + 52 q^{62} + 63 q^{63} + 67 q^{64} - 2 q^{65} - 9 q^{66} + 70 q^{67} + 30 q^{68} + 21 q^{69} + 30 q^{70} + 104 q^{71} + 90 q^{72} - 31 q^{73} + 52 q^{74} + 69 q^{75} + 36 q^{76} + 48 q^{77} + 41 q^{78} + 79 q^{79} + 21 q^{80} + 123 q^{81} + 46 q^{82} + 41 q^{83} + 20 q^{84} + 6 q^{85} + 61 q^{86} + 19 q^{87} + 56 q^{88} + 58 q^{89} + 23 q^{90} + 31 q^{91} + 65 q^{92} + 13 q^{93} + 51 q^{94} + 77 q^{95} + 21 q^{96} - 8 q^{97} + 81 q^{98} + 129 q^{99}+O(q^{100}) \) 67 * q + 67 * q^2 + 21 * q^3 + 67 * q^4 + 21 * q^5 + 21 * q^6 + 38 * q^7 + 67 * q^8 + 90 * q^9 + 21 * q^10 + 56 * q^11 + 21 * q^12 + 33 * q^13 + 38 * q^14 + 25 * q^15 + 67 * q^16 + 30 * q^17 + 90 * q^18 + 36 * q^19 + 21 * q^20 + 20 * q^21 + 56 * q^22 + 65 * q^23 + 21 * q^24 + 72 * q^25 + 33 * q^26 + 57 * q^27 + 38 * q^28 + 84 * q^29 + 25 * q^30 + 52 * q^31 + 67 * q^32 - 9 * q^33 + 30 * q^34 + 30 * q^35 + 90 * q^36 + 52 * q^37 + 36 * q^38 + 41 * q^39 + 21 * q^40 + 46 * q^41 + 20 * q^42 + 61 * q^43 + 56 * q^44 + 23 * q^45 + 65 * q^46 + 51 * q^47 + 21 * q^48 + 81 * q^49 + 72 * q^50 + 33 * q^51 + 33 * q^52 + 72 * q^53 + 57 * q^54 + 14 * q^55 + 38 * q^56 - 26 * q^57 + 84 * q^58 + 71 * q^59 + 25 * q^60 + 42 * q^61 + 52 * q^62 + 63 * q^63 + 67 * q^64 - 2 * q^65 - 9 * q^66 + 70 * q^67 + 30 * q^68 + 21 * q^69 + 30 * q^70 + 104 * q^71 + 90 * q^72 - 31 * q^73 + 52 * q^74 + 69 * q^75 + 36 * q^76 + 48 * q^77 + 41 * q^78 + 79 * q^79 + 21 * q^80 + 123 * q^81 + 46 * q^82 + 41 * q^83 + 20 * q^84 + 6 * q^85 + 61 * q^86 + 19 * q^87 + 56 * q^88 + 58 * q^89 + 23 * q^90 + 31 * q^91 + 65 * q^92 + 13 * q^93 + 51 * q^94 + 77 * q^95 + 21 * q^96 - 8 * q^97 + 81 * q^98 + 129 * q^99

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