LMFDB - Embedded newform 6046.2.a.e.1.5

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:	$1$
Dimension:	$56$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
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)

$f(q)$	$=$	$q+1.00000 q^{2} -2.93762 q^{3} +1.00000 q^{4} +1.81603 q^{5} -2.93762 q^{6} -0.742842 q^{7} +1.00000 q^{8} +5.62960 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.93762 q^{3} +1.00000 q^{4} +1.81603 q^{5} -2.93762 q^{6} -0.742842 q^{7} +1.00000 q^{8} +5.62960 q^{9} +1.81603 q^{10} -0.741853 q^{11} -2.93762 q^{12} -2.46739 q^{13} -0.742842 q^{14} -5.33479 q^{15} +1.00000 q^{16} +6.26947 q^{17} +5.62960 q^{18} -1.20682 q^{19} +1.81603 q^{20} +2.18219 q^{21} -0.741853 q^{22} -8.28947 q^{23} -2.93762 q^{24} -1.70205 q^{25} -2.46739 q^{26} -7.72476 q^{27} -0.742842 q^{28} +3.47853 q^{29} -5.33479 q^{30} -3.68364 q^{31} +1.00000 q^{32} +2.17928 q^{33} +6.26947 q^{34} -1.34902 q^{35} +5.62960 q^{36} +9.46819 q^{37} -1.20682 q^{38} +7.24825 q^{39} +1.81603 q^{40} +2.72015 q^{41} +2.18219 q^{42} -3.67111 q^{43} -0.741853 q^{44} +10.2235 q^{45} -8.28947 q^{46} -8.70463 q^{47} -2.93762 q^{48} -6.44819 q^{49} -1.70205 q^{50} -18.4173 q^{51} -2.46739 q^{52} -6.50521 q^{53} -7.72476 q^{54} -1.34722 q^{55} -0.742842 q^{56} +3.54518 q^{57} +3.47853 q^{58} +11.2706 q^{59} -5.33479 q^{60} -14.8705 q^{61} -3.68364 q^{62} -4.18191 q^{63} +1.00000 q^{64} -4.48084 q^{65} +2.17928 q^{66} -1.51826 q^{67} +6.26947 q^{68} +24.3513 q^{69} -1.34902 q^{70} -0.651591 q^{71} +5.62960 q^{72} +14.6951 q^{73} +9.46819 q^{74} +4.99997 q^{75} -1.20682 q^{76} +0.551080 q^{77} +7.24825 q^{78} -0.852830 q^{79} +1.81603 q^{80} +5.80361 q^{81} +2.72015 q^{82} -8.67456 q^{83} +2.18219 q^{84} +11.3855 q^{85} -3.67111 q^{86} -10.2186 q^{87} -0.741853 q^{88} -4.60991 q^{89} +10.2235 q^{90} +1.83288 q^{91} -8.28947 q^{92} +10.8211 q^{93} -8.70463 q^{94} -2.19162 q^{95} -2.93762 q^{96} +2.59641 q^{97} -6.44819 q^{98} -4.17633 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q + 56 q^{2} - 18 q^{3} + 56 q^{4} - 17 q^{5} - 18 q^{6} - 35 q^{7} + 56 q^{8} + 34 q^{9}+O(q^{10}) $ 56 * q + 56 * q^2 - 18 * q^3 + 56 * q^4 - 17 * q^5 - 18 * q^6 - 35 * q^7 + 56 * q^8 + 34 * q^9	\( 56 q + 56 q^{2} - 18 q^{3} + 56 q^{4} - 17 q^{5} - 18 q^{6} - 35 q^{7} + 56 q^{8} + 34 q^{9} - 17 q^{10} - 53 q^{11} - 18 q^{12} - 21 q^{13} - 35 q^{14} - 36 q^{15} + 56 q^{16} - 22 q^{17} + 34 q^{18} - 31 q^{19} - 17 q^{20} - 23 q^{21} - 53 q^{22} - 59 q^{23} - 18 q^{24} + 41 q^{25} - 21 q^{26} - 63 q^{27} - 35 q^{28} - 88 q^{29} - 36 q^{30} - 44 q^{31} + 56 q^{32} + 4 q^{33} - 22 q^{34} - 51 q^{35} + 34 q^{36} - 60 q^{37} - 31 q^{38} - 62 q^{39} - 17 q^{40} - 39 q^{41} - 23 q^{42} - 66 q^{43} - 53 q^{44} - 34 q^{45} - 59 q^{46} - 51 q^{47} - 18 q^{48} + 41 q^{49} + 41 q^{50} - 48 q^{51} - 21 q^{52} - 75 q^{53} - 63 q^{54} - 41 q^{55} - 35 q^{56} - 12 q^{57} - 88 q^{58} - 77 q^{59} - 36 q^{60} - 43 q^{61} - 44 q^{62} - 88 q^{63} + 56 q^{64} - 54 q^{65} + 4 q^{66} - 62 q^{67} - 22 q^{68} - 48 q^{69} - 51 q^{70} - 122 q^{71} + 34 q^{72} - 7 q^{73} - 60 q^{74} - 63 q^{75} - 31 q^{76} - 39 q^{77} - 62 q^{78} - 91 q^{79} - 17 q^{80} + 8 q^{81} - 39 q^{82} - 51 q^{83} - 23 q^{84} - 72 q^{85} - 66 q^{86} - 19 q^{87} - 53 q^{88} - 62 q^{89} - 34 q^{90} - 48 q^{91} - 59 q^{92} - 41 q^{93} - 51 q^{94} - 120 q^{95} - 18 q^{96} + 6 q^{97} + 41 q^{98} - 128 q^{99}+O(q^{100}) \) 56 * q + 56 * q^2 - 18 * q^3 + 56 * q^4 - 17 * q^5 - 18 * q^6 - 35 * q^7 + 56 * q^8 + 34 * q^9 - 17 * q^10 - 53 * q^11 - 18 * q^12 - 21 * q^13 - 35 * q^14 - 36 * q^15 + 56 * q^16 - 22 * q^17 + 34 * q^18 - 31 * q^19 - 17 * q^20 - 23 * q^21 - 53 * q^22 - 59 * q^23 - 18 * q^24 + 41 * q^25 - 21 * q^26 - 63 * q^27 - 35 * q^28 - 88 * q^29 - 36 * q^30 - 44 * q^31 + 56 * q^32 + 4 * q^33 - 22 * q^34 - 51 * q^35 + 34 * q^36 - 60 * q^37 - 31 * q^38 - 62 * q^39 - 17 * q^40 - 39 * q^41 - 23 * q^42 - 66 * q^43 - 53 * q^44 - 34 * q^45 - 59 * q^46 - 51 * q^47 - 18 * q^48 + 41 * q^49 + 41 * q^50 - 48 * q^51 - 21 * q^52 - 75 * q^53 - 63 * q^54 - 41 * q^55 - 35 * q^56 - 12 * q^57 - 88 * q^58 - 77 * q^59 - 36 * q^60 - 43 * q^61 - 44 * q^62 - 88 * q^63 + 56 * q^64 - 54 * q^65 + 4 * q^66 - 62 * q^67 - 22 * q^68 - 48 * q^69 - 51 * q^70 - 122 * q^71 + 34 * q^72 - 7 * q^73 - 60 * q^74 - 63 * q^75 - 31 * q^76 - 39 * q^77 - 62 * q^78 - 91 * q^79 - 17 * q^80 + 8 * q^81 - 39 * q^82 - 51 * q^83 - 23 * q^84 - 72 * q^85 - 66 * q^86 - 19 * q^87 - 53 * q^88 - 62 * q^89 - 34 * q^90 - 48 * q^91 - 59 * q^92 - 41 * q^93 - 51 * q^94 - 120 * q^95 - 18 * q^96 + 6 * q^97 + 41 * q^98 - 128 * 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
$3$	−2.93762	−1.69603	−0.848017	−	0.529968i	$-0.822204\pi$
$3$	−2.93762	−1.69603	−0.848017	+	0.529968i	$0.822204\pi$
$4$	1.00000	0.500000
$5$	1.81603	0.812152	0.406076	−	0.913839i	$-0.366897\pi$
$5$	1.81603	0.812152	0.406076	+	0.913839i	$0.366897\pi$
$6$	−2.93762	−1.19928
$7$	−0.742842	−0.280768	−0.140384	−	0.990097i	$-0.544834\pi$
$7$	−0.742842	−0.280768	−0.140384	+	0.990097i	$0.544834\pi$
$8$	1.00000	0.353553
$9$	5.62960	1.87653
$10$	1.81603	0.574278
$11$	−0.741853	−0.223677	−0.111838	−	0.993726i	$-0.535674\pi$
$11$	−0.741853	−0.223677	−0.111838	+	0.993726i	$0.535674\pi$
$12$	−2.93762	−0.848017
$13$	−2.46739	−0.684331	−0.342165	−	0.939640i	$-0.611160\pi$
$13$	−2.46739	−0.684331	−0.342165	+	0.939640i	$0.611160\pi$
$14$	−0.742842	−0.198533
$15$	−5.33479	−1.37744
$16$	1.00000	0.250000
$17$	6.26947	1.52057	0.760285	−	0.649590i	$-0.225059\pi$
$17$	6.26947	1.52057	0.760285	+	0.649590i	$0.225059\pi$
$18$	5.62960	1.32691
$19$	−1.20682	−0.276864	−0.138432	−	0.990372i	$-0.544206\pi$
$19$	−1.20682	−0.276864	−0.138432	+	0.990372i	$0.544206\pi$
$20$	1.81603	0.406076
$21$	2.18219	0.476192
$22$	−0.741853	−0.158164
$23$	−8.28947	−1.72847	−0.864237	−	0.503084i	$-0.832199\pi$
$23$	−8.28947	−1.72847	−0.864237	+	0.503084i	$0.832199\pi$
$24$	−2.93762	−0.599639
$25$	−1.70205	−0.340410
$26$	−2.46739	−0.483895
$27$	−7.72476	−1.48663
$28$	−0.742842	−0.140384
$29$	3.47853	0.645946	0.322973	−	0.946408i	$-0.395318\pi$
$29$	3.47853	0.645946	0.322973	+	0.946408i	$0.395318\pi$
$30$	−5.33479	−0.973995
$31$	−3.68364	−0.661602	−0.330801	−	0.943701i	$-0.607319\pi$
$31$	−3.68364	−0.661602	−0.330801	+	0.943701i	$0.607319\pi$
$32$	1.00000	0.176777
$33$	2.17928	0.379364
$34$	6.26947	1.07521
$35$	−1.34902	−0.228026
$36$	5.62960	0.938267
$37$	9.46819	1.55656	0.778281	−	0.627917i	$-0.216092\pi$
$37$	9.46819	1.55656	0.778281	+	0.627917i	$0.216092\pi$
$38$	−1.20682	−0.195772
$39$	7.24825	1.16065
$40$	1.81603	0.287139
$41$	2.72015	0.424816	0.212408	−	0.977181i	$-0.431869\pi$
$41$	2.72015	0.424816	0.212408	+	0.977181i	$0.431869\pi$
$42$	2.18219	0.336719
$43$	−3.67111	−0.559839	−0.279920	−	0.960023i	$-0.590308\pi$
$43$	−3.67111	−0.559839	−0.279920	+	0.960023i	$0.590308\pi$
$44$	−0.741853	−0.111838
$45$	10.2235	1.52403
$46$	−8.28947	−1.22222
$47$	−8.70463	−1.26970	−0.634850	−	0.772635i	$-0.718938\pi$
$47$	−8.70463	−1.26970	−0.634850	+	0.772635i	$0.718938\pi$
$48$	−2.93762	−0.424009
$49$	−6.44819	−0.921169
$50$	−1.70205	−0.240706
$51$	−18.4173	−2.57894
$52$	−2.46739	−0.342165
$53$	−6.50521	−0.893560	−0.446780	−	0.894644i	$-0.647429\pi$
$53$	−6.50521	−0.893560	−0.446780	+	0.894644i	$0.647429\pi$
$54$	−7.72476	−1.05121
$55$	−1.34722	−0.181660
$56$	−0.742842	−0.0992665
$57$	3.54518	0.469571
$58$	3.47853	0.456753
$59$	11.2706	1.46731	0.733657	−	0.679520i	$-0.237812\pi$
$59$	11.2706	1.46731	0.733657	+	0.679520i	$0.237812\pi$
$60$	−5.33479	−0.688719
$61$	−14.8705	−1.90398	−0.951989	−	0.306133i	$-0.900965\pi$
$61$	−14.8705	−1.90398	−0.951989	+	0.306133i	$0.900965\pi$
$62$	−3.68364	−0.467823
$63$	−4.18191	−0.526871
$64$	1.00000	0.125000
$65$	−4.48084	−0.555780
$66$	2.17928	0.268251
$67$	−1.51826	−0.185486	−0.0927428	−	0.995690i	$-0.529563\pi$
$67$	−1.51826	−0.185486	−0.0927428	+	0.995690i	$0.529563\pi$
$68$	6.26947	0.760285
$69$	24.3513	2.93155
$70$	−1.34902	−0.161239
$71$	−0.651591	−0.0773297	−0.0386648	−	0.999252i	$-0.512310\pi$
$71$	−0.651591	−0.0773297	−0.0386648	+	0.999252i	$0.512310\pi$
$72$	5.62960	0.663455
$73$	14.6951	1.71993	0.859964	−	0.510354i	$-0.170486\pi$
$73$	14.6951	1.71993	0.859964	+	0.510354i	$0.170486\pi$
$74$	9.46819	1.10066
$75$	4.99997	0.577346
$76$	−1.20682	−0.138432
$77$	0.551080	0.0628013
$78$	7.24825	0.820702
$79$	−0.852830	−0.0959509	−0.0479755	−	0.998849i	$-0.515277\pi$
$79$	−0.852830	−0.0959509	−0.0479755	+	0.998849i	$0.515277\pi$
$80$	1.81603	0.203038
$81$	5.80361	0.644845
$82$	2.72015	0.300390
$83$	−8.67456	−0.952157	−0.476079	−	0.879403i	$-0.657942\pi$
$83$	−8.67456	−0.952157	−0.476079	+	0.879403i	$0.657942\pi$
$84$	2.18219	0.238096
$85$	11.3855	1.23493
$86$	−3.67111	−0.395866
$87$	−10.2186	−1.09555
$88$	−0.741853	−0.0790818
$89$	−4.60991	−0.488650	−0.244325	−	0.969693i	$-0.578566\pi$
$89$	−4.60991	−0.488650	−0.244325	+	0.969693i	$0.578566\pi$
$90$	10.2235	1.07765
$91$	1.83288	0.192138
$92$	−8.28947	−0.864237
$93$	10.8211	1.12210
$94$	−8.70463	−0.897814
$95$	−2.19162	−0.224855
$96$	−2.93762	−0.299819
$97$	2.59641	0.263626	0.131813	−	0.991275i	$-0.457920\pi$
$97$	2.59641	0.263626	0.131813	+	0.991275i	$0.457920\pi$
$98$	−6.44819	−0.651365
$99$	−4.17633	−0.419737
$100$	−1.70205	−0.170205
$101$	7.42035	0.738353	0.369176	−	0.929359i	$-0.379640\pi$
$101$	7.42035	0.738353	0.369176	+	0.929359i	$0.379640\pi$
$102$	−18.4173	−1.82359
$103$	−0.0982244	−0.00967833	−0.00483917	−	0.999988i	$-0.501540\pi$
$103$	−0.0982244	−0.00967833	−0.00483917	+	0.999988i	$0.501540\pi$
$104$	−2.46739	−0.241947
$105$	3.96291	0.386740
$106$	−6.50521	−0.631842
$107$	12.2160	1.18097	0.590485	−	0.807049i	$-0.298937\pi$
$107$	12.2160	1.18097	0.590485	+	0.807049i	$0.298937\pi$
$108$	−7.72476	−0.743316
$109$	−15.7520	−1.50877	−0.754386	−	0.656431i	$-0.772065\pi$
$109$	−15.7520	−1.50877	−0.754386	+	0.656431i	$0.772065\pi$
$110$	−1.34722	−0.128453
$111$	−27.8139	−2.63998
$112$	−0.742842	−0.0701920
$113$	10.4938	0.987177	0.493589	−	0.869695i	$-0.335685\pi$
$113$	10.4938	0.987177	0.493589	+	0.869695i	$0.335685\pi$
$114$	3.54518	0.332037
$115$	−15.0539	−1.40378
$116$	3.47853	0.322973
$117$	−13.8904	−1.28417
$118$	11.2706	1.03755
$119$	−4.65723	−0.426927
$120$	−5.33479	−0.486998
$121$	−10.4497	−0.949969
$122$	−14.8705	−1.34632
$123$	−7.99076	−0.720503
$124$	−3.68364	−0.330801
$125$	−12.1711	−1.08862
$126$	−4.18191	−0.372554
$127$	11.6733	1.03584	0.517918	−	0.855430i	$-0.326707\pi$
$127$	11.6733	1.03584	0.517918	+	0.855430i	$0.326707\pi$
$128$	1.00000	0.0883883
$129$	10.7843	0.949507
$130$	−4.48084	−0.392996
$131$	6.94260	0.606578	0.303289	−	0.952899i	$-0.401915\pi$
$131$	6.94260	0.606578	0.303289	+	0.952899i	$0.401915\pi$
$132$	2.17928	0.189682
$133$	0.896478	0.0777345
$134$	−1.51826	−0.131158
$135$	−14.0284	−1.20737
$136$	6.26947	0.537603
$137$	−5.01929	−0.428827	−0.214413	−	0.976743i	$-0.568784\pi$
$137$	−5.01929	−0.428827	−0.214413	+	0.976743i	$0.568784\pi$
$138$	24.3513	2.07292
$139$	−15.9130	−1.34972	−0.674860	−	0.737946i	$-0.735796\pi$
$139$	−15.9130	−1.34972	−0.674860	+	0.737946i	$0.735796\pi$
$140$	−1.34902	−0.114013
$141$	25.5709	2.15346
$142$	−0.651591	−0.0546803
$143$	1.83044	0.153069
$144$	5.62960	0.469133
$145$	6.31710	0.524606
$146$	14.6951	1.21617
$147$	18.9423	1.56234
$148$	9.46819	0.778281
$149$	−3.02247	−0.247611	−0.123805	−	0.992307i	$-0.539510\pi$
$149$	−3.02247	−0.247611	−0.123805	+	0.992307i	$0.539510\pi$
$150$	4.99997	0.408246
$151$	8.12872	0.661506	0.330753	−	0.943717i	$-0.392697\pi$
$151$	8.12872	0.661506	0.330753	+	0.943717i	$0.392697\pi$
$152$	−1.20682	−0.0978862
$153$	35.2946	2.85340
$154$	0.551080	0.0444073
$155$	−6.68959	−0.537321
$156$	7.24825	0.580324
$157$	−12.5940	−1.00511	−0.502554	−	0.864546i	$-0.667606\pi$
$157$	−12.5940	−1.00511	−0.502554	+	0.864546i	$0.667606\pi$
$158$	−0.852830	−0.0678475
$159$	19.1098	1.51551
$160$	1.81603	0.143569
$161$	6.15777	0.485300
$162$	5.80361	0.455974
$163$	8.32650	0.652181	0.326091	−	0.945338i	$-0.394268\pi$
$163$	8.32650	0.652181	0.326091	+	0.945338i	$0.394268\pi$
$164$	2.72015	0.212408
$165$	3.95763	0.308101
$166$	−8.67456	−0.673277
$167$	−12.3033	−0.952054	−0.476027	−	0.879431i	$-0.657924\pi$
$167$	−12.3033	−0.952054	−0.476027	+	0.879431i	$0.657924\pi$
$168$	2.18219	0.168359
$169$	−6.91199	−0.531692
$170$	11.3855	0.873230
$171$	−6.79392	−0.519544
$172$	−3.67111	−0.279920
$173$	6.23300	0.473886	0.236943	−	0.971523i	$-0.423854\pi$
$173$	6.23300	0.473886	0.236943	+	0.971523i	$0.423854\pi$
$174$	−10.2186	−0.774669
$175$	1.26435	0.0955761
$176$	−0.741853	−0.0559192
$177$	−33.1089	−2.48861
$178$	−4.60991	−0.345528
$179$	−5.20364	−0.388938	−0.194469	−	0.980909i	$-0.562298\pi$
$179$	−5.20364	−0.388938	−0.194469	+	0.980909i	$0.562298\pi$
$180$	10.2235	0.762015
$181$	11.7292	0.871822	0.435911	−	0.899990i	$-0.356426\pi$
$181$	11.7292	0.871822	0.435911	+	0.899990i	$0.356426\pi$
$182$	1.83288	0.135862
$183$	43.6840	3.22921
$184$	−8.28947	−0.611108
$185$	17.1945	1.26416
$186$	10.8211	0.793444
$187$	−4.65102	−0.340116
$188$	−8.70463	−0.634850
$189$	5.73828	0.417399
$190$	−2.19162	−0.158997
$191$	−13.6489	−0.987602	−0.493801	−	0.869575i	$-0.664393\pi$
$191$	−13.6489	−0.987602	−0.493801	+	0.869575i	$0.664393\pi$
$192$	−2.93762	−0.212004
$193$	−10.0140	−0.720823	−0.360412	−	0.932793i	$-0.617364\pi$
$193$	−10.0140	−0.720823	−0.360412	+	0.932793i	$0.617364\pi$
$194$	2.59641	0.186412
$195$	13.1630	0.942623
$196$	−6.44819	−0.460585
$197$	1.37889	0.0982420	0.0491210	−	0.998793i	$-0.484358\pi$
$197$	1.37889	0.0982420	0.0491210	+	0.998793i	$0.484358\pi$
$198$	−4.17633	−0.296799
$199$	−6.81218	−0.482903	−0.241451	−	0.970413i	$-0.577623\pi$
$199$	−6.81218	−0.482903	−0.241451	+	0.970413i	$0.577623\pi$
$200$	−1.70205	−0.120353
$201$	4.46008	0.314590
$202$	7.42035	0.522094
$203$	−2.58400	−0.181361
$204$	−18.4173	−1.28947
$205$	4.93986	0.345015
$206$	−0.0982244	−0.00684362
$207$	−46.6664	−3.24354
$208$	−2.46739	−0.171083
$209$	0.895284	0.0619281
$210$	3.96291	0.273467
$211$	−12.1268	−0.834845	−0.417422	−	0.908713i	$-0.637066\pi$
$211$	−12.1268	−0.834845	−0.417422	+	0.908713i	$0.637066\pi$
$212$	−6.50521	−0.446780
$213$	1.91413	0.131154
$214$	12.2160	0.835072
$215$	−6.66684	−0.454675
$216$	−7.72476	−0.525604
$217$	2.73637	0.185757
$218$	−15.7520	−1.06686
$219$	−43.1685	−2.91706
$220$	−1.34722	−0.0908298
$221$	−15.4692	−1.04057
$222$	−27.8139	−1.86675
$223$	5.29135	0.354335	0.177167	−	0.984181i	$-0.443307\pi$
$223$	5.29135	0.354335	0.177167	+	0.984181i	$0.443307\pi$
$224$	−0.742842	−0.0496332
$225$	−9.58185	−0.638790
$226$	10.4938	0.698040
$227$	−4.38707	−0.291180	−0.145590	−	0.989345i	$-0.546508\pi$
$227$	−4.38707	−0.291180	−0.145590	+	0.989345i	$0.546508\pi$
$228$	3.54518	0.234785
$229$	−18.9426	−1.25176	−0.625882	−	0.779918i	$-0.715261\pi$
$229$	−18.9426	−1.25176	−0.625882	+	0.779918i	$0.715261\pi$
$230$	−15.0539	−0.992625
$231$	−1.61886	−0.106513
$232$	3.47853	0.228376
$233$	20.5837	1.34848	0.674241	−	0.738511i	$-0.264471\pi$
$233$	20.5837	1.34848	0.674241	+	0.738511i	$0.264471\pi$
$234$	−13.8904	−0.908045
$235$	−15.8078	−1.03119
$236$	11.2706	0.733657
$237$	2.50529	0.162736
$238$	−4.65723	−0.301883
$239$	−3.53922	−0.228933	−0.114467	−	0.993427i	$-0.536516\pi$
$239$	−3.53922	−0.228933	−0.114467	+	0.993427i	$0.536516\pi$
$240$	−5.33479	−0.344359
$241$	−24.3835	−1.57068	−0.785340	−	0.619065i	$-0.787512\pi$
$241$	−24.3835	−1.57068	−0.785340	+	0.619065i	$0.787512\pi$
$242$	−10.4497	−0.671729
$243$	6.12551	0.392952
$244$	−14.8705	−0.951989
$245$	−11.7101	−0.748129
$246$	−7.99076	−0.509472
$247$	2.97770	0.189466
$248$	−3.68364	−0.233912
$249$	25.4826	1.61489
$250$	−12.1711	−0.769768
$251$	10.6885	0.674651	0.337325	−	0.941388i	$-0.390478\pi$
$251$	10.6885	0.674651	0.337325	+	0.941388i	$0.390478\pi$
$252$	−4.18191	−0.263435
$253$	6.14957	0.386620
$254$	11.6733	0.732446
$255$	−33.4463	−2.09449
$256$	1.00000	0.0625000
$257$	−5.27193	−0.328854	−0.164427	−	0.986389i	$-0.552577\pi$
$257$	−5.27193	−0.328854	−0.164427	+	0.986389i	$0.552577\pi$
$258$	10.7843	0.671403
$259$	−7.03337	−0.437033
$260$	−4.48084	−0.277890
$261$	19.5827	1.21214
$262$	6.94260	0.428916
$263$	2.99963	0.184965	0.0924826	−	0.995714i	$-0.470520\pi$
$263$	2.99963	0.184965	0.0924826	+	0.995714i	$0.470520\pi$
$264$	2.17928	0.134125
$265$	−11.8136	−0.725706
$266$	0.896478	0.0549666
$267$	13.5422	0.828767
$268$	−1.51826	−0.0927428
$269$	−9.88093	−0.602451	−0.301225	−	0.953553i	$-0.597396\pi$
$269$	−9.88093	−0.602451	−0.301225	+	0.953553i	$0.597396\pi$
$270$	−14.0284	−0.853740
$271$	−16.7585	−1.01800	−0.509002	−	0.860765i	$-0.669985\pi$
$271$	−16.7585	−1.01800	−0.509002	+	0.860765i	$0.669985\pi$
$272$	6.26947	0.380142
$273$	−5.38430	−0.325873
$274$	−5.01929	−0.303226
$275$	1.26267	0.0761418
$276$	24.3513	1.46578
$277$	−19.9298	−1.19747	−0.598734	−	0.800948i	$-0.704329\pi$
$277$	−19.9298	−1.19747	−0.598734	+	0.800948i	$0.704329\pi$
$278$	−15.9130	−0.954396
$279$	−20.7374	−1.24152
$280$	−1.34902	−0.0806194
$281$	−17.8070	−1.06228	−0.531140	−	0.847284i	$-0.678236\pi$
$281$	−17.8070	−1.06228	−0.531140	+	0.847284i	$0.678236\pi$
$282$	25.5709	1.52272
$283$	−17.2027	−1.02260	−0.511298	−	0.859403i	$-0.670835\pi$
$283$	−17.2027	−1.02260	−0.511298	+	0.859403i	$0.670835\pi$
$284$	−0.651591	−0.0386648
$285$	6.43814	0.381363
$286$	1.83044	0.108236
$287$	−2.02064	−0.119275
$288$	5.62960	0.331727
$289$	22.3063	1.31213
$290$	6.31710	0.370953
$291$	−7.62727	−0.447118
$292$	14.6951	0.859964
$293$	4.48289	0.261893	0.130947	−	0.991389i	$-0.458198\pi$
$293$	4.48289	0.261893	0.130947	+	0.991389i	$0.458198\pi$
$294$	18.9423	1.10474
$295$	20.4678	1.19168
$296$	9.46819	0.550328
$297$	5.73064	0.332525
$298$	−3.02247	−0.175087
$299$	20.4534	1.18285
$300$	4.99997	0.288673
$301$	2.72706	0.157185
$302$	8.12872	0.467755
$303$	−21.7982	−1.25227
$304$	−1.20682	−0.0692160
$305$	−27.0053	−1.54632
$306$	35.2946	2.01766
$307$	−21.0641	−1.20219	−0.601095	−	0.799177i	$-0.705269\pi$
$307$	−21.0641	−1.20219	−0.601095	+	0.799177i	$0.705269\pi$
$308$	0.551080	0.0314007
$309$	0.288546	0.0164148
$310$	−6.68959	−0.379943
$311$	−27.0859	−1.53590	−0.767950	−	0.640510i	$-0.778723\pi$
$311$	−27.0859	−1.53590	−0.767950	+	0.640510i	$0.778723\pi$
$312$	7.24825	0.410351
$313$	1.30920	0.0740002	0.0370001	−	0.999315i	$-0.488220\pi$
$313$	1.30920	0.0740002	0.0370001	+	0.999315i	$0.488220\pi$
$314$	−12.5940	−0.710718
$315$	−7.59445	−0.427899
$316$	−0.852830	−0.0479755
$317$	−4.77664	−0.268283	−0.134141	−	0.990962i	$-0.542828\pi$
$317$	−4.77664	−0.268283	−0.134141	+	0.990962i	$0.542828\pi$
$318$	19.1098	1.07163
$319$	−2.58055	−0.144483
$320$	1.81603	0.101519
$321$	−35.8861	−2.00297
$322$	6.15777	0.343159
$323$	−7.56613	−0.420991
$324$	5.80361	0.322423
$325$	4.19961	0.232953
$326$	8.32650	0.461162
$327$	46.2735	2.55893
$328$	2.72015	0.150195
$329$	6.46617	0.356491
$330$	3.95763	0.217860
$331$	22.8304	1.25487	0.627435	−	0.778669i	$-0.284105\pi$
$331$	22.8304	1.25487	0.627435	+	0.778669i	$0.284105\pi$
$332$	−8.67456	−0.476079
$333$	53.3021	2.92094
$334$	−12.3033	−0.673204
$335$	−2.75721	−0.150642
$336$	2.18219	0.119048
$337$	−23.7527	−1.29389	−0.646945	−	0.762536i	$-0.723954\pi$
$337$	−23.7527	−1.29389	−0.646945	+	0.762536i	$0.723954\pi$
$338$	−6.91199	−0.375963
$339$	−30.8269	−1.67429
$340$	11.3855	0.617467
$341$	2.73272	0.147985
$342$	−6.79392	−0.367373
$343$	9.98988	0.539403
$344$	−3.67111	−0.197933
$345$	44.2226	2.38087
$346$	6.23300	0.335088
$347$	−22.4596	−1.20569	−0.602847	−	0.797857i	$-0.705967\pi$
$347$	−22.4596	−1.20569	−0.602847	+	0.797857i	$0.705967\pi$
$348$	−10.2186	−0.547773
$349$	−3.35445	−0.179560	−0.0897798	−	0.995962i	$-0.528616\pi$
$349$	−3.35445	−0.179560	−0.0897798	+	0.995962i	$0.528616\pi$
$350$	1.26435	0.0675825
$351$	19.0600	1.01735
$352$	−0.741853	−0.0395409
$353$	36.9273	1.96544	0.982720	−	0.185098i	$-0.0592604\pi$
$353$	36.9273	1.96544	0.982720	+	0.185098i	$0.0592604\pi$
$354$	−33.1089	−1.75972
$355$	−1.18331	−0.0628034
$356$	−4.60991	−0.244325
$357$	13.6812	0.724084
$358$	−5.20364	−0.275021
$359$	19.9030	1.05044	0.525221	−	0.850966i	$-0.323983\pi$
$359$	19.9030	1.05044	0.525221	+	0.850966i	$0.323983\pi$
$360$	10.2235	0.538826
$361$	−17.5436	−0.923346
$362$	11.7292	0.616471
$363$	30.6971	1.61118
$364$	1.83288	0.0960691
$365$	26.6866	1.39684
$366$	43.6840	2.28340
$367$	3.23953	0.169102	0.0845510	−	0.996419i	$-0.473054\pi$
$367$	3.23953	0.169102	0.0845510	+	0.996419i	$0.473054\pi$
$368$	−8.28947	−0.432119
$369$	15.3134	0.797181
$370$	17.1945	0.893899
$371$	4.83235	0.250883
$372$	10.8211	0.561050
$373$	−26.3219	−1.36289	−0.681447	−	0.731867i	$-0.738649\pi$
$373$	−26.3219	−1.36289	−0.681447	+	0.731867i	$0.738649\pi$
$374$	−4.65102	−0.240499
$375$	35.7540	1.84633
$376$	−8.70463	−0.448907
$377$	−8.58288	−0.442041
$378$	5.73828	0.295145
$379$	−12.4852	−0.641324	−0.320662	−	0.947194i	$-0.603905\pi$
$379$	−12.4852	−0.641324	−0.320662	+	0.947194i	$0.603905\pi$
$380$	−2.19162	−0.112428
$381$	−34.2916	−1.75681
$382$	−13.6489	−0.698340
$383$	−10.8864	−0.556271	−0.278135	−	0.960542i	$-0.589716\pi$
$383$	−10.8864	−0.556271	−0.278135	+	0.960542i	$0.589716\pi$
$384$	−2.93762	−0.149910
$385$	1.00077	0.0510042
$386$	−10.0140	−0.509699
$387$	−20.6669	−1.05056
$388$	2.59641	0.131813
$389$	−16.1275	−0.817696	−0.408848	−	0.912603i	$-0.634069\pi$
$389$	−16.1275	−0.817696	−0.408848	+	0.912603i	$0.634069\pi$
$390$	13.1630	0.666535
$391$	−51.9706	−2.62827
$392$	−6.44819	−0.325683
$393$	−20.3947	−1.02878
$394$	1.37889	0.0694676
$395$	−1.54876	−0.0779267
$396$	−4.17633	−0.209869
$397$	18.0258	0.904690	0.452345	−	0.891843i	$-0.350588\pi$
$397$	18.0258	0.904690	0.452345	+	0.891843i	$0.350588\pi$
$398$	−6.81218	−0.341464
$399$	−2.63351	−0.131840
$400$	−1.70205	−0.0851024
$401$	−12.7352	−0.635967	−0.317984	−	0.948096i	$-0.603006\pi$
$401$	−12.7352	−0.635967	−0.317984	+	0.948096i	$0.603006\pi$
$402$	4.46008	0.222449
$403$	9.08898	0.452754
$404$	7.42035	0.369176
$405$	10.5395	0.523712
$406$	−2.58400	−0.128242
$407$	−7.02400	−0.348167
$408$	−18.4173	−0.911793
$409$	36.4606	1.80286	0.901430	−	0.432926i	$-0.142519\pi$
$409$	36.4606	1.80286	0.901430	+	0.432926i	$0.142519\pi$
$410$	4.93986	0.243962
$411$	14.7448	0.727305
$412$	−0.0982244	−0.00483917
$413$	−8.37231	−0.411975
$414$	−46.6664	−2.29353
$415$	−15.7532	−0.773296
$416$	−2.46739	−0.120974
$417$	46.7462	2.28917
$418$	0.895284	0.0437898
$419$	−13.0560	−0.637827	−0.318913	−	0.947784i	$-0.603318\pi$
$419$	−13.0560	−0.637827	−0.318913	+	0.947784i	$0.603318\pi$
$420$	3.96291	0.193370
$421$	−33.5800	−1.63659	−0.818296	−	0.574798i	$-0.805081\pi$
$421$	−33.5800	−1.63659	−0.818296	+	0.574798i	$0.805081\pi$
$422$	−12.1268	−0.590324
$423$	−49.0036	−2.38264
$424$	−6.50521	−0.315921
$425$	−10.6709	−0.517617
$426$	1.91413	0.0927397
$427$	11.0465	0.534576
$428$	12.2160	0.590485
$429$	−5.37713	−0.259610
$430$	−6.66684	−0.321503
$431$	−13.7164	−0.660694	−0.330347	−	0.943860i	$-0.607166\pi$
$431$	−13.7164	−0.660694	−0.330347	+	0.943860i	$0.607166\pi$
$432$	−7.72476	−0.371658
$433$	−9.03167	−0.434035	−0.217017	−	0.976168i	$-0.569633\pi$
$433$	−9.03167	−0.434035	−0.217017	+	0.976168i	$0.569633\pi$
$434$	2.73637	0.131350
$435$	−18.5572	−0.889750
$436$	−15.7520	−0.754386
$437$	10.0039	0.478552
$438$	−43.1685	−2.06267
$439$	0.769849	0.0367429	0.0183714	−	0.999831i	$-0.494152\pi$
$439$	0.769849	0.0367429	0.0183714	+	0.999831i	$0.494152\pi$
$440$	−1.34722	−0.0642264
$441$	−36.3007	−1.72861
$442$	−15.4692	−0.735796
$443$	−19.3142	−0.917647	−0.458824	−	0.888527i	$-0.651729\pi$
$443$	−19.3142	−0.917647	−0.458824	+	0.888527i	$0.651729\pi$
$444$	−27.8139	−1.31999
$445$	−8.37173	−0.396858
$446$	5.29135	0.250553
$447$	8.87888	0.419956
$448$	−0.742842	−0.0350960
$449$	−9.19662	−0.434015	−0.217008	−	0.976170i	$-0.569630\pi$
$449$	−9.19662	−0.434015	−0.217008	+	0.976170i	$0.569630\pi$
$450$	−9.58185	−0.451693
$451$	−2.01795	−0.0950215
$452$	10.4938	0.493589
$453$	−23.8791	−1.12194
$454$	−4.38707	−0.205895
$455$	3.32856	0.156045
$456$	3.54518	0.166018
$457$	−35.3208	−1.65224	−0.826120	−	0.563495i	$-0.809456\pi$
$457$	−35.3208	−1.65224	−0.826120	+	0.563495i	$0.809456\pi$
$458$	−18.9426	−0.885131
$459$	−48.4302	−2.26053
$460$	−15.0539	−0.701892
$461$	−33.7440	−1.57162	−0.785808	−	0.618470i	$-0.787753\pi$
$461$	−33.7440	−1.57162	−0.785808	+	0.618470i	$0.787753\pi$
$462$	−1.61886	−0.0753162
$463$	4.19507	0.194961	0.0974807	−	0.995237i	$-0.468922\pi$
$463$	4.19507	0.194961	0.0974807	+	0.995237i	$0.468922\pi$
$464$	3.47853	0.161487
$465$	19.6515	0.911315
$466$	20.5837	0.953521
$467$	34.3443	1.58926	0.794632	−	0.607092i	$-0.207664\pi$
$467$	34.3443	1.58926	0.794632	+	0.607092i	$0.207664\pi$
$468$	−13.8904	−0.642085
$469$	1.12783	0.0520784
$470$	−15.8078	−0.729161
$471$	36.9962	1.70470
$472$	11.2706	0.518774
$473$	2.72342	0.125223
$474$	2.50529	0.115072
$475$	2.05407	0.0942471
$476$	−4.65723	−0.213464
$477$	−36.6218	−1.67679
$478$	−3.53922	−0.161880
$479$	40.3713	1.84461	0.922307	−	0.386459i	$-0.126302\pi$
$479$	40.3713	1.84461	0.922307	+	0.386459i	$0.126302\pi$
$480$	−5.33479	−0.243499
$481$	−23.3617	−1.06520
$482$	−24.3835	−1.11064
$483$	−18.0892	−0.823086
$484$	−10.4497	−0.474984
$485$	4.71515	0.214104
$486$	6.12551	0.277859
$487$	6.49060	0.294117	0.147059	−	0.989128i	$-0.453019\pi$
$487$	6.49060	0.294117	0.147059	+	0.989128i	$0.453019\pi$
$488$	−14.8705	−0.673158
$489$	−24.4601	−1.10612
$490$	−11.7101	−0.529007
$491$	−8.71631	−0.393361	−0.196681	−	0.980468i	$-0.563016\pi$
$491$	−8.71631	−0.393361	−0.196681	+	0.980468i	$0.563016\pi$
$492$	−7.99076	−0.360251
$493$	21.8085	0.982206
$494$	2.97770	0.133973
$495$	−7.58433	−0.340890
$496$	−3.68364	−0.165400
$497$	0.484030	0.0217117
$498$	25.4826	1.14190
$499$	3.21645	0.143988	0.0719940	−	0.997405i	$-0.477064\pi$
$499$	3.21645	0.143988	0.0719940	+	0.997405i	$0.477064\pi$
$500$	−12.1711	−0.544308
$501$	36.1423	1.61472
$502$	10.6885	0.477050
$503$	26.4696	1.18022	0.590111	−	0.807322i	$-0.299084\pi$
$503$	26.4696	1.18022	0.590111	+	0.807322i	$0.299084\pi$
$504$	−4.18191	−0.186277
$505$	13.4756	0.599654
$506$	6.14957	0.273382
$507$	20.3048	0.901767
$508$	11.6733	0.517918
$509$	10.3724	0.459748	0.229874	−	0.973220i	$-0.426169\pi$
$509$	10.3724	0.459748	0.229874	+	0.973220i	$0.426169\pi$
$510$	−33.4463	−1.48103
$511$	−10.9161	−0.482901
$512$	1.00000	0.0441942
$513$	9.32241	0.411595
$514$	−5.27193	−0.232535
$515$	−0.178378	−0.00786028
$516$	10.7843	0.474754
$517$	6.45755	0.284003
$518$	−7.03337	−0.309029
$519$	−18.3102	−0.803728
$520$	−4.48084	−0.196498
$521$	20.9192	0.916486	0.458243	−	0.888827i	$-0.348479\pi$
$521$	20.9192	0.916486	0.458243	+	0.888827i	$0.348479\pi$
$522$	19.5827	0.857112
$523$	22.4282	0.980716	0.490358	−	0.871521i	$-0.336866\pi$
$523$	22.4282	0.980716	0.490358	+	0.871521i	$0.336866\pi$
$524$	6.94260	0.303289
$525$	−3.71419	−0.162100
$526$	2.99963	0.130790
$527$	−23.0945	−1.00601
$528$	2.17928	0.0948410
$529$	45.7154	1.98762
$530$	−11.8136	−0.513152
$531$	63.4492	2.75346
$532$	0.896478	0.0388673
$533$	−6.71167	−0.290715
$534$	13.5422	0.586027
$535$	22.1847	0.959126
$536$	−1.51826	−0.0655790
$537$	15.2863	0.659652
$538$	−9.88093	−0.425997
$539$	4.78360	0.206044
$540$	−14.0284	−0.603685
$541$	−32.9709	−1.41753	−0.708765	−	0.705445i	$-0.750747\pi$
$541$	−32.9709	−1.41753	−0.708765	+	0.705445i	$0.750747\pi$
$542$	−16.7585	−0.719838
$543$	−34.4558	−1.47864
$544$	6.26947	0.268801
$545$	−28.6061	−1.22535
$546$	−5.38430	−0.230427
$547$	−26.2140	−1.12083	−0.560414	−	0.828213i	$-0.689358\pi$
$547$	−26.2140	−1.12083	−0.560414	+	0.828213i	$0.689358\pi$
$548$	−5.01929	−0.214413
$549$	−83.7152	−3.57288
$550$	1.26267	0.0538404
$551$	−4.19796	−0.178839
$552$	24.3513	1.03646
$553$	0.633518	0.0269399
$554$	−19.9298	−0.846737
$555$	−50.5108	−2.14407
$556$	−15.9130	−0.674860
$557$	−15.0784	−0.638891	−0.319445	−	0.947605i	$-0.603497\pi$
$557$	−15.0784	−0.638891	−0.319445	+	0.947605i	$0.603497\pi$
$558$	−20.7374	−0.877886
$559$	9.05806	0.383115
$560$	−1.34902	−0.0570066
$561$	13.6629	0.576849
$562$	−17.8070	−0.751145
$563$	22.4736	0.947150	0.473575	−	0.880753i	$-0.342963\pi$
$563$	22.4736	0.947150	0.473575	+	0.880753i	$0.342963\pi$
$564$	25.5709	1.07673
$565$	19.0571	0.801738
$566$	−17.2027	−0.723085
$567$	−4.31116	−0.181052
$568$	−0.651591	−0.0273402
$569$	37.5821	1.57552	0.787761	−	0.615980i	$-0.211240\pi$
$569$	37.5821	1.57552	0.787761	+	0.615980i	$0.211240\pi$
$570$	6.43814	0.269664
$571$	−39.3818	−1.64808	−0.824039	−	0.566533i	$-0.808284\pi$
$571$	−39.3818	−1.64808	−0.824039	+	0.566533i	$0.808284\pi$
$572$	1.83044	0.0765345
$573$	40.0954	1.67501
$574$	−2.02064	−0.0843400
$575$	14.1091	0.588389
$576$	5.62960	0.234567
$577$	38.4857	1.60218	0.801089	−	0.598545i	$-0.204254\pi$
$577$	38.4857	1.60218	0.801089	+	0.598545i	$0.204254\pi$
$578$	22.3063	0.927818
$579$	29.4173	1.22254
$580$	6.31710	0.262303
$581$	6.44383	0.267335
$582$	−7.62727	−0.316160
$583$	4.82591	0.199869
$584$	14.6951	0.608087
$585$	−25.2254	−1.04294
$586$	4.48289	0.185187
$587$	−11.1574	−0.460514	−0.230257	−	0.973130i	$-0.573957\pi$
$587$	−11.1574	−0.460514	−0.230257	+	0.973130i	$0.573957\pi$
$588$	18.9423	0.781168
$589$	4.44550	0.183174
$590$	20.4678	0.842646
$591$	−4.05066	−0.166622
$592$	9.46819	0.389140
$593$	11.8691	0.487407	0.243704	−	0.969850i	$-0.421638\pi$
$593$	11.8691	0.487407	0.243704	+	0.969850i	$0.421638\pi$
$594$	5.73064	0.235131
$595$	−8.45765	−0.346730
$596$	−3.02247	−0.123805
$597$	20.0116	0.819020
$598$	20.4534	0.836400
$599$	−19.7628	−0.807484	−0.403742	−	0.914873i	$-0.632291\pi$
$599$	−19.7628	−0.807484	−0.403742	+	0.914873i	$0.632291\pi$
$600$	4.99997	0.204123
$601$	10.9820	0.447967	0.223984	−	0.974593i	$-0.428094\pi$
$601$	10.9820	0.447967	0.223984	+	0.974593i	$0.428094\pi$
$602$	2.72706	0.111147
$603$	−8.54722	−0.348070
$604$	8.12872	0.330753
$605$	−18.9768	−0.771519
$606$	−21.7982	−0.885490
$607$	29.6604	1.20388	0.601940	−	0.798541i	$-0.294395\pi$
$607$	29.6604	1.20388	0.601940	+	0.798541i	$0.294395\pi$
$608$	−1.20682	−0.0489431
$609$	7.59079	0.307595
$610$	−27.0053	−1.09341
$611$	21.4777	0.868895
$612$	35.2946	1.42670
$613$	−8.02965	−0.324315	−0.162157	−	0.986765i	$-0.551845\pi$
$613$	−8.02965	−0.324315	−0.162157	+	0.986765i	$0.551845\pi$
$614$	−21.0641	−0.850077
$615$	−14.5114	−0.585157
$616$	0.551080	0.0222036
$617$	43.5305	1.75247	0.876236	−	0.481882i	$-0.160047\pi$
$617$	43.5305	1.75247	0.876236	+	0.481882i	$0.160047\pi$
$618$	0.288546	0.0116070
$619$	10.4816	0.421292	0.210646	−	0.977562i	$-0.432443\pi$
$619$	10.4816	0.421292	0.210646	+	0.977562i	$0.432443\pi$
$620$	−6.68959	−0.268660
$621$	64.0342	2.56961
$622$	−27.0859	−1.08604
$623$	3.42444	0.137197
$624$	7.24825	0.290162
$625$	−13.5928	−0.543712
$626$	1.30920	0.0523260
$627$	−2.63000	−0.105032
$628$	−12.5940	−0.502554
$629$	59.3606	2.36686
$630$	−7.59445	−0.302570
$631$	30.2500	1.20423	0.602117	−	0.798408i	$-0.294324\pi$
$631$	30.2500	1.20423	0.602117	+	0.798408i	$0.294324\pi$
$632$	−0.852830	−0.0339238
$633$	35.6240	1.41593
$634$	−4.77664	−0.189704
$635$	21.1990	0.841256
$636$	19.1098	0.757754
$637$	15.9102	0.630384
$638$	−2.58055	−0.102165
$639$	−3.66820	−0.145112
$640$	1.81603	0.0717847
$641$	−1.46520	−0.0578719	−0.0289359	−	0.999581i	$-0.509212\pi$
$641$	−1.46520	−0.0578719	−0.0289359	+	0.999581i	$0.509212\pi$
$642$	−35.8861	−1.41631
$643$	−0.679021	−0.0267780	−0.0133890	−	0.999910i	$-0.504262\pi$
$643$	−0.679021	−0.0267780	−0.0133890	+	0.999910i	$0.504262\pi$
$644$	6.15777	0.242650
$645$	19.5846	0.771144
$646$	−7.56613	−0.297685
$647$	−38.4355	−1.51105	−0.755527	−	0.655117i	$-0.772619\pi$
$647$	−38.4355	−1.51105	−0.755527	+	0.655117i	$0.772619\pi$
$648$	5.80361	0.227987
$649$	−8.36116	−0.328204
$650$	4.19961	0.164722
$651$	−8.03840	−0.315050
$652$	8.32650	0.326091
$653$	−19.9252	−0.779732	−0.389866	−	0.920872i	$-0.627479\pi$
$653$	−19.9252	−0.779732	−0.389866	+	0.920872i	$0.627479\pi$
$654$	46.2735	1.80944
$655$	12.6080	0.492633
$656$	2.72015	0.106204
$657$	82.7274	3.22750
$658$	6.46617	0.252077
$659$	50.5616	1.96960	0.984800	−	0.173695i	$-0.0555706\pi$
$659$	50.5616	1.96960	0.984800	+	0.173695i	$0.0555706\pi$
$660$	3.95763	0.154051
$661$	−13.2843	−0.516700	−0.258350	−	0.966051i	$-0.583179\pi$
$661$	−13.2843	−0.516700	−0.258350	+	0.966051i	$0.583179\pi$
$662$	22.8304	0.887328
$663$	45.4427	1.76485
$664$	−8.67456	−0.336638
$665$	1.62803	0.0631322
$666$	53.3021	2.06542
$667$	−28.8351	−1.11650
$668$	−12.3033	−0.476027
$669$	−15.5440	−0.600964
$670$	−2.75721	−0.106520
$671$	11.0317	0.425876
$672$	2.18219	0.0841797
$673$	−21.4395	−0.826431	−0.413216	−	0.910633i	$-0.635594\pi$
$673$	−21.4395	−0.826431	−0.413216	+	0.910633i	$0.635594\pi$
$674$	−23.7527	−0.914919
$675$	13.1479	0.506064
$676$	−6.91199	−0.265846
$677$	−22.0218	−0.846368	−0.423184	−	0.906044i	$-0.639088\pi$
$677$	−22.0218	−0.846368	−0.423184	+	0.906044i	$0.639088\pi$
$678$	−30.8269	−1.18390
$679$	−1.92872	−0.0740177
$680$	11.3855	0.436615
$681$	12.8875	0.493851
$682$	2.73272	0.104641
$683$	−12.3605	−0.472963	−0.236482	−	0.971636i	$-0.575994\pi$
$683$	−12.3605	−0.472963	−0.236482	+	0.971636i	$0.575994\pi$
$684$	−6.79392	−0.259772
$685$	−9.11517	−0.348273
$686$	9.98988	0.381415
$687$	55.6462	2.12304
$688$	−3.67111	−0.139960
$689$	16.0509	0.611490
$690$	44.2226	1.68353
$691$	−11.7636	−0.447510	−0.223755	−	0.974645i	$-0.571832\pi$
$691$	−11.7636	−0.447510	−0.223755	+	0.974645i	$0.571832\pi$
$692$	6.23300	0.236943
$693$	3.10236	0.117849
$694$	−22.4596	−0.852554
$695$	−28.8983	−1.09618
$696$	−10.2186	−0.387334
$697$	17.0539	0.645962
$698$	−3.35445	−0.126968
$699$	−60.4670	−2.28707
$700$	1.26435	0.0477881
$701$	1.41980	0.0536250	0.0268125	−	0.999640i	$-0.491464\pi$
$701$	1.41980	0.0536250	0.0268125	+	0.999640i	$0.491464\pi$
$702$	19.0600	0.719373
$703$	−11.4264	−0.430956
$704$	−0.741853	−0.0279596
$705$	46.4374	1.74893
$706$	36.9273	1.38978
$707$	−5.51215	−0.207306
$708$	−33.1089	−1.24431
$709$	−14.0205	−0.526551	−0.263275	−	0.964721i	$-0.584803\pi$
$709$	−14.0205	−0.526551	−0.263275	+	0.964721i	$0.584803\pi$
$710$	−1.18331	−0.0444087
$711$	−4.80110	−0.180055
$712$	−4.60991	−0.172764
$713$	30.5355	1.14356
$714$	13.6812	0.512004
$715$	3.32413	0.124315
$716$	−5.20364	−0.194469
$717$	10.3969	0.388279
$718$	19.9030	0.742775
$719$	−44.9568	−1.67661	−0.838303	−	0.545204i	$-0.816452\pi$
$719$	−44.9568	−1.67661	−0.838303	+	0.545204i	$0.816452\pi$
$720$	10.2235	0.381008
$721$	0.0729652	0.00271737
$722$	−17.5436	−0.652905
$723$	71.6295	2.66393
$724$	11.7292	0.435911
$725$	−5.92062	−0.219886
$726$	30.6971	1.13928
$727$	24.2560	0.899605	0.449803	−	0.893128i	$-0.351494\pi$
$727$	24.2560	0.899605	0.449803	+	0.893128i	$0.351494\pi$
$728$	1.83288	0.0679311
$729$	−35.4052	−1.31131
$730$	26.6866	0.987717
$731$	−23.0159	−0.851275
$732$	43.6840	1.61461
$733$	−9.33837	−0.344921	−0.172460	−	0.985016i	$-0.555172\pi$
$733$	−9.33837	−0.344921	−0.172460	+	0.985016i	$0.555172\pi$
$734$	3.23953	0.119573
$735$	34.3997	1.26885
$736$	−8.28947	−0.305554
$737$	1.12633	0.0414889
$738$	15.3134	0.563692
$739$	35.4317	1.30338	0.651688	−	0.758487i	$-0.274061\pi$
$739$	35.4317	1.30338	0.651688	+	0.758487i	$0.274061\pi$
$740$	17.1945	0.632082
$741$	−8.74734	−0.321342
$742$	4.83235	0.177401
$743$	8.51374	0.312339	0.156169	−	0.987730i	$-0.450085\pi$
$743$	8.51374	0.312339	0.156169	+	0.987730i	$0.450085\pi$
$744$	10.8211	0.396722
$745$	−5.48889	−0.201098
$746$	−26.3219	−0.963712
$747$	−48.8343	−1.78675
$748$	−4.65102	−0.170058
$749$	−9.07459	−0.331578
$750$	35.7540	1.30555
$751$	22.7801	0.831256	0.415628	−	0.909535i	$-0.363562\pi$
$751$	22.7801	0.831256	0.415628	+	0.909535i	$0.363562\pi$
$752$	−8.70463	−0.317425
$753$	−31.3987	−1.14423
$754$	−8.58288	−0.312570
$755$	14.7620	0.537243
$756$	5.73828	0.208699
$757$	−9.18184	−0.333720	−0.166860	−	0.985981i	$-0.553363\pi$
$757$	−9.18184	−0.333720	−0.166860	+	0.985981i	$0.553363\pi$
$758$	−12.4852	−0.453485
$759$	−18.0651	−0.655721
$760$	−2.19162	−0.0794984
$761$	32.0724	1.16262	0.581311	−	0.813681i	$-0.302540\pi$
$761$	32.0724	1.16262	0.581311	+	0.813681i	$0.302540\pi$
$762$	−34.2916	−1.24225
$763$	11.7013	0.423615
$764$	−13.6489	−0.493801
$765$	64.0960	2.31739
$766$	−10.8864	−0.393343
$767$	−27.8091	−1.00413
$768$	−2.93762	−0.106002
$769$	52.4738	1.89225	0.946127	−	0.323795i	$-0.104959\pi$
$769$	52.4738	1.89225	0.946127	+	0.323795i	$0.104959\pi$
$770$	1.00077	0.0360654
$771$	15.4869	0.557747
$772$	−10.0140	−0.360412
$773$	−35.1330	−1.26365	−0.631824	−	0.775112i	$-0.717693\pi$
$773$	−35.1330	−1.26365	−0.631824	+	0.775112i	$0.717693\pi$
$774$	−20.6669	−0.742856
$775$	6.26974	0.225216
$776$	2.59641	0.0932058
$777$	20.6614	0.741222
$778$	−16.1275	−0.578198
$779$	−3.28273	−0.117616
$780$	13.1630	0.471311
$781$	0.483385	0.0172969
$782$	−51.9706	−1.85847
$783$	−26.8708	−0.960284
$784$	−6.44819	−0.230292
$785$	−22.8710	−0.816300
$786$	−20.3947	−0.727456
$787$	−13.0932	−0.466722	−0.233361	−	0.972390i	$-0.574972\pi$
$787$	−13.0932	−0.466722	−0.233361	+	0.972390i	$0.574972\pi$
$788$	1.37889	0.0491210
$789$	−8.81178	−0.313708
$790$	−1.54876	−0.0551025
$791$	−7.79527	−0.277168
$792$	−4.17633	−0.148400
$793$	36.6914	1.30295
$794$	18.0258	0.639712
$795$	34.7040	1.23082
$796$	−6.81218	−0.241451
$797$	−11.7482	−0.416142	−0.208071	−	0.978114i	$-0.566718\pi$
$797$	−11.7482	−0.416142	−0.208071	+	0.978114i	$0.566718\pi$
$798$	−2.63351	−0.0932253
$799$	−54.5734	−1.93067
$800$	−1.70205	−0.0601765
$801$	−25.9520	−0.916968
$802$	−12.7352	−0.449697
$803$	−10.9016	−0.384708
$804$	4.46008	0.157295
$805$	11.1827	0.394138
$806$	9.08898	0.320146
$807$	29.0264	1.02178
$808$	7.42035	0.261047
$809$	30.4039	1.06894	0.534471	−	0.845187i	$-0.320511\pi$
$809$	30.4039	1.06894	0.534471	+	0.845187i	$0.320511\pi$
$810$	10.5395	0.370320
$811$	−28.9298	−1.01586	−0.507931	−	0.861398i	$-0.669590\pi$
$811$	−28.9298	−1.01586	−0.507931	+	0.861398i	$0.669590\pi$
$812$	−2.58400	−0.0906805
$813$	49.2300	1.72657
$814$	−7.02400	−0.246191
$815$	15.1211	0.529670
$816$	−18.4173	−0.644735
$817$	4.43038	0.154999
$818$	36.4606	1.27481
$819$	10.3184	0.360554
$820$	4.93986	0.172507
$821$	22.3286	0.779275	0.389637	−	0.920968i	$-0.372600\pi$
$821$	22.3286	0.779275	0.389637	+	0.920968i	$0.372600\pi$
$822$	14.7448	0.514283
$823$	43.9850	1.53322	0.766610	−	0.642113i	$-0.221942\pi$
$823$	43.9850	1.53322	0.766610	+	0.642113i	$0.221942\pi$
$824$	−0.0982244	−0.00342181
$825$	−3.70924	−0.129139
$826$	−8.37231	−0.291310
$827$	−4.90241	−0.170473	−0.0852367	−	0.996361i	$-0.527165\pi$
$827$	−4.90241	−0.170473	−0.0852367	+	0.996361i	$0.527165\pi$
$828$	−46.6664	−1.62177
$829$	−21.9559	−0.762560	−0.381280	−	0.924460i	$-0.624517\pi$
$829$	−21.9559	−0.762560	−0.381280	+	0.924460i	$0.624517\pi$
$830$	−15.7532	−0.546803
$831$	58.5462	2.03095
$832$	−2.46739	−0.0855413
$833$	−40.4267	−1.40070
$834$	46.7462	1.61869
$835$	−22.3430	−0.773213
$836$	0.895284	0.0309640
$837$	28.4553	0.983558
$838$	−13.0560	−0.451012
$839$	−34.1365	−1.17852	−0.589261	−	0.807943i	$-0.700581\pi$
$839$	−34.1365	−1.17852	−0.589261	+	0.807943i	$0.700581\pi$
$840$	3.96291	0.136733
$841$	−16.8999	−0.582754
$842$	−33.5800	−1.15724
$843$	52.3103	1.80166
$844$	−12.1268	−0.417422
$845$	−12.5524	−0.431814
$846$	−49.0036	−1.68478
$847$	7.76245	0.266721
$848$	−6.50521	−0.223390
$849$	50.5351	1.73436
$850$	−10.6709	−0.366010
$851$	−78.4863	−2.69048
$852$	1.91413	0.0655769
$853$	42.5420	1.45661	0.728305	−	0.685253i	$-0.240308\pi$
$853$	42.5420	1.45661	0.728305	+	0.685253i	$0.240308\pi$
$854$	11.0465	0.378002
$855$	−12.3379	−0.421949
$856$	12.2160	0.417536
$857$	24.0891	0.822869	0.411435	−	0.911439i	$-0.365028\pi$
$857$	24.0891	0.822869	0.411435	+	0.911439i	$0.365028\pi$
$858$	−5.37713	−0.183572
$859$	−29.5461	−1.00810	−0.504050	−	0.863675i	$-0.668157\pi$
$859$	−29.5461	−1.00810	−0.504050	+	0.863675i	$0.668157\pi$
$860$	−6.66684	−0.227337
$861$	5.93587	0.202294
$862$	−13.7164	−0.467181
$863$	24.2046	0.823934	0.411967	−	0.911199i	$-0.364842\pi$
$863$	24.2046	0.823934	0.411967	+	0.911199i	$0.364842\pi$
$864$	−7.72476	−0.262802
$865$	11.3193	0.384868
$866$	−9.03167	−0.306909
$867$	−65.5273	−2.22542
$868$	2.73637	0.0928783
$869$	0.632675	0.0214620
$870$	−18.5572	−0.629148
$871$	3.74615	0.126933
$872$	−15.7520	−0.533432
$873$	14.6168	0.494703
$874$	10.0039	0.338387
$875$	9.04120	0.305649
$876$	−43.1685	−1.45853
$877$	−3.73356	−0.126073	−0.0630366	−	0.998011i	$-0.520078\pi$
$877$	−3.73356	−0.126073	−0.0630366	+	0.998011i	$0.520078\pi$
$878$	0.769849	0.0259811
$879$	−13.1690	−0.444180
$880$	−1.34722	−0.0454149
$881$	53.0820	1.78838	0.894188	−	0.447691i	$-0.147754\pi$
$881$	53.0820	1.78838	0.894188	+	0.447691i	$0.147754\pi$
$882$	−36.3007	−1.22231
$883$	28.4583	0.957698	0.478849	−	0.877897i	$-0.341054\pi$
$883$	28.4583	0.957698	0.478849	+	0.877897i	$0.341054\pi$
$884$	−15.4692	−0.520286
$885$	−60.1266	−2.02113
$886$	−19.3142	−0.648875
$887$	−11.3975	−0.382691	−0.191345	−	0.981523i	$-0.561285\pi$
$887$	−11.3975	−0.382691	−0.191345	+	0.981523i	$0.561285\pi$
$888$	−27.8139	−0.933375
$889$	−8.67140	−0.290829
$890$	−8.37173	−0.280621
$891$	−4.30542	−0.144237
$892$	5.29135	0.177167
$893$	10.5049	0.351534
$894$	8.87888	0.296954
$895$	−9.44994	−0.315877
$896$	−0.742842	−0.0248166
$897$	−60.0842	−2.00615
$898$	−9.19662	−0.306895
$899$	−12.8136	−0.427359
$900$	−9.58185	−0.319395
$901$	−40.7842	−1.35872
$902$	−2.01795	−0.0671904
$903$	−8.01105	−0.266591
$904$	10.4938	0.349020
$905$	21.3005	0.708052
$906$	−23.8791	−0.793330
$907$	48.8251	1.62121	0.810606	−	0.585592i	$-0.199138\pi$
$907$	48.8251	1.62121	0.810606	+	0.585592i	$0.199138\pi$
$908$	−4.38707	−0.145590
$909$	41.7736	1.38554
$910$	3.32856	0.110341
$911$	−32.0646	−1.06235	−0.531173	−	0.847263i	$-0.678249\pi$
$911$	−32.0646	−1.06235	−0.531173	+	0.847263i	$0.678249\pi$
$912$	3.54518	0.117393
$913$	6.43525	0.212976
$914$	−35.3208	−1.16831
$915$	79.3312	2.62261
$916$	−18.9426	−0.625882
$917$	−5.15726	−0.170308
$918$	−48.4302	−1.59843
$919$	31.4897	1.03875	0.519375	−	0.854546i	$-0.326165\pi$
$919$	31.4897	1.03875	0.519375	+	0.854546i	$0.326165\pi$
$920$	−15.0539	−0.496313
$921$	61.8782	2.03896
$922$	−33.7440	−1.11130
$923$	1.60773	0.0529190
$924$	−1.61886	−0.0532566
$925$	−16.1153	−0.529868
$926$	4.19507	0.137859
$927$	−0.552964	−0.0181617
$928$	3.47853	0.114188
$929$	18.9290	0.621041	0.310520	−	0.950567i	$-0.399497\pi$
$929$	18.9290	0.621041	0.310520	+	0.950567i	$0.399497\pi$
$930$	19.6515	0.644397
$931$	7.78181	0.255038
$932$	20.5837	0.674241
$933$	79.5680	2.60494
$934$	34.3443	1.12378
$935$	−8.44638	−0.276226
$936$	−13.8904	−0.454022
$937$	48.9334	1.59858	0.799292	−	0.600943i	$-0.205208\pi$
$937$	48.9334	1.59858	0.799292	+	0.600943i	$0.205208\pi$
$938$	1.12783	0.0368250
$939$	−3.84592	−0.125507
$940$	−15.8078	−0.515595
$941$	43.5606	1.42004	0.710018	−	0.704184i	$-0.248687\pi$
$941$	43.5606	1.42004	0.710018	+	0.704184i	$0.248687\pi$
$942$	36.9962	1.20540
$943$	−22.5486	−0.734284
$944$	11.2706	0.366828
$945$	10.4209	0.338991
$946$	2.72342	0.0885462
$947$	−14.8560	−0.482756	−0.241378	−	0.970431i	$-0.577599\pi$
$947$	−14.8560	−0.482756	−0.241378	+	0.970431i	$0.577599\pi$
$948$	2.50529	0.0813680
$949$	−36.2585	−1.17700
$950$	2.05407	0.0666428
$951$	14.0319	0.455017
$952$	−4.65723	−0.150942
$953$	−56.9454	−1.84464	−0.922322	−	0.386423i	$-0.873710\pi$
$953$	−56.9454	−1.84464	−0.922322	+	0.386423i	$0.873710\pi$
$954$	−36.6218	−1.18567
$955$	−24.7868	−0.802083
$956$	−3.53922	−0.114467
$957$	7.58068	0.245049
$958$	40.3713	1.30434
$959$	3.72854	0.120401
$960$	−5.33479	−0.172180
$961$	−17.4308	−0.562283
$962$	−23.3617	−0.753212
$963$	68.7715	2.21613
$964$	−24.3835	−0.785340
$965$	−18.1857	−0.585418
$966$	−18.0892	−0.582010
$967$	25.6876	0.826058	0.413029	−	0.910718i	$-0.364471\pi$
$967$	25.6876	0.826058	0.413029	+	0.910718i	$0.364471\pi$
$968$	−10.4497	−0.335865
$969$	22.2264	0.714015
$970$	4.71515	0.151394
$971$	23.1475	0.742838	0.371419	−	0.928465i	$-0.378871\pi$
$971$	23.1475	0.742838	0.371419	+	0.928465i	$0.378871\pi$
$972$	6.12551	0.196476
$973$	11.8208	0.378958
$974$	6.49060	0.207972
$975$	−12.3369	−0.395096
$976$	−14.8705	−0.475994
$977$	−15.7712	−0.504566	−0.252283	−	0.967654i	$-0.581181\pi$
$977$	−15.7712	−0.504566	−0.252283	+	0.967654i	$0.581181\pi$
$978$	−24.4601	−0.782147
$979$	3.41988	0.109300
$980$	−11.7101	−0.374065
$981$	−88.6777	−2.83126
$982$	−8.71631	−0.278149
$983$	−18.3368	−0.584853	−0.292427	−	0.956288i	$-0.594463\pi$
$983$	−18.3368	−0.584853	−0.292427	+	0.956288i	$0.594463\pi$
$984$	−7.99076	−0.254736
$985$	2.50410	0.0797874
$986$	21.8085	0.694525
$987$	−18.9951	−0.604622
$988$	2.97770	0.0947332
$989$	30.4316	0.967668
$990$	−7.58433	−0.241046
$991$	−4.56872	−0.145130	−0.0725651	−	0.997364i	$-0.523119\pi$
$991$	−4.56872	−0.145130	−0.0725651	+	0.997364i	$0.523119\pi$
$992$	−3.68364	−0.116956
$993$	−67.0669	−2.12830
$994$	0.484030	0.0153525
$995$	−12.3711	−0.392190
$996$	25.4826	0.807446
$997$	1.93148	0.0611705	0.0305852	−	0.999532i	$-0.490263\pi$
$997$	1.93148	0.0611705	0.0305852	+	0.999532i	$0.490263\pi$
$998$	3.21645	0.101815
$999$	−73.1396	−2.31403

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6046.2.a.e.1.5	✓	56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6046.2.a.e.1.5	✓	56	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.93762 q^{3} +1.00000 q^{4} +1.81603 q^{5} -2.93762 q^{6} -0.742842 q^{7} +1.00000 q^{8} +5.62960 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.93762 q^{3} +1.00000 q^{4} +1.81603 q^{5} -2.93762 q^{6} -0.742842 q^{7} +1.00000 q^{8} +5.62960 q^{9} +1.81603 q^{10} -0.741853 q^{11} -2.93762 q^{12} -2.46739 q^{13} -0.742842 q^{14} -5.33479 q^{15} +1.00000 q^{16} +6.26947 q^{17} +5.62960 q^{18} -1.20682 q^{19} +1.81603 q^{20} +2.18219 q^{21} -0.741853 q^{22} -8.28947 q^{23} -2.93762 q^{24} -1.70205 q^{25} -2.46739 q^{26} -7.72476 q^{27} -0.742842 q^{28} +3.47853 q^{29} -5.33479 q^{30} -3.68364 q^{31} +1.00000 q^{32} +2.17928 q^{33} +6.26947 q^{34} -1.34902 q^{35} +5.62960 q^{36} +9.46819 q^{37} -1.20682 q^{38} +7.24825 q^{39} +1.81603 q^{40} +2.72015 q^{41} +2.18219 q^{42} -3.67111 q^{43} -0.741853 q^{44} +10.2235 q^{45} -8.28947 q^{46} -8.70463 q^{47} -2.93762 q^{48} -6.44819 q^{49} -1.70205 q^{50} -18.4173 q^{51} -2.46739 q^{52} -6.50521 q^{53} -7.72476 q^{54} -1.34722 q^{55} -0.742842 q^{56} +3.54518 q^{57} +3.47853 q^{58} +11.2706 q^{59} -5.33479 q^{60} -14.8705 q^{61} -3.68364 q^{62} -4.18191 q^{63} +1.00000 q^{64} -4.48084 q^{65} +2.17928 q^{66} -1.51826 q^{67} +6.26947 q^{68} +24.3513 q^{69} -1.34902 q^{70} -0.651591 q^{71} +5.62960 q^{72} +14.6951 q^{73} +9.46819 q^{74} +4.99997 q^{75} -1.20682 q^{76} +0.551080 q^{77} +7.24825 q^{78} -0.852830 q^{79} +1.81603 q^{80} +5.80361 q^{81} +2.72015 q^{82} -8.67456 q^{83} +2.18219 q^{84} +11.3855 q^{85} -3.67111 q^{86} -10.2186 q^{87} -0.741853 q^{88} -4.60991 q^{89} +10.2235 q^{90} +1.83288 q^{91} -8.28947 q^{92} +10.8211 q^{93} -8.70463 q^{94} -2.19162 q^{95} -2.93762 q^{96} +2.59641 q^{97} -6.44819 q^{98} -4.17633 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q + 56 q^{2} - 18 q^{3} + 56 q^{4} - 17 q^{5} - 18 q^{6} - 35 q^{7} + 56 q^{8} + 34 q^{9}+O(q^{10}) \) 56 * q + 56 * q^2 - 18 * q^3 + 56 * q^4 - 17 * q^5 - 18 * q^6 - 35 * q^7 + 56 * q^8 + 34 * q^9	\( 56 q + 56 q^{2} - 18 q^{3} + 56 q^{4} - 17 q^{5} - 18 q^{6} - 35 q^{7} + 56 q^{8} + 34 q^{9} - 17 q^{10} - 53 q^{11} - 18 q^{12} - 21 q^{13} - 35 q^{14} - 36 q^{15} + 56 q^{16} - 22 q^{17} + 34 q^{18} - 31 q^{19} - 17 q^{20} - 23 q^{21} - 53 q^{22} - 59 q^{23} - 18 q^{24} + 41 q^{25} - 21 q^{26} - 63 q^{27} - 35 q^{28} - 88 q^{29} - 36 q^{30} - 44 q^{31} + 56 q^{32} + 4 q^{33} - 22 q^{34} - 51 q^{35} + 34 q^{36} - 60 q^{37} - 31 q^{38} - 62 q^{39} - 17 q^{40} - 39 q^{41} - 23 q^{42} - 66 q^{43} - 53 q^{44} - 34 q^{45} - 59 q^{46} - 51 q^{47} - 18 q^{48} + 41 q^{49} + 41 q^{50} - 48 q^{51} - 21 q^{52} - 75 q^{53} - 63 q^{54} - 41 q^{55} - 35 q^{56} - 12 q^{57} - 88 q^{58} - 77 q^{59} - 36 q^{60} - 43 q^{61} - 44 q^{62} - 88 q^{63} + 56 q^{64} - 54 q^{65} + 4 q^{66} - 62 q^{67} - 22 q^{68} - 48 q^{69} - 51 q^{70} - 122 q^{71} + 34 q^{72} - 7 q^{73} - 60 q^{74} - 63 q^{75} - 31 q^{76} - 39 q^{77} - 62 q^{78} - 91 q^{79} - 17 q^{80} + 8 q^{81} - 39 q^{82} - 51 q^{83} - 23 q^{84} - 72 q^{85} - 66 q^{86} - 19 q^{87} - 53 q^{88} - 62 q^{89} - 34 q^{90} - 48 q^{91} - 59 q^{92} - 41 q^{93} - 51 q^{94} - 120 q^{95} - 18 q^{96} + 6 q^{97} + 41 q^{98} - 128 q^{99}+O(q^{100}) \) 56 * q + 56 * q^2 - 18 * q^3 + 56 * q^4 - 17 * q^5 - 18 * q^6 - 35 * q^7 + 56 * q^8 + 34 * q^9 - 17 * q^10 - 53 * q^11 - 18 * q^12 - 21 * q^13 - 35 * q^14 - 36 * q^15 + 56 * q^16 - 22 * q^17 + 34 * q^18 - 31 * q^19 - 17 * q^20 - 23 * q^21 - 53 * q^22 - 59 * q^23 - 18 * q^24 + 41 * q^25 - 21 * q^26 - 63 * q^27 - 35 * q^28 - 88 * q^29 - 36 * q^30 - 44 * q^31 + 56 * q^32 + 4 * q^33 - 22 * q^34 - 51 * q^35 + 34 * q^36 - 60 * q^37 - 31 * q^38 - 62 * q^39 - 17 * q^40 - 39 * q^41 - 23 * q^42 - 66 * q^43 - 53 * q^44 - 34 * q^45 - 59 * q^46 - 51 * q^47 - 18 * q^48 + 41 * q^49 + 41 * q^50 - 48 * q^51 - 21 * q^52 - 75 * q^53 - 63 * q^54 - 41 * q^55 - 35 * q^56 - 12 * q^57 - 88 * q^58 - 77 * q^59 - 36 * q^60 - 43 * q^61 - 44 * q^62 - 88 * q^63 + 56 * q^64 - 54 * q^65 + 4 * q^66 - 62 * q^67 - 22 * q^68 - 48 * q^69 - 51 * q^70 - 122 * q^71 + 34 * q^72 - 7 * q^73 - 60 * q^74 - 63 * q^75 - 31 * q^76 - 39 * q^77 - 62 * q^78 - 91 * q^79 - 17 * q^80 + 8 * q^81 - 39 * q^82 - 51 * q^83 - 23 * q^84 - 72 * q^85 - 66 * q^86 - 19 * q^87 - 53 * q^88 - 62 * q^89 - 34 * q^90 - 48 * q^91 - 59 * q^92 - 41 * q^93 - 51 * q^94 - 120 * q^95 - 18 * q^96 + 6 * q^97 + 41 * q^98 - 128 * q^99

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