LMFDB - Embedded newform 845.2.a.m.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [845,2,Mod(1,845)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(845, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("845.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 845 = 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	845.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:	$6.74735897080$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.4752.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 3x^{2} + 4x + 1 $ x^4 - 2x^3 - 3x^2 + 4*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 65)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.49551$ of defining polynomial
Character	$\chi$	$=$	845.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.49551 q^{2} -2.82684 q^{3} +4.22756 q^{4} -1.00000 q^{5} -7.05440 q^{6} +1.90521 q^{7} +5.55889 q^{8} +4.99102 q^{9} +O(q^{10})$	\(q+2.49551 q^{2} -2.82684 q^{3} +4.22756 q^{4} -1.00000 q^{5} -7.05440 q^{6} +1.90521 q^{7} +5.55889 q^{8} +4.99102 q^{9} -2.49551 q^{10} +1.06939 q^{11} -11.9506 q^{12} +4.75447 q^{14} +2.82684 q^{15} +5.41713 q^{16} +0.637263 q^{17} +12.4551 q^{18} +5.73205 q^{19} -4.22756 q^{20} -5.38573 q^{21} +2.66867 q^{22} +3.81785 q^{23} -15.7141 q^{24} +1.00000 q^{25} -5.62828 q^{27} +8.05440 q^{28} +9.45512 q^{29} +7.05440 q^{30} -1.46410 q^{31} +2.40072 q^{32} -3.02299 q^{33} +1.59030 q^{34} -1.90521 q^{35} +21.0998 q^{36} +0.757449 q^{37} +14.3044 q^{38} -5.55889 q^{40} +0.267949 q^{41} -13.4401 q^{42} +0.637263 q^{43} +4.52091 q^{44} -4.99102 q^{45} +9.52748 q^{46} -9.44613 q^{47} -15.3134 q^{48} -3.37017 q^{49} +2.49551 q^{50} -1.80144 q^{51} -6.99102 q^{53} -14.0454 q^{54} -1.06939 q^{55} +10.5909 q^{56} -16.2036 q^{57} +23.5953 q^{58} +0.741035 q^{59} +11.9506 q^{60} +4.19856 q^{61} -3.65368 q^{62} +9.50894 q^{63} -4.84325 q^{64} -7.54390 q^{66} +8.09479 q^{67} +2.69407 q^{68} -10.7925 q^{69} -4.75447 q^{70} +9.76488 q^{71} +27.7445 q^{72} -3.71649 q^{73} +1.89022 q^{74} -2.82684 q^{75} +24.2326 q^{76} +2.03741 q^{77} -9.31937 q^{79} -5.41713 q^{80} +0.937188 q^{81} +0.668669 q^{82} +5.11778 q^{83} -22.7685 q^{84} -0.637263 q^{85} +1.59030 q^{86} -26.7281 q^{87} +5.94462 q^{88} -12.5783 q^{89} -12.4551 q^{90} +16.1402 q^{92} +4.13878 q^{93} -23.5729 q^{94} -5.73205 q^{95} -6.78645 q^{96} -4.22155 q^{97} -8.41027 q^{98} +5.33734 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 4 q^{5} - 4 q^{6} + 10 q^{7} + 6 q^{8} + 4 q^{9}+O(q^{10}) $ 4 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 4 * q^5 - 4 * q^6 + 10 * q^7 + 6 * q^8 + 4 * q^9	\( 4 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 4 q^{5} - 4 q^{6} + 10 q^{7} + 6 q^{8} + 4 q^{9} - 2 q^{10} - 10 q^{12} + 2 q^{14} + 2 q^{15} + 2 q^{16} - 2 q^{17} + 20 q^{18} + 16 q^{19} - 2 q^{20} + 4 q^{21} + 12 q^{22} - 10 q^{23} - 24 q^{24} + 4 q^{25} - 2 q^{27} + 8 q^{28} + 8 q^{29} + 4 q^{30} + 8 q^{31} + 4 q^{32} + 18 q^{33} - 4 q^{34} - 10 q^{35} + 20 q^{36} - 2 q^{37} + 8 q^{38} - 6 q^{40} + 8 q^{41} - 4 q^{42} - 2 q^{43} + 12 q^{44} - 4 q^{45} + 16 q^{46} + 8 q^{47} - 28 q^{48} + 12 q^{49} + 2 q^{50} + 4 q^{51} - 12 q^{53} - 16 q^{54} + 12 q^{56} - 14 q^{57} + 22 q^{58} + 12 q^{59} + 10 q^{60} + 28 q^{61} + 4 q^{62} + 4 q^{63} + 4 q^{64} + 6 q^{66} + 30 q^{67} + 14 q^{68} - 16 q^{69} - 2 q^{70} + 4 q^{71} + 12 q^{72} - 8 q^{73} - 10 q^{74} - 2 q^{75} + 20 q^{76} + 18 q^{77} - 8 q^{79} - 2 q^{80} - 8 q^{81} + 4 q^{82} - 12 q^{83} - 28 q^{84} + 2 q^{85} - 4 q^{86} - 22 q^{87} - 18 q^{88} - 12 q^{89} - 20 q^{90} + 22 q^{92} + 8 q^{93} - 32 q^{94} - 16 q^{95} + 4 q^{96} + 2 q^{97} - 24 q^{98} + 24 q^{99}+O(q^{100}) \) 4 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 4 * q^5 - 4 * q^6 + 10 * q^7 + 6 * q^8 + 4 * q^9 - 2 * q^10 - 10 * q^12 + 2 * q^14 + 2 * q^15 + 2 * q^16 - 2 * q^17 + 20 * q^18 + 16 * q^19 - 2 * q^20 + 4 * q^21 + 12 * q^22 - 10 * q^23 - 24 * q^24 + 4 * q^25 - 2 * q^27 + 8 * q^28 + 8 * q^29 + 4 * q^30 + 8 * q^31 + 4 * q^32 + 18 * q^33 - 4 * q^34 - 10 * q^35 + 20 * q^36 - 2 * q^37 + 8 * q^38 - 6 * q^40 + 8 * q^41 - 4 * q^42 - 2 * q^43 + 12 * q^44 - 4 * q^45 + 16 * q^46 + 8 * q^47 - 28 * q^48 + 12 * q^49 + 2 * q^50 + 4 * q^51 - 12 * q^53 - 16 * q^54 + 12 * q^56 - 14 * q^57 + 22 * q^58 + 12 * q^59 + 10 * q^60 + 28 * q^61 + 4 * q^62 + 4 * q^63 + 4 * q^64 + 6 * q^66 + 30 * q^67 + 14 * q^68 - 16 * q^69 - 2 * q^70 + 4 * q^71 + 12 * q^72 - 8 * q^73 - 10 * q^74 - 2 * q^75 + 20 * q^76 + 18 * q^77 - 8 * q^79 - 2 * q^80 - 8 * q^81 + 4 * q^82 - 12 * q^83 - 28 * q^84 + 2 * q^85 - 4 * q^86 - 22 * q^87 - 18 * q^88 - 12 * q^89 - 20 * q^90 + 22 * q^92 + 8 * q^93 - 32 * q^94 - 16 * q^95 + 4 * q^96 + 2 * q^97 - 24 * q^98 + 24 * 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 $

$2$	2.49551	1.76459	0.882295	−	0.470696i	$-0.155997\pi$
$2$	2.49551	1.76459	0.882295	+	0.470696i	$0.155997\pi$
$3$	−2.82684	−1.63208	−0.816038	−	0.577998i	$-0.803834\pi$
$3$	−2.82684	−1.63208	−0.816038	+	0.577998i	$0.803834\pi$
$4$	4.22756	2.11378
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−7.05440	−2.87995
$7$	1.90521	0.720103	0.360051	−	0.932933i	$-0.382759\pi$
$7$	1.90521	0.720103	0.360051	+	0.932933i	$0.382759\pi$
$8$	5.55889	1.96536
$9$	4.99102	1.66367
$10$	−2.49551	−0.789149
$11$	1.06939	0.322433	0.161217	−	0.986919i	$-0.448458\pi$
$11$	1.06939	0.322433	0.161217	+	0.986919i	$0.448458\pi$
$12$	−11.9506	−3.44985
$13$	0	0
$13$	0	0
$14$	4.75447	1.27069
$15$	2.82684	0.729887
$16$	5.41713	1.35428
$17$	0.637263	0.154559	0.0772795	−	0.997009i	$-0.475377\pi$
$17$	0.637263	0.154559	0.0772795	+	0.997009i	$0.475377\pi$
$18$	12.4551	2.93570
$19$	5.73205	1.31502	0.657511	−	0.753445i	$-0.271609\pi$
$19$	5.73205	1.31502	0.657511	+	0.753445i	$0.271609\pi$
$20$	−4.22756	−0.945311
$21$	−5.38573	−1.17526
$22$	2.66867	0.568962
$23$	3.81785	0.796078	0.398039	−	0.917369i	$-0.369691\pi$
$23$	3.81785	0.796078	0.398039	+	0.917369i	$0.369691\pi$
$24$	−15.7141	−3.20762
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.62828	−1.08316
$28$	8.05440	1.52214
$29$	9.45512	1.75577	0.877886	−	0.478870i	$-0.158954\pi$
$29$	9.45512	1.75577	0.877886	+	0.478870i	$0.158954\pi$
$30$	7.05440	1.28795
$31$	−1.46410	−0.262960	−0.131480	−	0.991319i	$-0.541973\pi$
$31$	−1.46410	−0.262960	−0.131480	+	0.991319i	$0.541973\pi$
$32$	2.40072	0.424391
$33$	−3.02299	−0.526235
$34$	1.59030	0.272733
$35$	−1.90521	−0.322040
$36$	21.0998	3.51663
$37$	0.757449	0.124524	0.0622619	−	0.998060i	$-0.480169\pi$
$37$	0.757449	0.124524	0.0622619	+	0.998060i	$0.480169\pi$
$38$	14.3044	2.32048
$39$	0	0
$40$	−5.55889	−0.878938
$41$	0.267949	0.0418466	0.0209233	−	0.999781i	$-0.493339\pi$
$41$	0.267949	0.0418466	0.0209233	+	0.999781i	$0.493339\pi$
$42$	−13.4401	−2.07386
$43$	0.637263	0.0971817	0.0485909	−	0.998819i	$-0.484527\pi$
$43$	0.637263	0.0971817	0.0485909	+	0.998819i	$0.484527\pi$
$44$	4.52091	0.681552
$45$	−4.99102	−0.744017
$46$	9.52748	1.40475
$47$	−9.44613	−1.37786	−0.688930	−	0.724828i	$-0.741919\pi$
$47$	−9.44613	−1.37786	−0.688930	+	0.724828i	$0.741919\pi$
$48$	−15.3134	−2.21029
$49$	−3.37017	−0.481452
$50$	2.49551	0.352918
$51$	−1.80144	−0.252252
$52$	0	0
$53$	−6.99102	−0.960290	−0.480145	−	0.877189i	$-0.659416\pi$
$53$	−6.99102	−0.960290	−0.480145	+	0.877189i	$0.659416\pi$
$54$	−14.0454	−1.91134
$55$	−1.06939	−0.144196
$56$	10.5909	1.41526
$57$	−16.2036	−2.14622
$58$	23.5953	3.09822
$59$	0.741035	0.0964746	0.0482373	−	0.998836i	$-0.484640\pi$
$59$	0.741035	0.0964746	0.0482373	+	0.998836i	$0.484640\pi$
$60$	11.9506	1.54282
$61$	4.19856	0.537571	0.268785	−	0.963200i	$-0.413378\pi$
$61$	4.19856	0.537571	0.268785	+	0.963200i	$0.413378\pi$
$62$	−3.65368	−0.464017
$63$	9.50894	1.19801
$64$	−4.84325	−0.605406
$65$	0	0
$66$	−7.54390	−0.928589
$67$	8.09479	0.988936	0.494468	−	0.869196i	$-0.335363\pi$
$67$	8.09479	0.988936	0.494468	+	0.869196i	$0.335363\pi$
$68$	2.69407	0.326704
$69$	−10.7925	−1.29926
$70$	−4.75447	−0.568268
$71$	9.76488	1.15888	0.579439	−	0.815016i	$-0.303272\pi$
$71$	9.76488	1.15888	0.579439	+	0.815016i	$0.303272\pi$
$72$	27.7445	3.26972
$73$	−3.71649	−0.434982	−0.217491	−	0.976062i	$-0.569787\pi$
$73$	−3.71649	−0.434982	−0.217491	+	0.976062i	$0.569787\pi$
$74$	1.89022	0.219734
$75$	−2.82684	−0.326415
$76$	24.2326	2.77967
$77$	2.03741	0.232185
$78$	0	0
$79$	−9.31937	−1.04851	−0.524255	−	0.851561i	$-0.675656\pi$
$79$	−9.31937	−1.04851	−0.524255	+	0.851561i	$0.675656\pi$
$80$	−5.41713	−0.605654
$81$	0.937188	0.104132
$82$	0.668669	0.0738422
$83$	5.11778	0.561749	0.280875	−	0.959744i	$-0.409376\pi$
$83$	5.11778	0.561749	0.280875	+	0.959744i	$0.409376\pi$
$84$	−22.7685	−2.48424
$85$	−0.637263	−0.0691209
$86$	1.59030	0.171486
$87$	−26.7281	−2.86555
$88$	5.94462	0.633698
$89$	−12.5783	−1.33330	−0.666650	−	0.745371i	$-0.732273\pi$
$89$	−12.5783	−1.33330	−0.666650	+	0.745371i	$0.732273\pi$
$90$	−12.4551	−1.31288
$91$	0	0
$92$	16.1402	1.68273
$93$	4.13878	0.429171
$94$	−23.5729	−2.43136
$95$	−5.73205	−0.588096
$96$	−6.78645	−0.692639
$97$	−4.22155	−0.428634	−0.214317	−	0.976764i	$-0.568752\pi$
$97$	−4.22155	−0.428634	−0.214317	+	0.976764i	$0.568752\pi$
$98$	−8.41027	−0.849566
$99$	5.33734	0.536423
$100$	4.22756	0.422756
$101$	−15.2476	−1.51719	−0.758595	−	0.651562i	$-0.774114\pi$
$101$	−15.2476	−1.51719	−0.758595	+	0.651562i	$0.774114\pi$
$102$	−4.49551	−0.445122
$103$	−13.5269	−1.33285	−0.666423	−	0.745574i	$-0.732176\pi$
$103$	−13.5269	−1.33285	−0.666423	+	0.745574i	$0.732176\pi$
$104$	0	0
$105$	5.38573	0.525593
$106$	−17.4461	−1.69452
$107$	−7.36274	−0.711783	−0.355891	−	0.934527i	$-0.615823\pi$
$107$	−7.36274	−0.711783	−0.355891	+	0.934527i	$0.615823\pi$
$108$	−23.7939	−2.28957
$109$	10.0760	0.965103	0.482551	−	0.875868i	$-0.339710\pi$
$109$	10.0760	0.965103	0.482551	+	0.875868i	$0.339710\pi$
$110$	−2.66867	−0.254448
$111$	−2.14119	−0.203232
$112$	10.3208	0.975223
$113$	−6.68806	−0.629160	−0.314580	−	0.949231i	$-0.601864\pi$
$113$	−6.68806	−0.629160	−0.314580	+	0.949231i	$0.601864\pi$
$114$	−40.4362	−3.78719
$115$	−3.81785	−0.356017
$116$	39.9721	3.71131
$117$	0	0
$118$	1.84926	0.170238
$119$	1.21412	0.111298
$120$	15.7141	1.43449
$121$	−9.85641	−0.896037
$122$	10.4775	0.948592
$123$	−0.757449	−0.0682969
$124$	−6.18958	−0.555840
$125$	−1.00000	−0.0894427
$126$	23.7296	2.11400
$127$	1.48950	0.132172	0.0660859	−	0.997814i	$-0.478949\pi$
$127$	1.48950	0.132172	0.0660859	+	0.997814i	$0.478949\pi$
$128$	−16.8878	−1.49269
$129$	−1.80144	−0.158608
$130$	0	0
$131$	4.12676	0.360557	0.180278	−	0.983616i	$-0.442300\pi$
$131$	4.12676	0.360557	0.180278	+	0.983616i	$0.442300\pi$
$132$	−12.7799	−1.11234
$133$	10.9208	0.946951
$134$	20.2006	1.74507
$135$	5.62828	0.484405
$136$	3.54248	0.303765
$137$	−20.1096	−1.71808	−0.859041	−	0.511906i	$-0.828940\pi$
$137$	−20.1096	−1.71808	−0.859041	+	0.511906i	$0.828940\pi$
$138$	−26.9327	−2.29266
$139$	20.8253	1.76638	0.883189	−	0.469018i	$-0.155392\pi$
$139$	20.8253	1.76638	0.883189	+	0.469018i	$0.155392\pi$
$140$	−8.05440	−0.680721
$141$	26.7027	2.24877
$142$	24.3683	2.04494
$143$	0	0
$144$	27.0370	2.25308
$145$	−9.45512	−0.785205
$146$	−9.27453	−0.767565
$147$	9.52691	0.785767
$148$	3.20216	0.263216
$149$	13.3678	1.09513	0.547565	−	0.836763i	$-0.315555\pi$
$149$	13.3678	1.09513	0.547565	+	0.836763i	$0.315555\pi$
$150$	−7.05440	−0.575989
$151$	18.2984	1.48910	0.744550	−	0.667567i	$-0.232664\pi$
$151$	18.2984	1.48910	0.744550	+	0.667567i	$0.232664\pi$
$152$	31.8638	2.58450
$153$	3.18059	0.257136
$154$	5.08438	0.409711
$155$	1.46410	0.117599
$156$	0	0
$157$	2.42229	0.193320	0.0966599	−	0.995317i	$-0.469184\pi$
$157$	2.42229	0.193320	0.0966599	+	0.995317i	$0.469184\pi$
$158$	−23.2566	−1.85019
$159$	19.7625	1.56727
$160$	−2.40072	−0.189794
$161$	7.27382	0.573258
$162$	2.33876	0.183750
$163$	15.9829	1.25188	0.625938	−	0.779873i	$-0.284716\pi$
$163$	15.9829	1.25188	0.625938	+	0.779873i	$0.284716\pi$
$164$	1.13277	0.0884545
$165$	3.02299	0.235340
$166$	12.7715	0.991257
$167$	−14.3932	−1.11378	−0.556888	−	0.830588i	$-0.688005\pi$
$167$	−14.3932	−1.11378	−0.556888	+	0.830588i	$0.688005\pi$
$168$	−29.9387	−2.30982
$169$	0	0
$170$	−1.59030	−0.121970
$171$	28.6088	2.18777
$172$	2.69407	0.205421
$173$	−24.3489	−1.85122	−0.925608	−	0.378484i	$-0.876445\pi$
$173$	−24.3489	−1.85122	−0.925608	+	0.378484i	$0.876445\pi$
$174$	−66.7001	−5.05653
$175$	1.90521	0.144021
$176$	5.79302	0.436666
$177$	−2.09479	−0.157454
$178$	−31.3893	−2.35273
$179$	3.78829	0.283150	0.141575	−	0.989928i	$-0.454783\pi$
$179$	3.78829	0.283150	0.141575	+	0.989928i	$0.454783\pi$
$180$	−21.0998	−1.57269
$181$	−8.48794	−0.630904	−0.315452	−	0.948942i	$-0.602156\pi$
$181$	−8.48794	−0.630904	−0.315452	+	0.948942i	$0.602156\pi$
$182$	0	0
$183$	−11.8687	−0.877356
$184$	21.2230	1.56458
$185$	−0.757449	−0.0556888
$186$	10.3284	0.757312
$187$	0.681482	0.0498349
$188$	−39.9341	−2.91249
$189$	−10.7231	−0.779988
$190$	−14.3044	−1.03775
$191$	−5.44310	−0.393849	−0.196924	−	0.980419i	$-0.563095\pi$
$191$	−5.44310	−0.393849	−0.196924	+	0.980419i	$0.563095\pi$
$192$	13.6911	0.988069
$193$	12.1576	0.875123	0.437562	−	0.899188i	$-0.355842\pi$
$193$	12.1576	0.875123	0.437562	+	0.899188i	$0.355842\pi$
$194$	−10.5349	−0.756363
$195$	0	0
$196$	−14.2476	−1.01768
$197$	−4.37830	−0.311941	−0.155970	−	0.987762i	$-0.549850\pi$
$197$	−4.37830	−0.311941	−0.155970	+	0.987762i	$0.549850\pi$
$198$	13.3194	0.946566
$199$	20.8373	1.47712	0.738558	−	0.674189i	$-0.235507\pi$
$199$	20.8373	1.47712	0.738558	+	0.674189i	$0.235507\pi$
$200$	5.55889	0.393073
$201$	−22.8827	−1.61402
$202$	−38.0504	−2.67722
$203$	18.0140	1.26434
$204$	−7.61569	−0.533205
$205$	−0.267949	−0.0187144
$206$	−33.7565	−2.35193
$207$	19.0550	1.32441
$208$	0	0
$209$	6.12979	0.424007
$210$	13.4401	0.927457
$211$	−10.6537	−0.733429	−0.366715	−	0.930333i	$-0.619517\pi$
$211$	−10.6537	−0.733429	−0.366715	+	0.930333i	$0.619517\pi$
$212$	−29.5549	−2.02984
$213$	−27.6037	−1.89138
$214$	−18.3738	−1.25600
$215$	−0.637263	−0.0434610
$216$	−31.2870	−2.12881
$217$	−2.78942	−0.189358
$218$	25.1447	1.70301
$219$	10.5059	0.709924
$220$	−4.52091	−0.304799
$221$	0	0
$222$	−5.34335	−0.358622
$223$	21.3393	1.42899	0.714494	−	0.699642i	$-0.246657\pi$
$223$	21.3393	1.42899	0.714494	+	0.699642i	$0.246657\pi$
$224$	4.57388	0.305605
$225$	4.99102	0.332734
$226$	−16.6901	−1.11021
$227$	15.6857	1.04109	0.520547	−	0.853833i	$-0.325728\pi$
$227$	15.6857	1.04109	0.520547	+	0.853833i	$0.325728\pi$
$228$	−68.5016	−4.53663
$229$	−7.62085	−0.503600	−0.251800	−	0.967779i	$-0.581023\pi$
$229$	−7.62085	−0.503600	−0.251800	+	0.967779i	$0.581023\pi$
$230$	−9.52748	−0.628224
$231$	−5.75944	−0.378943
$232$	52.5599	3.45073
$233$	−19.0550	−1.24833	−0.624166	−	0.781292i	$-0.714561\pi$
$233$	−19.0550	−1.24833	−0.624166	+	0.781292i	$0.714561\pi$
$234$	0	0
$235$	9.44613	0.616198
$236$	3.13277	0.203926
$237$	26.3444	1.71125
$238$	3.02985	0.196396
$239$	12.7535	0.824954	0.412477	−	0.910968i	$-0.364664\pi$
$239$	12.7535	0.824954	0.412477	+	0.910968i	$0.364664\pi$
$240$	15.3134	0.988473
$241$	25.9288	1.67022	0.835111	−	0.550081i	$-0.185403\pi$
$241$	25.9288	1.67022	0.835111	+	0.550081i	$0.185403\pi$
$242$	−24.5967	−1.58114
$243$	14.2356	0.913211
$244$	17.7497	1.13631
$245$	3.37017	0.215312
$246$	−1.89022	−0.120516
$247$	0	0
$248$	−8.13878	−0.516813
$249$	−14.4671	−0.916817
$250$	−2.49551	−0.157830
$251$	7.61186	0.480457	0.240228	−	0.970716i	$-0.422778\pi$
$251$	7.61186	0.480457	0.240228	+	0.970716i	$0.422778\pi$
$252$	40.1996	2.53234
$253$	4.08277	0.256682
$254$	3.71706	0.233229
$255$	1.80144	0.112811
$256$	−32.4572	−2.02857
$257$	0.335783	0.0209456	0.0104728	−	0.999945i	$-0.496666\pi$
$257$	0.335783	0.0209456	0.0104728	+	0.999945i	$0.496666\pi$
$258$	−4.49551	−0.279878
$259$	1.44310	0.0896700
$260$	0	0
$261$	47.1906	2.92103
$262$	10.2984	0.636235
$263$	−5.37589	−0.331492	−0.165746	−	0.986169i	$-0.553003\pi$
$263$	−5.37589	−0.331492	−0.165746	+	0.986169i	$0.553003\pi$
$264$	−16.8045	−1.03424
$265$	6.99102	0.429455
$266$	27.2529	1.67098
$267$	35.5569	2.17605
$268$	34.2212	2.09039
$269$	−1.31038	−0.0798956	−0.0399478	−	0.999202i	$-0.512719\pi$
$269$	−1.31038	−0.0798956	−0.0399478	+	0.999202i	$0.512719\pi$
$270$	14.0454	0.854777
$271$	−11.6453	−0.707403	−0.353701	−	0.935358i	$-0.615077\pi$
$271$	−11.6453	−0.707403	−0.353701	+	0.935358i	$0.615077\pi$
$272$	3.45214	0.209317
$273$	0	0
$274$	−50.1838	−3.03171
$275$	1.06939	0.0644866
$276$	−45.6257	−2.74635
$277$	−20.3161	−1.22068	−0.610338	−	0.792141i	$-0.708967\pi$
$277$	−20.3161	−1.22068	−0.610338	+	0.792141i	$0.708967\pi$
$278$	51.9697	3.11693
$279$	−7.30735	−0.437480
$280$	−10.5909	−0.632925
$281$	−11.8744	−0.708366	−0.354183	−	0.935176i	$-0.615241\pi$
$281$	−11.8744	−0.708366	−0.354183	+	0.935176i	$0.615241\pi$
$282$	66.6368	3.96816
$283$	−22.6521	−1.34653	−0.673264	−	0.739402i	$-0.735108\pi$
$283$	−22.6521	−1.34653	−0.673264	+	0.739402i	$0.735108\pi$
$284$	41.2816	2.44961
$285$	16.2036	0.959817
$286$	0	0
$287$	0.510500	0.0301339
$288$	11.9820	0.706048
$289$	−16.5939	−0.976112
$290$	−23.5953	−1.38556
$291$	11.9336	0.699562
$292$	−15.7117	−0.919456
$293$	18.6127	1.08737	0.543683	−	0.839290i	$-0.317029\pi$
$293$	18.6127	1.08737	0.543683	+	0.839290i	$0.317029\pi$
$294$	23.7745	1.38656
$295$	−0.741035	−0.0431448
$296$	4.21058	0.244735
$297$	−6.01882	−0.349247
$298$	33.3593	1.93245
$299$	0	0
$300$	−11.9506	−0.689970
$301$	1.21412	0.0699808
$302$	45.6637	2.62765
$303$	43.1024	2.47617
$304$	31.0513	1.78091
$305$	−4.19856	−0.240409
$306$	7.93719	0.453739
$307$	3.14776	0.179652	0.0898262	−	0.995957i	$-0.471369\pi$
$307$	3.14776	0.179652	0.0898262	+	0.995957i	$0.471369\pi$
$308$	8.61329	0.490788
$309$	38.2384	2.17531
$310$	3.65368	0.207515
$311$	−3.18059	−0.180355	−0.0901774	−	0.995926i	$-0.528743\pi$
$311$	−3.18059	−0.180355	−0.0901774	+	0.995926i	$0.528743\pi$
$312$	0	0
$313$	35.3533	1.99829	0.999144	−	0.0413596i	$-0.0131689\pi$
$313$	35.3533	1.99829	0.999144	+	0.0413596i	$0.0131689\pi$
$314$	6.04484	0.341130
$315$	−9.50894	−0.535768
$316$	−39.3982	−2.21632
$317$	−13.6357	−0.765858	−0.382929	−	0.923778i	$-0.625085\pi$
$317$	−13.6357	−0.765858	−0.382929	+	0.923778i	$0.625085\pi$
$318$	49.3174	2.76558
$319$	10.1112	0.566119
$320$	4.84325	0.270746
$321$	20.8133	1.16168
$322$	18.1519	1.01156
$323$	3.65283	0.203249
$324$	3.96202	0.220112
$325$	0	0
$326$	39.8854	2.20905
$327$	−28.4831	−1.57512
$328$	1.48950	0.0822439
$329$	−17.9969	−0.992201
$330$	7.54390	0.415278
$331$	−28.7959	−1.58277	−0.791383	−	0.611320i	$-0.790639\pi$
$331$	−28.7959	−1.58277	−0.791383	+	0.611320i	$0.790639\pi$
$332$	21.6357	1.18741
$333$	3.78044	0.207167
$334$	−35.9182	−1.96536
$335$	−8.09479	−0.442265
$336$	−29.1752	−1.59164
$337$	11.7493	0.640026	0.320013	−	0.947413i	$-0.396313\pi$
$337$	11.7493	0.640026	0.320013	+	0.947413i	$0.396313\pi$
$338$	0	0
$339$	18.9061	1.02684
$340$	−2.69407	−0.146106
$341$	−1.56569	−0.0847871
$342$	71.3934	3.86051
$343$	−19.7574	−1.06680
$344$	3.54248	0.190997
$345$	10.7925	0.581046
$346$	−60.7630	−3.26664
$347$	1.89977	0.101985	0.0509926	−	0.998699i	$-0.483762\pi$
$347$	1.89977	0.101985	0.0509926	+	0.998699i	$0.483762\pi$
$348$	−112.995	−6.05714
$349$	−10.2691	−0.549692	−0.274846	−	0.961488i	$-0.588627\pi$
$349$	−10.2691	−0.549692	−0.274846	+	0.961488i	$0.588627\pi$
$350$	4.75447	0.254137
$351$	0	0
$352$	2.56730	0.136838
$353$	0.800589	0.0426110	0.0213055	−	0.999773i	$-0.493218\pi$
$353$	0.800589	0.0426110	0.0213055	+	0.999773i	$0.493218\pi$
$354$	−5.22756	−0.277842
$355$	−9.76488	−0.518266
$356$	−53.1756	−2.81830
$357$	−3.43213	−0.181647
$358$	9.45370	0.499643
$359$	8.13272	0.429228	0.214614	−	0.976699i	$-0.431151\pi$
$359$	8.13272	0.429228	0.214614	+	0.976699i	$0.431151\pi$
$360$	−27.7445	−1.46226
$361$	13.8564	0.729285
$362$	−21.1817	−1.11329
$363$	27.8625	1.46240
$364$	0	0
$365$	3.71649	0.194530
$366$	−29.6183	−1.54817
$367$	−20.5265	−1.07147	−0.535737	−	0.844385i	$-0.679966\pi$
$367$	−20.5265	−1.07147	−0.535737	+	0.844385i	$0.679966\pi$
$368$	20.6818	1.07811
$369$	1.33734	0.0696191
$370$	−1.89022	−0.0982679
$371$	−13.3194	−0.691507
$372$	17.4969	0.907173
$373$	−17.8058	−0.921951	−0.460976	−	0.887413i	$-0.652500\pi$
$373$	−17.8058	−0.921951	−0.460976	+	0.887413i	$0.652500\pi$
$374$	1.70064	0.0879382
$375$	2.82684	0.145977
$376$	−52.5100	−2.70800
$377$	0	0
$378$	−26.7595	−1.37636
$379$	2.04555	0.105073	0.0525363	−	0.998619i	$-0.483269\pi$
$379$	2.04555	0.105073	0.0525363	+	0.998619i	$0.483269\pi$
$380$	−24.2326	−1.24311
$381$	−4.21058	−0.215714
$382$	−13.5833	−0.694982
$383$	−7.90521	−0.403937	−0.201969	−	0.979392i	$-0.564734\pi$
$383$	−7.90521	−0.403937	−0.201969	+	0.979392i	$0.564734\pi$
$384$	47.7391	2.43618
$385$	−2.03741	−0.103836
$386$	30.3394	1.54423
$387$	3.18059	0.161679
$388$	−17.8469	−0.906037
$389$	−9.21171	−0.467052	−0.233526	−	0.972351i	$-0.575026\pi$
$389$	−9.21171	−0.467052	−0.233526	+	0.972351i	$0.575026\pi$
$390$	0	0
$391$	2.43298	0.123041
$392$	−18.7344	−0.946229
$393$	−11.6657	−0.588456
$394$	−10.9261	−0.550448
$395$	9.31937	0.468908
$396$	22.5639	1.13388
$397$	−6.35438	−0.318917	−0.159458	−	0.987205i	$-0.550975\pi$
$397$	−6.35438	−0.318917	−0.159458	+	0.987205i	$0.550975\pi$
$398$	51.9996	2.60651
$399$	−30.8713	−1.54550
$400$	5.41713	0.270857
$401$	4.16920	0.208200	0.104100	−	0.994567i	$-0.466804\pi$
$401$	4.16920	0.208200	0.104100	+	0.994567i	$0.466804\pi$
$402$	−57.1038	−2.84808
$403$	0	0
$404$	−64.4600	−3.20701
$405$	−0.937188	−0.0465692
$406$	44.9541	2.23103
$407$	0.810008	0.0401506
$408$	−10.0140	−0.495767
$409$	10.1681	0.502778	0.251389	−	0.967886i	$-0.419113\pi$
$409$	10.1681	0.502778	0.251389	+	0.967886i	$0.419113\pi$
$410$	−0.668669	−0.0330232
$411$	56.8467	2.80404
$412$	−57.1858	−2.81734
$413$	1.41183	0.0694716
$414$	47.5518	2.33704
$415$	−5.11778	−0.251222
$416$	0	0
$417$	−58.8697	−2.88286
$418$	15.2969	0.748198
$419$	28.5909	1.39676	0.698378	−	0.715730i	$-0.253906\pi$
$419$	28.5909	1.39676	0.698378	+	0.715730i	$0.253906\pi$
$420$	22.7685	1.11099
$421$	2.01797	0.0983498	0.0491749	−	0.998790i	$-0.484341\pi$
$421$	2.01797	0.0983498	0.0491749	+	0.998790i	$0.484341\pi$
$422$	−26.5863	−1.29420
$423$	−47.1458	−2.29231
$424$	−38.8623	−1.88732
$425$	0.637263	0.0309118
$426$	−68.8853	−3.33750
$427$	7.99915	0.387106
$428$	−31.1264	−1.50455
$429$	0	0
$430$	−1.59030	−0.0766908
$431$	−20.6123	−0.992860	−0.496430	−	0.868077i	$-0.665356\pi$
$431$	−20.6123	−0.992860	−0.496430	+	0.868077i	$0.665356\pi$
$432$	−30.4891	−1.46691
$433$	29.4356	1.41458	0.707292	−	0.706921i	$-0.249917\pi$
$433$	29.4356	1.41458	0.707292	+	0.706921i	$0.249917\pi$
$434$	−6.96103	−0.334140
$435$	26.7281	1.28151
$436$	42.5967	2.04001
$437$	21.8841	1.04686
$438$	26.2176	1.25272
$439$	16.9520	0.809077	0.404538	−	0.914521i	$-0.367432\pi$
$439$	16.9520	0.809077	0.404538	+	0.914521i	$0.367432\pi$
$440$	−5.94462	−0.283398
$441$	−16.8205	−0.800978
$442$	0	0
$443$	−24.1399	−1.14692	−0.573461	−	0.819233i	$-0.694400\pi$
$443$	−24.1399	−1.14692	−0.573461	+	0.819233i	$0.694400\pi$
$444$	−9.05199	−0.429588
$445$	12.5783	0.596270
$446$	53.2525	2.52158
$447$	−37.7885	−1.78733
$448$	−9.22742	−0.435955
$449$	−20.8630	−0.984585	−0.492293	−	0.870430i	$-0.663841\pi$
$449$	−20.8630	−0.984585	−0.492293	+	0.870430i	$0.663841\pi$
$450$	12.4551	0.587140
$451$	0.286542	0.0134927
$452$	−28.2742	−1.32990
$453$	−51.7265	−2.43032
$454$	39.1437	1.83710
$455$	0	0
$456$	−90.0739	−4.21810
$457$	−30.5659	−1.42981	−0.714906	−	0.699220i	$-0.753531\pi$
$457$	−30.5659	−1.42981	−0.714906	+	0.699220i	$0.753531\pi$
$458$	−19.0179	−0.888648
$459$	−3.58669	−0.167413
$460$	−16.1402	−0.752541
$461$	4.67822	0.217887	0.108943	−	0.994048i	$-0.465253\pi$
$461$	4.67822	0.217887	0.108943	+	0.994048i	$0.465253\pi$
$462$	−14.3727	−0.668680
$463$	−14.0011	−0.650688	−0.325344	−	0.945596i	$-0.605480\pi$
$463$	−14.0011	−0.650688	−0.325344	+	0.945596i	$0.605480\pi$
$464$	51.2196	2.37781
$465$	−4.13878	−0.191931
$466$	−47.5518	−2.20280
$467$	−6.98506	−0.323230	−0.161615	−	0.986854i	$-0.551670\pi$
$467$	−6.98506	−0.323230	−0.161615	+	0.986854i	$0.551670\pi$
$468$	0	0
$469$	15.4223	0.712135
$470$	23.5729	1.08734
$471$	−6.84742	−0.315513
$472$	4.11933	0.189608
$473$	0.681482	0.0313346
$474$	65.7425	3.01965
$475$	5.73205	0.263005
$476$	5.13277	0.235260
$477$	−34.8923	−1.59761
$478$	31.8264	1.45571
$479$	−16.2888	−0.744252	−0.372126	−	0.928182i	$-0.621371\pi$
$479$	−16.2888	−0.744252	−0.372126	+	0.928182i	$0.621371\pi$
$480$	6.78645	0.309758
$481$	0	0
$482$	64.7056	2.94726
$483$	−20.5619	−0.935600
$484$	−41.6685	−1.89402
$485$	4.22155	0.191691
$486$	35.5249	1.61144
$487$	20.0409	0.908139	0.454069	−	0.890966i	$-0.349972\pi$
$487$	20.0409	0.908139	0.454069	+	0.890966i	$0.349972\pi$
$488$	23.3393	1.05652
$489$	−45.1810	−2.04316
$490$	8.41027	0.379937
$491$	−15.7983	−0.712969	−0.356484	−	0.934301i	$-0.616025\pi$
$491$	−15.7983	−0.712969	−0.356484	+	0.934301i	$0.616025\pi$
$492$	−3.20216	−0.144365
$493$	6.02540	0.271370
$494$	0	0
$495$	−5.33734	−0.239896
$496$	−7.93123	−0.356123
$497$	18.6042	0.834511
$498$	−36.1028	−1.61781
$499$	1.24651	0.0558016	0.0279008	−	0.999611i	$-0.491118\pi$
$499$	1.24651	0.0558016	0.0279008	+	0.999611i	$0.491118\pi$
$500$	−4.22756	−0.189062
$501$	40.6871	1.81777
$502$	18.9955	0.847809
$503$	−7.65345	−0.341250	−0.170625	−	0.985336i	$-0.554579\pi$
$503$	−7.65345	−0.341250	−0.170625	+	0.985336i	$0.554579\pi$
$504$	52.8592	2.35453
$505$	15.2476	0.678508
$506$	10.1886	0.452938
$507$	0	0
$508$	6.29695	0.279382
$509$	25.7241	1.14020	0.570101	−	0.821575i	$-0.306904\pi$
$509$	25.7241	1.14020	0.570101	+	0.821575i	$0.306904\pi$
$510$	4.49551	0.199064
$511$	−7.08070	−0.313232
$512$	−47.2215	−2.08691
$513$	−32.2616	−1.42438
$514$	0.837948	0.0369603
$515$	13.5269	0.596067
$516$	−7.61569	−0.335262
$517$	−10.1016	−0.444268
$518$	3.60127	0.158231
$519$	68.8305	3.02132
$520$	0	0
$521$	−30.1519	−1.32098	−0.660490	−	0.750835i	$-0.729651\pi$
$521$	−30.1519	−1.32098	−0.660490	+	0.750835i	$0.729651\pi$
$522$	117.765	5.15442
$523$	3.93752	0.172176	0.0860880	−	0.996288i	$-0.472563\pi$
$523$	3.93752	0.172176	0.0860880	+	0.996288i	$0.472563\pi$
$524$	17.4461	0.762138
$525$	−5.38573	−0.235052
$526$	−13.4156	−0.584947
$527$	−0.933018	−0.0406429
$528$	−16.3759	−0.712672
$529$	−8.42399	−0.366261
$530$	17.4461	0.757812
$531$	3.69852	0.160502
$532$	46.1682	2.00165
$533$	0	0
$534$	88.7326	3.83983
$535$	7.36274	0.318319
$536$	44.9980	1.94362
$537$	−10.7089	−0.462122
$538$	−3.27007	−0.140983
$539$	−3.60402	−0.155236
$540$	23.7939	1.02393
$541$	−15.8881	−0.683083	−0.341541	−	0.939867i	$-0.610949\pi$
$541$	−15.8881	−0.683083	−0.341541	+	0.939867i	$0.610949\pi$
$542$	−29.0610	−1.24828
$543$	23.9940	1.02968
$544$	1.52989	0.0655935
$545$	−10.0760	−0.431607
$546$	0	0
$547$	−6.56107	−0.280531	−0.140266	−	0.990114i	$-0.544796\pi$
$547$	−6.56107	−0.280531	−0.140266	+	0.990114i	$0.544796\pi$
$548$	−85.0147	−3.63165
$549$	20.9551	0.894341
$550$	2.66867	0.113792
$551$	54.1972	2.30888
$552$	−59.9941	−2.55352
$553$	−17.7554	−0.755035
$554$	−50.6990	−2.15399
$555$	2.14119	0.0908883
$556$	88.0401	3.73373
$557$	7.85006	0.332618	0.166309	−	0.986074i	$-0.446815\pi$
$557$	7.85006	0.332618	0.166309	+	0.986074i	$0.446815\pi$
$558$	−18.2356	−0.771973
$559$	0	0
$560$	−10.3208	−0.436133
$561$	−1.92644	−0.0813344
$562$	−29.6326	−1.24998
$563$	−15.5595	−0.655755	−0.327878	−	0.944720i	$-0.606333\pi$
$563$	−15.5595	−0.655755	−0.327878	+	0.944720i	$0.606333\pi$
$564$	112.887	4.75341
$565$	6.68806	0.281369
$566$	−56.5285	−2.37607
$567$	1.78554	0.0749857
$568$	54.2819	2.27762
$569$	3.47915	0.145853	0.0729267	−	0.997337i	$-0.476766\pi$
$569$	3.47915	0.145853	0.0729267	+	0.997337i	$0.476766\pi$
$570$	40.4362	1.69368
$571$	−21.5118	−0.900240	−0.450120	−	0.892968i	$-0.648619\pi$
$571$	−21.5118	−0.900240	−0.450120	+	0.892968i	$0.648619\pi$
$572$	0	0
$573$	15.3868	0.642791
$574$	1.27396	0.0531739
$575$	3.81785	0.159216
$576$	−24.1727	−1.00720
$577$	9.97608	0.415310	0.207655	−	0.978202i	$-0.433417\pi$
$577$	9.97608	0.415310	0.207655	+	0.978202i	$0.433417\pi$
$578$	−41.4102	−1.72244
$579$	−34.3676	−1.42827
$580$	−39.9721	−1.65975
$581$	9.75045	0.404517
$582$	29.7805	1.23444
$583$	−7.47612	−0.309629
$584$	−20.6595	−0.854898
$585$	0	0
$586$	46.4482	1.91876
$587$	−24.0571	−0.992945	−0.496472	−	0.868053i	$-0.665372\pi$
$587$	−24.0571	−0.992945	−0.496472	+	0.868053i	$0.665372\pi$
$588$	40.2756	1.66094
$589$	−8.39230	−0.345799
$590$	−1.84926	−0.0761328
$591$	12.3767	0.509111
$592$	4.10320	0.168641
$593$	0.940219	0.0386102	0.0193051	−	0.999814i	$-0.493855\pi$
$593$	0.940219	0.0386102	0.0193051	+	0.999814i	$0.493855\pi$
$594$	−15.0200	−0.616279
$595$	−1.21412	−0.0497741
$596$	56.5130	2.31486
$597$	−58.9037	−2.41077
$598$	0	0
$599$	−11.4270	−0.466896	−0.233448	−	0.972369i	$-0.575001\pi$
$599$	−11.4270	−0.466896	−0.233448	+	0.972369i	$0.575001\pi$
$600$	−15.7141	−0.641525
$601$	−36.0431	−1.47023	−0.735114	−	0.677944i	$-0.762871\pi$
$601$	−36.0431	−1.47023	−0.735114	+	0.677944i	$0.762871\pi$
$602$	3.02985	0.123487
$603$	40.4012	1.64526
$604$	77.3574	3.14763
$605$	9.85641	0.400720
$606$	107.562	4.36942
$607$	39.8907	1.61911	0.809557	−	0.587041i	$-0.199707\pi$
$607$	39.8907	1.61911	0.809557	+	0.587041i	$0.199707\pi$
$608$	13.7610	0.558084
$609$	−50.9227	−2.06349
$610$	−10.4775	−0.424223
$611$	0	0
$612$	13.4461	0.543528
$613$	−0.345472	−0.0139535	−0.00697673	−	0.999976i	$-0.502221\pi$
$613$	−0.345472	−0.0139535	−0.00697673	+	0.999976i	$0.502221\pi$
$614$	7.85527	0.317013
$615$	0.757449	0.0305433
$616$	11.3258	0.456328
$617$	38.6850	1.55740	0.778700	−	0.627397i	$-0.215879\pi$
$617$	38.6850	1.55740	0.778700	+	0.627397i	$0.215879\pi$
$618$	95.4242	3.83852
$619$	14.8971	0.598764	0.299382	−	0.954133i	$-0.403219\pi$
$619$	14.8971	0.598764	0.299382	+	0.954133i	$0.403219\pi$
$620$	6.18958	0.248579
$621$	−21.4879	−0.862281
$622$	−7.93719	−0.318252
$623$	−23.9644	−0.960113
$624$	0	0
$625$	1.00000	0.0400000
$626$	88.2245	3.52616
$627$	−17.3279	−0.692011
$628$	10.2404	0.408635
$629$	0.482694	0.0192463
$630$	−23.7296	−0.945412
$631$	−38.8450	−1.54640	−0.773198	−	0.634165i	$-0.781344\pi$
$631$	−38.8450	−1.54640	−0.773198	+	0.634165i	$0.781344\pi$
$632$	−51.8053	−2.06071
$633$	30.1162	1.19701
$634$	−34.0280	−1.35143
$635$	−1.48950	−0.0591090
$636$	83.5470	3.31285
$637$	0	0
$638$	25.2326	0.998967
$639$	48.7367	1.92799
$640$	16.8878	0.667549
$641$	37.1816	1.46859	0.734293	−	0.678832i	$-0.237514\pi$
$641$	37.1816	1.46859	0.734293	+	0.678832i	$0.237514\pi$
$642$	51.9397	2.04990
$643$	9.10377	0.359018	0.179509	−	0.983756i	$-0.442549\pi$
$643$	9.10377	0.359018	0.179509	+	0.983756i	$0.442549\pi$
$644$	30.7505	1.21174
$645$	1.80144	0.0709316
$646$	9.11565	0.358651
$647$	−19.1224	−0.751778	−0.375889	−	0.926665i	$-0.622663\pi$
$647$	−19.1224	−0.751778	−0.375889	+	0.926665i	$0.622663\pi$
$648$	5.20972	0.204657
$649$	0.792455	0.0311066
$650$	0	0
$651$	7.88525	0.309047
$652$	67.5686	2.64619
$653$	34.6324	1.35527	0.677636	−	0.735397i	$-0.263004\pi$
$653$	34.6324	1.35527	0.677636	+	0.735397i	$0.263004\pi$
$654$	−71.0799	−2.77944
$655$	−4.12676	−0.161246
$656$	1.45152	0.0566722
$657$	−18.5491	−0.723667
$658$	−44.9114	−1.75083
$659$	−6.69852	−0.260937	−0.130469	−	0.991452i	$-0.541648\pi$
$659$	−6.69852	−0.260937	−0.130469	+	0.991452i	$0.541648\pi$
$660$	12.7799	0.497456
$661$	6.02758	0.234446	0.117223	−	0.993106i	$-0.462601\pi$
$661$	6.02758	0.234446	0.117223	+	0.993106i	$0.462601\pi$
$662$	−71.8604	−2.79294
$663$	0	0
$664$	28.4492	1.10404
$665$	−10.9208	−0.423489
$666$	9.43412	0.365565
$667$	36.0983	1.39773
$668$	−60.8479	−2.35428
$669$	−60.3228	−2.33222
$670$	−20.2006	−0.780417
$671$	4.48990	0.173330
$672$	−12.9296	−0.498771
$673$	23.3568	0.900338	0.450169	−	0.892943i	$-0.351364\pi$
$673$	23.3568	0.900338	0.450169	+	0.892943i	$0.351364\pi$
$674$	29.3205	1.12938
$675$	−5.62828	−0.216633
$676$	0	0
$677$	−45.4042	−1.74503	−0.872513	−	0.488590i	$-0.837511\pi$
$677$	−45.4042	−1.74503	−0.872513	+	0.488590i	$0.837511\pi$
$678$	47.1802	1.81195
$679$	−8.04295	−0.308660
$680$	−3.54248	−0.135848
$681$	−44.3408	−1.69914
$682$	−3.90720	−0.149615
$683$	−25.4978	−0.975645	−0.487823	−	0.872943i	$-0.662209\pi$
$683$	−25.4978	−0.975645	−0.487823	+	0.872943i	$0.662209\pi$
$684$	120.945	4.62445
$685$	20.1096	0.768350
$686$	−49.3047	−1.88246
$687$	21.5429	0.821913
$688$	3.45214	0.131612
$689$	0	0
$690$	26.9327	1.02531
$691$	−6.59630	−0.250935	−0.125468	−	0.992098i	$-0.540043\pi$
$691$	−6.59630	−0.250935	−0.125468	+	0.992098i	$0.540043\pi$
$692$	−102.937	−3.91306
$693$	10.1688	0.386279
$694$	4.74090	0.179962
$695$	−20.8253	−0.789948
$696$	−148.578	−5.63185
$697$	0.170754	0.00646778
$698$	−25.6266	−0.969981
$699$	53.8653	2.03737
$700$	8.05440	0.304428
$701$	29.2474	1.10466	0.552329	−	0.833626i	$-0.313739\pi$
$701$	29.2474	1.10466	0.552329	+	0.833626i	$0.313739\pi$
$702$	0	0
$703$	4.34174	0.163752
$704$	−5.17932	−0.195203
$705$	−26.7027	−1.00568
$706$	1.99787	0.0751910
$707$	−29.0499	−1.09253
$708$	−8.85584	−0.332823
$709$	10.9335	0.410614	0.205307	−	0.978698i	$-0.434181\pi$
$709$	10.9335	0.410614	0.205307	+	0.978698i	$0.434181\pi$
$710$	−24.3683	−0.914527
$711$	−46.5131	−1.74438
$712$	−69.9216	−2.62042
$713$	−5.58973	−0.209337
$714$	−8.56490	−0.320533
$715$	0	0
$716$	16.0152	0.598516
$717$	−36.0520	−1.34639
$718$	20.2953	0.757412
$719$	16.0598	0.598929	0.299464	−	0.954107i	$-0.403192\pi$
$719$	16.0598	0.598929	0.299464	+	0.954107i	$0.403192\pi$
$720$	−27.0370	−1.00761
$721$	−25.7716	−0.959786
$722$	34.5788	1.28689
$723$	−73.2966	−2.72593
$724$	−35.8833	−1.33359
$725$	9.45512	0.351154
$726$	69.5310	2.58054
$727$	51.3754	1.90541	0.952704	−	0.303900i	$-0.0982889\pi$
$727$	51.3754	1.90541	0.952704	+	0.303900i	$0.0982889\pi$
$728$	0	0
$729$	−43.0532	−1.59456
$730$	9.27453	0.343266
$731$	0.406104	0.0150203
$732$	−50.1754	−1.85454
$733$	−9.82358	−0.362842	−0.181421	−	0.983406i	$-0.558070\pi$
$733$	−9.82358	−0.362842	−0.181421	+	0.983406i	$0.558070\pi$
$734$	−51.2240	−1.89071
$735$	−9.52691	−0.351406
$736$	9.16560	0.337848
$737$	8.65648	0.318866
$738$	3.33734	0.122849
$739$	49.0842	1.80559	0.902797	−	0.430068i	$-0.141510\pi$
$739$	49.0842	1.80559	0.902797	+	0.430068i	$0.141510\pi$
$740$	−3.20216	−0.117714
$741$	0	0
$742$	−33.2386	−1.22023
$743$	40.8375	1.49818	0.749091	−	0.662467i	$-0.230490\pi$
$743$	40.8375	1.49818	0.749091	+	0.662467i	$0.230490\pi$
$744$	23.0070	0.843478
$745$	−13.3678	−0.489757
$746$	−44.4346	−1.62687
$747$	25.5429	0.934566
$748$	2.88101	0.105340
$749$	−14.0276	−0.512557
$750$	7.05440	0.257590
$751$	−2.72680	−0.0995024	−0.0497512	−	0.998762i	$-0.515843\pi$
$751$	−2.72680	−0.0995024	−0.0497512	+	0.998762i	$0.515843\pi$
$752$	−51.1710	−1.86601
$753$	−21.5175	−0.784142
$754$	0	0
$755$	−18.2984	−0.665946
$756$	−45.3324	−1.64872
$757$	14.8060	0.538134	0.269067	−	0.963121i	$-0.413285\pi$
$757$	14.8060	0.538134	0.269067	+	0.963121i	$0.413285\pi$
$758$	5.10468	0.185410
$759$	−11.5413	−0.418924
$760$	−31.8638	−1.15582
$761$	−11.3689	−0.412122	−0.206061	−	0.978539i	$-0.566065\pi$
$761$	−11.3689	−0.412122	−0.206061	+	0.978539i	$0.566065\pi$
$762$	−10.5075	−0.380647
$763$	19.1969	0.694973
$764$	−23.0110	−0.832510
$765$	−3.18059	−0.114994
$766$	−19.7275	−0.712784
$767$	0	0
$768$	91.7512	3.31078
$769$	−21.0562	−0.759307	−0.379654	−	0.925129i	$-0.623957\pi$
$769$	−21.0562	−0.759307	−0.379654	+	0.925129i	$0.623957\pi$
$770$	−5.08438	−0.183228
$771$	−0.949203	−0.0341847
$772$	51.3970	1.84982
$773$	14.0829	0.506526	0.253263	−	0.967397i	$-0.418496\pi$
$773$	14.0829	0.506526	0.253263	+	0.967397i	$0.418496\pi$
$774$	7.93719	0.285296
$775$	−1.46410	−0.0525921
$776$	−23.4671	−0.842421
$777$	−4.07941	−0.146348
$778$	−22.9879	−0.824156
$779$	1.53590	0.0550293
$780$	0	0
$781$	10.4425	0.373660
$782$	6.07151	0.217117
$783$	−53.2160	−1.90179
$784$	−18.2566	−0.652023
$785$	−2.42229	−0.0864552
$786$	−29.1118	−1.03838
$787$	33.0242	1.17719	0.588593	−	0.808429i	$-0.299682\pi$
$787$	33.0242	1.17719	0.588593	+	0.808429i	$0.299682\pi$
$788$	−18.5095	−0.659374
$789$	15.1968	0.541020
$790$	23.2566	0.827431
$791$	−12.7422	−0.453060
$792$	29.6697	1.05427
$793$	0	0
$794$	−15.8574	−0.562758
$795$	−19.7625	−0.700903
$796$	88.0909	3.12230
$797$	−16.9416	−0.600102	−0.300051	−	0.953923i	$-0.597004\pi$
$797$	−16.9416	−0.600102	−0.300051	+	0.953923i	$0.597004\pi$
$798$	−77.0395	−2.72717
$799$	−6.01967	−0.212961
$800$	2.40072	0.0848783
$801$	−62.7787	−2.21817
$802$	10.4043	0.367387
$803$	−3.97437	−0.140253
$804$	−96.7378	−3.41168
$805$	−7.27382	−0.256369
$806$	0	0
$807$	3.70425	0.130396
$808$	−84.7596	−2.98183
$809$	−51.7635	−1.81991	−0.909954	−	0.414708i	$-0.863884\pi$
$809$	−51.7635	−1.81991	−0.909954	+	0.414708i	$0.863884\pi$
$810$	−2.33876	−0.0821756
$811$	22.6699	0.796047	0.398023	−	0.917375i	$-0.369696\pi$
$811$	22.6699	0.796047	0.398023	+	0.917375i	$0.369696\pi$
$812$	76.1553	2.67253
$813$	32.9194	1.15453
$814$	2.02138	0.0708494
$815$	−15.9829	−0.559856
$816$	−9.75864	−0.341621
$817$	3.65283	0.127796
$818$	25.3745	0.887198
$819$	0	0
$820$	−1.13277	−0.0395581
$821$	28.6631	1.00035	0.500174	−	0.865925i	$-0.333269\pi$
$821$	28.6631	1.00035	0.500174	+	0.865925i	$0.333269\pi$
$822$	141.861	4.94799
$823$	−25.8327	−0.900472	−0.450236	−	0.892910i	$-0.648660\pi$
$823$	−25.8327	−0.900472	−0.450236	+	0.892910i	$0.648660\pi$
$824$	−75.1946	−2.61953
$825$	−3.02299	−0.105247
$826$	3.52323	0.122589
$827$	−16.0820	−0.559227	−0.279613	−	0.960113i	$-0.590206\pi$
$827$	−16.0820	−0.559227	−0.279613	+	0.960113i	$0.590206\pi$
$828$	80.5560	2.79951
$829$	−22.5818	−0.784298	−0.392149	−	0.919902i	$-0.628268\pi$
$829$	−22.5818	−0.784298	−0.392149	+	0.919902i	$0.628268\pi$
$830$	−12.7715	−0.443304
$831$	57.4304	1.99224
$832$	0	0
$833$	−2.14768	−0.0744128
$834$	−146.910	−5.08707
$835$	14.3932	0.498096
$836$	25.9141	0.896257
$837$	8.24037	0.284829
$838$	71.3487	2.46470
$839$	−17.8440	−0.616042	−0.308021	−	0.951380i	$-0.599667\pi$
$839$	−17.8440	−0.616042	−0.308021	+	0.951380i	$0.599667\pi$
$840$	29.9387	1.03298
$841$	60.3992	2.08273
$842$	5.03586	0.173547
$843$	33.5669	1.15611
$844$	−45.0390	−1.55031
$845$	0	0
$846$	−117.653	−4.04498
$847$	−18.7785	−0.645239
$848$	−37.8713	−1.30050
$849$	64.0339	2.19764
$850$	1.59030	0.0545467
$851$	2.89183	0.0991306
$852$	−116.696	−3.99795
$853$	19.7936	0.677720	0.338860	−	0.940837i	$-0.389959\pi$
$853$	19.7936	0.677720	0.338860	+	0.940837i	$0.389959\pi$
$854$	19.9619	0.683083
$855$	−28.6088	−0.978399
$856$	−40.9286	−1.39891
$857$	−11.7302	−0.400696	−0.200348	−	0.979725i	$-0.564207\pi$
$857$	−11.7302	−0.400696	−0.200348	+	0.979725i	$0.564207\pi$
$858$	0	0
$859$	5.37452	0.183376	0.0916882	−	0.995788i	$-0.470774\pi$
$859$	5.37452	0.183376	0.0916882	+	0.995788i	$0.470774\pi$
$860$	−2.69407	−0.0918669
$861$	−1.44310	−0.0491808
$862$	−51.4382	−1.75199
$863$	−25.3234	−0.862017	−0.431008	−	0.902348i	$-0.641842\pi$
$863$	−25.3234	−0.862017	−0.431008	+	0.902348i	$0.641842\pi$
$864$	−13.5119	−0.459685
$865$	24.3489	0.827889
$866$	73.4567	2.49616
$867$	46.9083	1.59309
$868$	−11.7925	−0.400262
$869$	−9.96603	−0.338075
$870$	66.7001	2.26135
$871$	0	0
$872$	56.0112	1.89678
$873$	−21.0698	−0.713106
$874$	54.6120	1.84728
$875$	−1.90521	−0.0644079
$876$	44.4144	1.50062
$877$	20.6915	0.698703	0.349352	−	0.936992i	$-0.386402\pi$
$877$	20.6915	0.698703	0.349352	+	0.936992i	$0.386402\pi$
$878$	42.3040	1.42769
$879$	−52.6151	−1.77466
$880$	−5.79302	−0.195283
$881$	48.3993	1.63061	0.815307	−	0.579029i	$-0.196568\pi$
$881$	48.3993	1.63061	0.815307	+	0.579029i	$0.196568\pi$
$882$	−41.9758	−1.41340
$883$	45.8550	1.54314	0.771572	−	0.636142i	$-0.219471\pi$
$883$	45.8550	1.54314	0.771572	+	0.636142i	$0.219471\pi$
$884$	0	0
$885$	2.09479	0.0704155
$886$	−60.2413	−2.02385
$887$	−1.08234	−0.0363413	−0.0181707	−	0.999835i	$-0.505784\pi$
$887$	−1.08234	−0.0363413	−0.0181707	+	0.999835i	$0.505784\pi$
$888$	−11.9026	−0.399426
$889$	2.83781	0.0951772
$890$	31.3893	1.05217
$891$	1.00222	0.0335756
$892$	90.2133	3.02056
$893$	−54.1457	−1.81192
$894$	−94.3015	−3.15391
$895$	−3.78829	−0.126628
$896$	−32.1749	−1.07489
$897$	0	0
$898$	−52.0637	−1.73739
$899$	−13.8433	−0.461698
$900$	21.0998	0.703327
$901$	−4.45512	−0.148421
$902$	0.715068	0.0238092
$903$	−3.43213	−0.114214
$904$	−37.1782	−1.23653
$905$	8.48794	0.282149
$906$	−129.084	−4.28853
$907$	−45.5307	−1.51182	−0.755910	−	0.654675i	$-0.772805\pi$
$907$	−45.5307	−1.51182	−0.755910	+	0.654675i	$0.772805\pi$
$908$	66.3120	2.20064
$909$	−76.1009	−2.52411
$910$	0	0
$911$	39.7417	1.31670	0.658350	−	0.752712i	$-0.271255\pi$
$911$	39.7417	1.31670	0.658350	+	0.752712i	$0.271255\pi$
$912$	−87.7770	−2.90659
$913$	5.47290	0.181126
$914$	−76.2774	−2.52303
$915$	11.8687	0.392365
$916$	−32.2176	−1.06450
$917$	7.86236	0.259638
$918$	−8.95062	−0.295415
$919$	46.9938	1.55018	0.775091	−	0.631850i	$-0.217704\pi$
$919$	46.9938	1.55018	0.775091	+	0.631850i	$0.217704\pi$
$920$	−21.2230	−0.699702
$921$	−8.89822	−0.293206
$922$	11.6745	0.384481
$923$	0	0
$924$	−24.3484	−0.801002
$925$	0.757449	0.0249048
$926$	−34.9399	−1.14820
$927$	−67.5130	−2.21742
$928$	22.6991	0.745134
$929$	−15.2213	−0.499395	−0.249698	−	0.968324i	$-0.580331\pi$
$929$	−15.2213	−0.499395	−0.249698	+	0.968324i	$0.580331\pi$
$930$	−10.3284	−0.338680
$931$	−19.3180	−0.633121
$932$	−80.5560	−2.63870
$933$	8.99102	0.294353
$934$	−17.4313	−0.570369
$935$	−0.681482	−0.0222869
$936$	0	0
$937$	6.07285	0.198392	0.0991958	−	0.995068i	$-0.468373\pi$
$937$	6.07285	0.198392	0.0991958	+	0.995068i	$0.468373\pi$
$938$	38.4864	1.25663
$939$	−99.9382	−3.26136
$940$	39.9341	1.30251
$941$	−0.0496576	−0.00161879	−0.000809396	−	1.00000i	$-0.500258\pi$
$941$	−0.0496576	−0.00161879	−0.000809396		1.00000i	$0.500258\pi$
$942$	−17.0878	−0.556750
$943$	1.02299	0.0333132
$944$	4.01429	0.130654
$945$	10.7231	0.348821
$946$	1.70064	0.0552927
$947$	−18.6581	−0.606308	−0.303154	−	0.952942i	$-0.598040\pi$
$947$	−18.6581	−0.606308	−0.303154	+	0.952942i	$0.598040\pi$
$948$	111.372	3.61720
$949$	0	0
$950$	14.3044	0.464095
$951$	38.5459	1.24994
$952$	6.74917	0.218742
$953$	−1.52953	−0.0495463	−0.0247731	−	0.999693i	$-0.507886\pi$
$953$	−1.52953	−0.0495463	−0.0247731	+	0.999693i	$0.507886\pi$
$954$	−87.0739	−2.81912
$955$	5.44310	0.176135
$956$	53.9161	1.74377
$957$	−28.5827	−0.923948
$958$	−40.6487	−1.31330
$959$	−38.3131	−1.23720
$960$	−13.6911	−0.441878
$961$	−28.8564	−0.930852
$962$	0	0
$963$	−36.7475	−1.18417
$964$	109.616	3.53048
$965$	−12.1576	−0.391367
$966$	−51.3124	−1.65095
$967$	32.1716	1.03457	0.517285	−	0.855813i	$-0.326943\pi$
$967$	32.1716	1.03457	0.517285	+	0.855813i	$0.326943\pi$
$968$	−54.7907	−1.76104
$969$	−10.3259	−0.331717
$970$	10.5349	0.338256
$971$	−17.2541	−0.553710	−0.276855	−	0.960912i	$-0.589292\pi$
$971$	−17.2541	−0.553710	−0.276855	+	0.960912i	$0.589292\pi$
$972$	60.1816	1.93033
$973$	39.6766	1.27197
$974$	50.0122	1.60249
$975$	0	0
$976$	22.7442	0.728023
$977$	15.7228	0.503018	0.251509	−	0.967855i	$-0.419073\pi$
$977$	15.7228	0.503018	0.251509	+	0.967855i	$0.419073\pi$
$978$	−112.750	−3.60533
$979$	−13.4511	−0.429900
$980$	14.2476	0.455122
$981$	50.2893	1.60561
$982$	−39.4248	−1.25810
$983$	38.5356	1.22910	0.614548	−	0.788880i	$-0.289338\pi$
$983$	38.5356	1.22910	0.614548	+	0.788880i	$0.289338\pi$
$984$	−4.21058	−0.134228
$985$	4.37830	0.139504
$986$	15.0364	0.478857
$987$	50.8743	1.61935
$988$	0	0
$989$	2.43298	0.0773642
$990$	−13.3194	−0.423317
$991$	8.59143	0.272916	0.136458	−	0.990646i	$-0.456428\pi$
$991$	8.59143	0.272916	0.136458	+	0.990646i	$0.456428\pi$
$992$	−3.51490	−0.111598
$993$	81.4014	2.58320
$994$	46.4268	1.47257
$995$	−20.8373	−0.660587
$996$	−61.1606	−1.93795
$997$	−20.5374	−0.650425	−0.325213	−	0.945641i	$-0.605436\pi$
$997$	−20.5374	−0.650425	−0.325213	+	0.945641i	$0.605436\pi$
$998$	3.11069	0.0984670
$999$	−4.26313	−0.134880

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	845.2.a.m.1.4		4
3.2	odd	2		7605.2.a.cf.1.1		4
5.4	even	2		4225.2.a.bi.1.1		4
13.2	odd	12		845.2.m.g.316.4		8
13.3	even	3		845.2.e.m.191.1		8
13.4	even	6		845.2.e.n.146.4		8
13.5	odd	4		845.2.c.g.506.1		8
13.6	odd	12		65.2.m.a.36.1	✓	8
13.7	odd	12		845.2.m.g.361.4		8
13.8	odd	4		845.2.c.g.506.8		8
13.9	even	3		845.2.e.m.146.1		8
13.10	even	6		845.2.e.n.191.4		8
13.11	odd	12		65.2.m.a.56.1	yes	8
13.12	even	2		845.2.a.l.1.1		4
39.11	even	12		585.2.bu.c.316.4		8
39.32	even	12		585.2.bu.c.361.4		8
39.38	odd	2		7605.2.a.cj.1.4		4
52.11	even	12		1040.2.da.b.641.1		8
52.19	even	12		1040.2.da.b.881.1		8
65.19	odd	12		325.2.n.d.101.4		8
65.24	odd	12		325.2.n.d.251.4		8
65.32	even	12		325.2.m.b.49.1		8
65.37	even	12		325.2.m.c.199.4		8
65.58	even	12		325.2.m.c.49.4		8
65.63	even	12		325.2.m.b.199.1		8
65.64	even	2		4225.2.a.bl.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.2.m.a.36.1	✓	8	13.6	odd	12
65.2.m.a.56.1	yes	8	13.11	odd	12
325.2.m.b.49.1		8	65.32	even	12
325.2.m.b.199.1		8	65.63	even	12
325.2.m.c.49.4		8	65.58	even	12
325.2.m.c.199.4		8	65.37	even	12
325.2.n.d.101.4		8	65.19	odd	12
325.2.n.d.251.4		8	65.24	odd	12
585.2.bu.c.316.4		8	39.11	even	12
585.2.bu.c.361.4		8	39.32	even	12
845.2.a.l.1.1		4	13.12	even	2
845.2.a.m.1.4		4	1.1	even	1	trivial
845.2.c.g.506.1		8	13.5	odd	4
845.2.c.g.506.8		8	13.8	odd	4
845.2.e.m.146.1		8	13.9	even	3
845.2.e.m.191.1		8	13.3	even	3
845.2.e.n.146.4		8	13.4	even	6
845.2.e.n.191.4		8	13.10	even	6
845.2.m.g.316.4		8	13.2	odd	12
845.2.m.g.361.4		8	13.7	odd	12
1040.2.da.b.641.1		8	52.11	even	12
1040.2.da.b.881.1		8	52.19	even	12
4225.2.a.bi.1.1		4	5.4	even	2
4225.2.a.bl.1.4		4	65.64	even	2
7605.2.a.cf.1.1		4	3.2	odd	2
7605.2.a.cj.1.4		4	39.38	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 845 = 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	845.a (trivial)

\(f(q)\)	\(=\)	\(q+2.49551 q^{2} -2.82684 q^{3} +4.22756 q^{4} -1.00000 q^{5} -7.05440 q^{6} +1.90521 q^{7} +5.55889 q^{8} +4.99102 q^{9} +O(q^{10})\)	\(q+2.49551 q^{2} -2.82684 q^{3} +4.22756 q^{4} -1.00000 q^{5} -7.05440 q^{6} +1.90521 q^{7} +5.55889 q^{8} +4.99102 q^{9} -2.49551 q^{10} +1.06939 q^{11} -11.9506 q^{12} +4.75447 q^{14} +2.82684 q^{15} +5.41713 q^{16} +0.637263 q^{17} +12.4551 q^{18} +5.73205 q^{19} -4.22756 q^{20} -5.38573 q^{21} +2.66867 q^{22} +3.81785 q^{23} -15.7141 q^{24} +1.00000 q^{25} -5.62828 q^{27} +8.05440 q^{28} +9.45512 q^{29} +7.05440 q^{30} -1.46410 q^{31} +2.40072 q^{32} -3.02299 q^{33} +1.59030 q^{34} -1.90521 q^{35} +21.0998 q^{36} +0.757449 q^{37} +14.3044 q^{38} -5.55889 q^{40} +0.267949 q^{41} -13.4401 q^{42} +0.637263 q^{43} +4.52091 q^{44} -4.99102 q^{45} +9.52748 q^{46} -9.44613 q^{47} -15.3134 q^{48} -3.37017 q^{49} +2.49551 q^{50} -1.80144 q^{51} -6.99102 q^{53} -14.0454 q^{54} -1.06939 q^{55} +10.5909 q^{56} -16.2036 q^{57} +23.5953 q^{58} +0.741035 q^{59} +11.9506 q^{60} +4.19856 q^{61} -3.65368 q^{62} +9.50894 q^{63} -4.84325 q^{64} -7.54390 q^{66} +8.09479 q^{67} +2.69407 q^{68} -10.7925 q^{69} -4.75447 q^{70} +9.76488 q^{71} +27.7445 q^{72} -3.71649 q^{73} +1.89022 q^{74} -2.82684 q^{75} +24.2326 q^{76} +2.03741 q^{77} -9.31937 q^{79} -5.41713 q^{80} +0.937188 q^{81} +0.668669 q^{82} +5.11778 q^{83} -22.7685 q^{84} -0.637263 q^{85} +1.59030 q^{86} -26.7281 q^{87} +5.94462 q^{88} -12.5783 q^{89} -12.4551 q^{90} +16.1402 q^{92} +4.13878 q^{93} -23.5729 q^{94} -5.73205 q^{95} -6.78645 q^{96} -4.22155 q^{97} -8.41027 q^{98} +5.33734 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 4 q^{5} - 4 q^{6} + 10 q^{7} + 6 q^{8} + 4 q^{9}+O(q^{10}) \) 4 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 4 * q^5 - 4 * q^6 + 10 * q^7 + 6 * q^8 + 4 * q^9	\( 4 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 4 q^{5} - 4 q^{6} + 10 q^{7} + 6 q^{8} + 4 q^{9} - 2 q^{10} - 10 q^{12} + 2 q^{14} + 2 q^{15} + 2 q^{16} - 2 q^{17} + 20 q^{18} + 16 q^{19} - 2 q^{20} + 4 q^{21} + 12 q^{22} - 10 q^{23} - 24 q^{24} + 4 q^{25} - 2 q^{27} + 8 q^{28} + 8 q^{29} + 4 q^{30} + 8 q^{31} + 4 q^{32} + 18 q^{33} - 4 q^{34} - 10 q^{35} + 20 q^{36} - 2 q^{37} + 8 q^{38} - 6 q^{40} + 8 q^{41} - 4 q^{42} - 2 q^{43} + 12 q^{44} - 4 q^{45} + 16 q^{46} + 8 q^{47} - 28 q^{48} + 12 q^{49} + 2 q^{50} + 4 q^{51} - 12 q^{53} - 16 q^{54} + 12 q^{56} - 14 q^{57} + 22 q^{58} + 12 q^{59} + 10 q^{60} + 28 q^{61} + 4 q^{62} + 4 q^{63} + 4 q^{64} + 6 q^{66} + 30 q^{67} + 14 q^{68} - 16 q^{69} - 2 q^{70} + 4 q^{71} + 12 q^{72} - 8 q^{73} - 10 q^{74} - 2 q^{75} + 20 q^{76} + 18 q^{77} - 8 q^{79} - 2 q^{80} - 8 q^{81} + 4 q^{82} - 12 q^{83} - 28 q^{84} + 2 q^{85} - 4 q^{86} - 22 q^{87} - 18 q^{88} - 12 q^{89} - 20 q^{90} + 22 q^{92} + 8 q^{93} - 32 q^{94} - 16 q^{95} + 4 q^{96} + 2 q^{97} - 24 q^{98} + 24 q^{99}+O(q^{100}) \) 4 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 4 * q^5 - 4 * q^6 + 10 * q^7 + 6 * q^8 + 4 * q^9 - 2 * q^10 - 10 * q^12 + 2 * q^14 + 2 * q^15 + 2 * q^16 - 2 * q^17 + 20 * q^18 + 16 * q^19 - 2 * q^20 + 4 * q^21 + 12 * q^22 - 10 * q^23 - 24 * q^24 + 4 * q^25 - 2 * q^27 + 8 * q^28 + 8 * q^29 + 4 * q^30 + 8 * q^31 + 4 * q^32 + 18 * q^33 - 4 * q^34 - 10 * q^35 + 20 * q^36 - 2 * q^37 + 8 * q^38 - 6 * q^40 + 8 * q^41 - 4 * q^42 - 2 * q^43 + 12 * q^44 - 4 * q^45 + 16 * q^46 + 8 * q^47 - 28 * q^48 + 12 * q^49 + 2 * q^50 + 4 * q^51 - 12 * q^53 - 16 * q^54 + 12 * q^56 - 14 * q^57 + 22 * q^58 + 12 * q^59 + 10 * q^60 + 28 * q^61 + 4 * q^62 + 4 * q^63 + 4 * q^64 + 6 * q^66 + 30 * q^67 + 14 * q^68 - 16 * q^69 - 2 * q^70 + 4 * q^71 + 12 * q^72 - 8 * q^73 - 10 * q^74 - 2 * q^75 + 20 * q^76 + 18 * q^77 - 8 * q^79 - 2 * q^80 - 8 * q^81 + 4 * q^82 - 12 * q^83 - 28 * q^84 + 2 * q^85 - 4 * q^86 - 22 * q^87 - 18 * q^88 - 12 * q^89 - 20 * q^90 + 22 * q^92 + 8 * q^93 - 32 * q^94 - 16 * q^95 + 4 * q^96 + 2 * q^97 - 24 * q^98 + 24 * q^99

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