LMFDB - Embedded newform 403.2.a.b.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [403,2,Mod(1,403)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(403, 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("403.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 403 = 13 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	403.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:	$3.21797120146$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.5748973.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 6x^{4} + 4x^{3} + 8x^{2} - 4x - 1 $ x^6 - x^5 - 6x^4 + 4x^3 + 8x^2 - 4x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$2.35805$ of defining polynomial
Character	$\chi$	$=$	403.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.543189 q^{2} +1.35805 q^{3} -1.70495 q^{4} -3.27954 q^{5} +0.737678 q^{6} +0.540055 q^{7} -2.01249 q^{8} -1.15570 q^{9} +O(q^{10})$	\(q+0.543189 q^{2} +1.35805 q^{3} -1.70495 q^{4} -3.27954 q^{5} +0.737678 q^{6} +0.540055 q^{7} -2.01249 q^{8} -1.15570 q^{9} -1.78141 q^{10} -5.84127 q^{11} -2.31540 q^{12} +1.00000 q^{13} +0.293352 q^{14} -4.45378 q^{15} +2.31673 q^{16} -3.80684 q^{17} -0.627762 q^{18} +7.74981 q^{19} +5.59144 q^{20} +0.733423 q^{21} -3.17291 q^{22} -3.38945 q^{23} -2.73306 q^{24} +5.75537 q^{25} +0.543189 q^{26} -5.64365 q^{27} -0.920765 q^{28} -5.32949 q^{29} -2.41924 q^{30} +1.00000 q^{31} +5.28339 q^{32} -7.93275 q^{33} -2.06783 q^{34} -1.77113 q^{35} +1.97040 q^{36} +1.39331 q^{37} +4.20961 q^{38} +1.35805 q^{39} +6.60002 q^{40} +11.6244 q^{41} +0.398387 q^{42} -9.00680 q^{43} +9.95905 q^{44} +3.79015 q^{45} -1.84111 q^{46} +3.30435 q^{47} +3.14624 q^{48} -6.70834 q^{49} +3.12625 q^{50} -5.16988 q^{51} -1.70495 q^{52} +1.34386 q^{53} -3.06557 q^{54} +19.1567 q^{55} -1.08685 q^{56} +10.5246 q^{57} -2.89492 q^{58} -10.4595 q^{59} +7.59345 q^{60} -2.47671 q^{61} +0.543189 q^{62} -0.624140 q^{63} -1.76359 q^{64} -3.27954 q^{65} -4.30898 q^{66} +5.80808 q^{67} +6.49046 q^{68} -4.60305 q^{69} -0.962059 q^{70} -6.48589 q^{71} +2.32582 q^{72} +4.59191 q^{73} +0.756830 q^{74} +7.81609 q^{75} -13.2130 q^{76} -3.15461 q^{77} +0.737678 q^{78} -13.8141 q^{79} -7.59781 q^{80} -4.19727 q^{81} +6.31423 q^{82} +4.05827 q^{83} -1.25045 q^{84} +12.4847 q^{85} -4.89239 q^{86} -7.23772 q^{87} +11.7555 q^{88} +4.02258 q^{89} +2.05877 q^{90} +0.540055 q^{91} +5.77884 q^{92} +1.35805 q^{93} +1.79489 q^{94} -25.4158 q^{95} +7.17512 q^{96} -11.8862 q^{97} -3.64390 q^{98} +6.75074 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{2} - 5 q^{3} + 6 q^{4} - 9 q^{5} - 3 q^{8} - q^{9}+O(q^{10}) $ 6 * q - 2 * q^2 - 5 * q^3 + 6 * q^4 - 9 * q^5 - 3 * q^8 - q^9	\( 6 q - 2 q^{2} - 5 q^{3} + 6 q^{4} - 9 q^{5} - 3 q^{8} - q^{9} - 8 q^{10} - 5 q^{11} - 13 q^{12} + 6 q^{13} - 17 q^{14} + 4 q^{15} + 14 q^{16} - 23 q^{17} + 9 q^{18} + 7 q^{19} - 10 q^{20} + 2 q^{21} + 2 q^{22} - 18 q^{23} - 13 q^{24} + 11 q^{25} - 2 q^{26} - 5 q^{27} - 25 q^{28} - 18 q^{29} + 25 q^{30} + 6 q^{31} + 2 q^{32} - 2 q^{33} - 16 q^{34} - q^{35} - 2 q^{36} - 13 q^{37} - 8 q^{38} - 5 q^{39} - 29 q^{40} - 5 q^{41} + 31 q^{42} - 7 q^{43} + 30 q^{44} - 5 q^{45} + 19 q^{46} - 9 q^{47} - 19 q^{48} + 16 q^{49} + 29 q^{50} + 26 q^{51} + 6 q^{52} - 31 q^{53} - 4 q^{54} + 7 q^{55} + 8 q^{56} - 5 q^{57} + 35 q^{58} - q^{59} + 33 q^{60} - 15 q^{61} - 2 q^{62} + 11 q^{63} - 5 q^{64} - 9 q^{65} - 29 q^{66} - 28 q^{67} - 12 q^{68} + 5 q^{69} + 73 q^{70} + q^{71} + 45 q^{72} - 20 q^{73} + 4 q^{74} + q^{75} + 38 q^{76} - 29 q^{77} - 15 q^{79} + 7 q^{80} + 2 q^{81} + 36 q^{82} + q^{83} + 68 q^{84} + 29 q^{85} + 3 q^{86} + 10 q^{87} + 9 q^{88} - q^{89} - 32 q^{90} - 60 q^{92} - 5 q^{93} + 54 q^{94} - 13 q^{95} + 36 q^{96} - 5 q^{97} + 20 q^{98} - 15 q^{99}+O(q^{100}) \) 6 * q - 2 * q^2 - 5 * q^3 + 6 * q^4 - 9 * q^5 - 3 * q^8 - q^9 - 8 * q^10 - 5 * q^11 - 13 * q^12 + 6 * q^13 - 17 * q^14 + 4 * q^15 + 14 * q^16 - 23 * q^17 + 9 * q^18 + 7 * q^19 - 10 * q^20 + 2 * q^21 + 2 * q^22 - 18 * q^23 - 13 * q^24 + 11 * q^25 - 2 * q^26 - 5 * q^27 - 25 * q^28 - 18 * q^29 + 25 * q^30 + 6 * q^31 + 2 * q^32 - 2 * q^33 - 16 * q^34 - q^35 - 2 * q^36 - 13 * q^37 - 8 * q^38 - 5 * q^39 - 29 * q^40 - 5 * q^41 + 31 * q^42 - 7 * q^43 + 30 * q^44 - 5 * q^45 + 19 * q^46 - 9 * q^47 - 19 * q^48 + 16 * q^49 + 29 * q^50 + 26 * q^51 + 6 * q^52 - 31 * q^53 - 4 * q^54 + 7 * q^55 + 8 * q^56 - 5 * q^57 + 35 * q^58 - q^59 + 33 * q^60 - 15 * q^61 - 2 * q^62 + 11 * q^63 - 5 * q^64 - 9 * q^65 - 29 * q^66 - 28 * q^67 - 12 * q^68 + 5 * q^69 + 73 * q^70 + q^71 + 45 * q^72 - 20 * q^73 + 4 * q^74 + q^75 + 38 * q^76 - 29 * q^77 - 15 * q^79 + 7 * q^80 + 2 * q^81 + 36 * q^82 + q^83 + 68 * q^84 + 29 * q^85 + 3 * q^86 + 10 * q^87 + 9 * q^88 - q^89 - 32 * q^90 - 60 * q^92 - 5 * q^93 + 54 * q^94 - 13 * q^95 + 36 * q^96 - 5 * q^97 + 20 * q^98 - 15 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0.543189	0.384092	0.192046	−	0.981386i	$-0.438488\pi$
$2$	0.543189	0.384092	0.192046	+	0.981386i	$0.438488\pi$
$3$	1.35805	0.784071	0.392036	−	0.919950i	$-0.371771\pi$
$3$	1.35805	0.784071	0.392036	+	0.919950i	$0.371771\pi$
$4$	−1.70495	−0.852473
$5$	−3.27954	−1.46665	−0.733327	−	0.679876i	$-0.762034\pi$
$5$	−3.27954	−1.46665	−0.733327	+	0.679876i	$0.762034\pi$
$6$	0.737678	0.301156
$7$	0.540055	0.204122	0.102061	−	0.994778i	$-0.467456\pi$
$7$	0.540055	0.204122	0.102061	+	0.994778i	$0.467456\pi$
$8$	−2.01249	−0.711521
$9$	−1.15570	−0.385232
$10$	−1.78141	−0.563331
$11$	−5.84127	−1.76121	−0.880605	−	0.473851i	$-0.842863\pi$
$11$	−5.84127	−1.76121	−0.880605	+	0.473851i	$0.842863\pi$
$12$	−2.31540	−0.668399
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0.293352	0.0784016
$15$	−4.45378	−1.14996
$16$	2.31673	0.579183
$17$	−3.80684	−0.923294	−0.461647	−	0.887064i	$-0.652741\pi$
$17$	−3.80684	−0.923294	−0.461647	+	0.887064i	$0.652741\pi$
$18$	−0.627762	−0.147965
$19$	7.74981	1.77793	0.888965	−	0.457976i	$-0.151425\pi$
$19$	7.74981	1.77793	0.888965	+	0.457976i	$0.151425\pi$
$20$	5.59144	1.25028
$21$	0.733423	0.160046
$22$	−3.17291	−0.676467
$23$	−3.38945	−0.706750	−0.353375	−	0.935482i	$-0.614966\pi$
$23$	−3.38945	−0.706750	−0.353375	+	0.935482i	$0.614966\pi$
$24$	−2.73306	−0.557883
$25$	5.75537	1.15107
$26$	0.543189	0.106528
$27$	−5.64365	−1.08612
$28$	−0.920765	−0.174008
$29$	−5.32949	−0.989661	−0.494830	−	0.868990i	$-0.664770\pi$
$29$	−5.32949	−0.989661	−0.494830	+	0.868990i	$0.664770\pi$
$30$	−2.41924	−0.441691
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	5.28339	0.933981
$33$	−7.93275	−1.38091
$34$	−2.06783	−0.354630
$35$	−1.77113	−0.299376
$36$	1.97040	0.328400
$37$	1.39331	0.229059	0.114529	−	0.993420i	$-0.463464\pi$
$37$	1.39331	0.229059	0.114529	+	0.993420i	$0.463464\pi$
$38$	4.20961	0.682889
$39$	1.35805	0.217462
$40$	6.60002	1.04356
$41$	11.6244	1.81542	0.907712	−	0.419595i	$-0.137828\pi$
$41$	11.6244	1.81542	0.907712	+	0.419595i	$0.137828\pi$
$42$	0.398387	0.0614725
$43$	−9.00680	−1.37352	−0.686762	−	0.726883i	$-0.740968\pi$
$43$	−9.00680	−1.37352	−0.686762	+	0.726883i	$0.740968\pi$
$44$	9.95905	1.50138
$45$	3.79015	0.565003
$46$	−1.84111	−0.271457
$47$	3.30435	0.481989	0.240994	−	0.970526i	$-0.422526\pi$
$47$	3.30435	0.481989	0.240994	+	0.970526i	$0.422526\pi$
$48$	3.14624	0.454121
$49$	−6.70834	−0.958334
$50$	3.12625	0.442119
$51$	−5.16988	−0.723928
$52$	−1.70495	−0.236433
$53$	1.34386	0.184594	0.0922970	−	0.995732i	$-0.470579\pi$
$53$	1.34386	0.184594	0.0922970	+	0.995732i	$0.470579\pi$
$54$	−3.06557	−0.417171
$55$	19.1567	2.58309
$56$	−1.08685	−0.145237
$57$	10.5246	1.39402
$58$	−2.89492	−0.380121
$59$	−10.4595	−1.36171	−0.680855	−	0.732418i	$-0.738392\pi$
$59$	−10.4595	−1.36171	−0.680855	+	0.732418i	$0.738392\pi$
$60$	7.59345	0.980311
$61$	−2.47671	−0.317110	−0.158555	−	0.987350i	$-0.550684\pi$
$61$	−2.47671	−0.317110	−0.158555	+	0.987350i	$0.550684\pi$
$62$	0.543189	0.0689850
$63$	−0.624140	−0.0786343
$64$	−1.76359	−0.220448
$65$	−3.27954	−0.406777
$66$	−4.30898	−0.530399
$67$	5.80808	0.709570	0.354785	−	0.934948i	$-0.384554\pi$
$67$	5.80808	0.709570	0.354785	+	0.934948i	$0.384554\pi$
$68$	6.49046	0.787083
$69$	−4.60305	−0.554142
$70$	−0.962059	−0.114988
$71$	−6.48589	−0.769734	−0.384867	−	0.922972i	$-0.625753\pi$
$71$	−6.48589	−0.769734	−0.384867	+	0.922972i	$0.625753\pi$
$72$	2.32582	0.274101
$73$	4.59191	0.537443	0.268721	−	0.963218i	$-0.413399\pi$
$73$	4.59191	0.537443	0.268721	+	0.963218i	$0.413399\pi$
$74$	0.756830	0.0879797
$75$	7.81609	0.902524
$76$	−13.2130	−1.51564
$77$	−3.15461	−0.359501
$78$	0.737678	0.0835256
$79$	−13.8141	−1.55421	−0.777106	−	0.629370i	$-0.783313\pi$
$79$	−13.8141	−1.55421	−0.777106	+	0.629370i	$0.783313\pi$
$80$	−7.59781	−0.849461
$81$	−4.19727	−0.466364
$82$	6.31423	0.697290
$83$	4.05827	0.445453	0.222726	−	0.974881i	$-0.428504\pi$
$83$	4.05827	0.445453	0.222726	+	0.974881i	$0.428504\pi$
$84$	−1.25045	−0.136435
$85$	12.4847	1.35415
$86$	−4.89239	−0.527560
$87$	−7.23772	−0.775965
$88$	11.7555	1.25314
$89$	4.02258	0.426393	0.213196	−	0.977009i	$-0.431613\pi$
$89$	4.02258	0.426393	0.213196	+	0.977009i	$0.431613\pi$
$90$	2.05877	0.217013
$91$	0.540055	0.0566132
$92$	5.77884	0.602485
$93$	1.35805	0.140823
$94$	1.79489	0.185128
$95$	−25.4158	−2.60761
$96$	7.17512	0.732307
$97$	−11.8862	−1.20686	−0.603432	−	0.797415i	$-0.706200\pi$
$97$	−11.8862	−1.20686	−0.603432	+	0.797415i	$0.706200\pi$
$98$	−3.64390	−0.368089
$99$	6.75074	0.678475
$100$	−9.81259	−0.981259
$101$	−0.508149	−0.0505627	−0.0252814	−	0.999680i	$-0.508048\pi$
$101$	−0.508149	−0.0505627	−0.0252814	+	0.999680i	$0.508048\pi$
$102$	−2.80822	−0.278055
$103$	8.21601	0.809547	0.404774	−	0.914417i	$-0.367350\pi$
$103$	8.21601	0.809547	0.404774	+	0.914417i	$0.367350\pi$
$104$	−2.01249	−0.197340
$105$	−2.40529	−0.234732
$106$	0.729972	0.0709011
$107$	0.0961025	0.00929058	0.00464529	−	0.999989i	$-0.498521\pi$
$107$	0.0961025	0.00929058	0.00464529	+	0.999989i	$0.498521\pi$
$108$	9.62212	0.925889
$109$	−14.1133	−1.35181	−0.675906	−	0.736988i	$-0.736247\pi$
$109$	−14.1133	−1.35181	−0.675906	+	0.736988i	$0.736247\pi$
$110$	10.4057	0.992144
$111$	1.89219	0.179598
$112$	1.25116	0.118224
$113$	14.9129	1.40288	0.701442	−	0.712726i	$-0.252540\pi$
$113$	14.9129	1.40288	0.701442	+	0.712726i	$0.252540\pi$
$114$	5.71687	0.535434
$115$	11.1158	1.03656
$116$	9.08649	0.843659
$117$	−1.15570	−0.106844
$118$	−5.68148	−0.523022
$119$	−2.05590	−0.188464
$120$	8.96317	0.818221
$121$	23.1205	2.10186
$122$	−1.34532	−0.121800
$123$	15.7865	1.42342
$124$	−1.70495	−0.153109
$125$	−2.47727	−0.221573
$126$	−0.339026	−0.0302028
$127$	−15.6054	−1.38475	−0.692377	−	0.721536i	$-0.743436\pi$
$127$	−15.6054	−1.38475	−0.692377	+	0.721536i	$0.743436\pi$
$128$	−11.5247	−1.01865
$129$	−12.2317	−1.07694
$130$	−1.78141	−0.156240
$131$	20.7522	1.81313	0.906566	−	0.422064i	$-0.138694\pi$
$131$	20.7522	1.81313	0.906566	+	0.422064i	$0.138694\pi$
$132$	13.5249	1.17719
$133$	4.18533	0.362914
$134$	3.15488	0.272540
$135$	18.5086	1.59296
$136$	7.66121	0.656943
$137$	−12.5544	−1.07259	−0.536296	−	0.844030i	$-0.680177\pi$
$137$	−12.5544	−1.07259	−0.536296	+	0.844030i	$0.680177\pi$
$138$	−2.50033	−0.212842
$139$	−13.0532	−1.10716	−0.553579	−	0.832797i	$-0.686738\pi$
$139$	−13.0532	−1.10716	−0.553579	+	0.832797i	$0.686738\pi$
$140$	3.01968	0.255210
$141$	4.48748	0.377914
$142$	−3.52306	−0.295649
$143$	−5.84127	−0.488472
$144$	−2.67744	−0.223120
$145$	17.4783	1.45149
$146$	2.49427	0.206428
$147$	−9.11027	−0.751402
$148$	−2.37552	−0.195266
$149$	2.12774	0.174311	0.0871556	−	0.996195i	$-0.472222\pi$
$149$	2.12774	0.174311	0.0871556	+	0.996195i	$0.472222\pi$
$150$	4.24561	0.346653
$151$	−16.5244	−1.34474	−0.672368	−	0.740217i	$-0.734723\pi$
$151$	−16.5244	−1.34474	−0.672368	+	0.740217i	$0.734723\pi$
$152$	−15.5964	−1.26503
$153$	4.39955	0.355683
$154$	−1.71355	−0.138082
$155$	−3.27954	−0.263419
$156$	−2.31540	−0.185381
$157$	−5.88654	−0.469797	−0.234898	−	0.972020i	$-0.575476\pi$
$157$	−5.88654	−0.469797	−0.234898	+	0.972020i	$0.575476\pi$
$158$	−7.50369	−0.596961
$159$	1.82504	0.144735
$160$	−17.3271	−1.36983
$161$	−1.83049	−0.144263
$162$	−2.27991	−0.179127
$163$	−1.42245	−0.111415	−0.0557073	−	0.998447i	$-0.517741\pi$
$163$	−1.42245	−0.111415	−0.0557073	+	0.998447i	$0.517741\pi$
$164$	−19.8189	−1.54760
$165$	26.0157	2.02532
$166$	2.20440	0.171095
$167$	8.43401	0.652643	0.326322	−	0.945259i	$-0.394191\pi$
$167$	8.43401	0.652643	0.326322	+	0.945259i	$0.394191\pi$
$168$	−1.47600	−0.113876
$169$	1.00000	0.0769231
$170$	6.78154	0.520120
$171$	−8.95644	−0.684916
$172$	15.3561	1.17089
$173$	−9.21931	−0.700931	−0.350466	−	0.936576i	$-0.613977\pi$
$173$	−9.21931	−0.700931	−0.350466	+	0.936576i	$0.613977\pi$
$174$	−3.93145	−0.298042
$175$	3.10822	0.234959
$176$	−13.5327	−1.02006
$177$	−14.2045	−1.06768
$178$	2.18502	0.163774
$179$	8.15093	0.609229	0.304614	−	0.952476i	$-0.401472\pi$
$179$	8.15093	0.609229	0.304614	+	0.952476i	$0.401472\pi$
$180$	−6.46201	−0.481649
$181$	−5.07586	−0.377285	−0.188643	−	0.982046i	$-0.560409\pi$
$181$	−5.07586	−0.377285	−0.188643	+	0.982046i	$0.560409\pi$
$182$	0.293352	0.0217447
$183$	−3.36350	−0.248637
$184$	6.82123	0.502867
$185$	−4.56941	−0.335950
$186$	0.737678	0.0540892
$187$	22.2368	1.62611
$188$	−5.63374	−0.410883
$189$	−3.04788	−0.221701
$190$	−13.8056	−1.00156
$191$	−1.76824	−0.127945	−0.0639727	−	0.997952i	$-0.520377\pi$
$191$	−1.76824	−0.127945	−0.0639727	+	0.997952i	$0.520377\pi$
$192$	−2.39504	−0.172847
$193$	27.1111	1.95150	0.975750	−	0.218888i	$-0.0702428\pi$
$193$	27.1111	1.95150	0.975750	+	0.218888i	$0.0702428\pi$
$194$	−6.45647	−0.463547
$195$	−4.45378	−0.318942
$196$	11.4374	0.816954
$197$	−27.1695	−1.93575	−0.967873	−	0.251441i	$-0.919096\pi$
$197$	−27.1695	−1.93575	−0.967873	+	0.251441i	$0.919096\pi$
$198$	3.66693	0.260597
$199$	−5.39641	−0.382541	−0.191271	−	0.981537i	$-0.561261\pi$
$199$	−5.39641	−0.382541	−0.191271	+	0.981537i	$0.561261\pi$
$200$	−11.5826	−0.819013
$201$	7.88767	0.556353
$202$	−0.276021	−0.0194208
$203$	−2.87822	−0.202011
$204$	8.81437	0.617129
$205$	−38.1226	−2.66260
$206$	4.46284	0.310941
$207$	3.91718	0.272263
$208$	2.31673	0.160636
$209$	−45.2688	−3.13131
$210$	−1.30653	−0.0901588
$211$	0.436975	0.0300826	0.0150413	−	0.999887i	$-0.495212\pi$
$211$	0.436975	0.0300826	0.0150413	+	0.999887i	$0.495212\pi$
$212$	−2.29122	−0.157361
$213$	−8.80817	−0.603526
$214$	0.0522018	0.00356844
$215$	29.5381	2.01448
$216$	11.3578	0.772798
$217$	0.540055	0.0366613
$218$	−7.66620	−0.519221
$219$	6.23605	0.421393
$220$	−32.6611	−2.20201
$221$	−3.80684	−0.256076
$222$	1.02781	0.0689824
$223$	−4.25367	−0.284847	−0.142423	−	0.989806i	$-0.545489\pi$
$223$	−4.25367	−0.284847	−0.142423	+	0.989806i	$0.545489\pi$
$224$	2.85332	0.190646
$225$	−6.65147	−0.443431
$226$	8.10050	0.538838
$227$	−20.3852	−1.35301	−0.676507	−	0.736436i	$-0.736507\pi$
$227$	−20.3852	−1.35301	−0.676507	+	0.736436i	$0.736507\pi$
$228$	−17.9439	−1.18837
$229$	−11.6105	−0.767244	−0.383622	−	0.923490i	$-0.625323\pi$
$229$	−11.6105	−0.767244	−0.383622	+	0.923490i	$0.625323\pi$
$230$	6.03800	0.398134
$231$	−4.28412	−0.281875
$232$	10.7255	0.704164
$233$	−1.03767	−0.0679800	−0.0339900	−	0.999422i	$-0.510821\pi$
$233$	−1.03767	−0.0679800	−0.0339900	+	0.999422i	$0.510821\pi$
$234$	−0.627762	−0.0410381
$235$	−10.8367	−0.706911
$236$	17.8329	1.16082
$237$	−18.7603	−1.21861
$238$	−1.11674	−0.0723878
$239$	13.4827	0.872124	0.436062	−	0.899917i	$-0.356373\pi$
$239$	13.4827	0.872124	0.436062	+	0.899917i	$0.356373\pi$
$240$	−10.3182	−0.666038
$241$	3.81558	0.245783	0.122891	−	0.992420i	$-0.460783\pi$
$241$	3.81558	0.245783	0.122891	+	0.992420i	$0.460783\pi$
$242$	12.5588	0.807309
$243$	11.2308	0.720459
$244$	4.22265	0.270328
$245$	22.0003	1.40554
$246$	8.57505	0.546725
$247$	7.74981	0.493109
$248$	−2.01249	−0.127793
$249$	5.51133	0.349267
$250$	−1.34562	−0.0851047
$251$	4.09281	0.258336	0.129168	−	0.991623i	$-0.458769\pi$
$251$	4.09281	0.258336	0.129168	+	0.991623i	$0.458769\pi$
$252$	1.06413	0.0670336
$253$	19.7987	1.24474
$254$	−8.47666	−0.531873
$255$	16.9548	1.06175
$256$	−2.73294	−0.170809
$257$	8.86277	0.552845	0.276422	−	0.961036i	$-0.410851\pi$
$257$	8.86277	0.552845	0.276422	+	0.961036i	$0.410851\pi$
$258$	−6.64412	−0.413645
$259$	0.752464	0.0467559
$260$	5.59144	0.346766
$261$	6.15927	0.381249
$262$	11.2724	0.696411
$263$	28.5273	1.75907	0.879535	−	0.475834i	$-0.157854\pi$
$263$	28.5273	1.75907	0.879535	+	0.475834i	$0.157854\pi$
$264$	15.9645	0.982549
$265$	−4.40725	−0.270735
$266$	2.27342	0.139393
$267$	5.46287	0.334322
$268$	−9.90246	−0.604889
$269$	−14.5391	−0.886463	−0.443232	−	0.896407i	$-0.646168\pi$
$269$	−14.5391	−0.886463	−0.443232	+	0.896407i	$0.646168\pi$
$270$	10.0536	0.611845
$271$	28.7186	1.74453	0.872265	−	0.489033i	$-0.162650\pi$
$271$	28.7186	1.74453	0.872265	+	0.489033i	$0.162650\pi$
$272$	−8.81943	−0.534756
$273$	0.733423	0.0443888
$274$	−6.81940	−0.411975
$275$	−33.6187	−2.02728
$276$	7.84796	0.472391
$277$	−11.0493	−0.663890	−0.331945	−	0.943299i	$-0.607705\pi$
$277$	−11.0493	−0.663890	−0.331945	+	0.943299i	$0.607705\pi$
$278$	−7.09035	−0.425251
$279$	−1.15570	−0.0691898
$280$	3.56438	0.213012
$281$	−25.9813	−1.54992	−0.774958	−	0.632012i	$-0.782229\pi$
$281$	−25.9813	−1.54992	−0.774958	+	0.632012i	$0.782229\pi$
$282$	2.43755	0.145154
$283$	−12.1836	−0.724237	−0.362118	−	0.932132i	$-0.617946\pi$
$283$	−12.1836	−0.724237	−0.362118	+	0.932132i	$0.617946\pi$
$284$	11.0581	0.656177
$285$	−34.5160	−2.04455
$286$	−3.17291	−0.187618
$287$	6.27781	0.370567
$288$	−6.10600	−0.359800
$289$	−2.50797	−0.147528
$290$	9.49399	0.557506
$291$	−16.1421	−0.946267
$292$	−7.82896	−0.458155
$293$	−22.1697	−1.29517	−0.647585	−	0.761994i	$-0.724221\pi$
$293$	−22.1697	−1.29517	−0.647585	+	0.761994i	$0.724221\pi$
$294$	−4.94860	−0.288608
$295$	34.3023	1.99716
$296$	−2.80402	−0.162980
$297$	32.9661	1.91289
$298$	1.15576	0.0669516
$299$	−3.38945	−0.196017
$300$	−13.3260	−0.769377
$301$	−4.86417	−0.280366
$302$	−8.97587	−0.516503
$303$	−0.690093	−0.0396448
$304$	17.9542	1.02975
$305$	8.12246	0.465091
$306$	2.38979	0.136615
$307$	12.8952	0.735969	0.367985	−	0.929832i	$-0.380048\pi$
$307$	12.8952	0.735969	0.367985	+	0.929832i	$0.380048\pi$
$308$	5.37844	0.306465
$309$	11.1578	0.634743
$310$	−1.78141	−0.101177
$311$	−4.85954	−0.275559	−0.137780	−	0.990463i	$-0.543997\pi$
$311$	−4.85954	−0.275559	−0.137780	+	0.990463i	$0.543997\pi$
$312$	−2.73306	−0.154729
$313$	−13.5194	−0.764161	−0.382081	−	0.924129i	$-0.624792\pi$
$313$	−13.5194	−0.764161	−0.382081	+	0.924129i	$0.624792\pi$
$314$	−3.19750	−0.180445
$315$	2.04689	0.115329
$316$	23.5524	1.32492
$317$	−5.97619	−0.335656	−0.167828	−	0.985816i	$-0.553675\pi$
$317$	−5.97619	−0.335656	−0.167828	+	0.985816i	$0.553675\pi$
$318$	0.991339	0.0555915
$319$	31.1310	1.74300
$320$	5.78374	0.323321
$321$	0.130512	0.00728447
$322$	−0.994303	−0.0554104
$323$	−29.5023	−1.64155
$324$	7.15612	0.397562
$325$	5.75537	0.319250
$326$	−0.772658	−0.0427935
$327$	−19.1666	−1.05992
$328$	−23.3939	−1.29171
$329$	1.78453	0.0983844
$330$	14.1315	0.777911
$331$	27.9988	1.53896	0.769478	−	0.638674i	$-0.220517\pi$
$331$	27.9988	1.53896	0.769478	+	0.638674i	$0.220517\pi$
$332$	−6.91912	−0.379736
$333$	−1.61024	−0.0882408
$334$	4.58126	0.250675
$335$	−19.0478	−1.04069
$336$	1.69914	0.0926959
$337$	−6.67062	−0.363372	−0.181686	−	0.983357i	$-0.558155\pi$
$337$	−6.67062	−0.363372	−0.181686	+	0.983357i	$0.558155\pi$
$338$	0.543189	0.0295456
$339$	20.2524	1.09996
$340$	−21.2857	−1.15438
$341$	−5.84127	−0.316323
$342$	−4.86504	−0.263071
$343$	−7.40326	−0.399739
$344$	18.1260	0.977291
$345$	15.0959	0.812735
$346$	−5.00783	−0.269222
$347$	19.8224	1.06412	0.532061	−	0.846706i	$-0.321418\pi$
$347$	19.8224	1.06412	0.532061	+	0.846706i	$0.321418\pi$
$348$	12.3399	0.661489
$349$	−33.1056	−1.77210	−0.886051	−	0.463588i	$-0.846562\pi$
$349$	−33.1056	−1.77210	−0.886051	+	0.463588i	$0.846562\pi$
$350$	1.68835	0.0902461
$351$	−5.64365	−0.301236
$352$	−30.8617	−1.64494
$353$	−7.62627	−0.405905	−0.202953	−	0.979189i	$-0.565054\pi$
$353$	−7.62627	−0.405905	−0.202953	+	0.979189i	$0.565054\pi$
$354$	−7.71574	−0.410087
$355$	21.2707	1.12893
$356$	−6.85828	−0.363488
$357$	−2.79202	−0.147770
$358$	4.42749	0.234000
$359$	19.7708	1.04346	0.521732	−	0.853109i	$-0.325286\pi$
$359$	19.7708	1.04346	0.521732	+	0.853109i	$0.325286\pi$
$360$	−7.62763	−0.402011
$361$	41.0596	2.16103
$362$	−2.75715	−0.144912
$363$	31.3988	1.64801
$364$	−0.920765	−0.0482612
$365$	−15.0593	−0.788242
$366$	−1.82701	−0.0954995
$367$	27.1488	1.41715	0.708577	−	0.705634i	$-0.249338\pi$
$367$	27.1488	1.41715	0.708577	+	0.705634i	$0.249338\pi$
$368$	−7.85246	−0.409338
$369$	−13.4343	−0.699360
$370$	−2.48205	−0.129036
$371$	0.725761	0.0376796
$372$	−2.31540	−0.120048
$373$	−17.5513	−0.908770	−0.454385	−	0.890806i	$-0.650141\pi$
$373$	−17.5513	−0.908770	−0.454385	+	0.890806i	$0.650141\pi$
$374$	12.0788	0.624578
$375$	−3.36425	−0.173729
$376$	−6.64996	−0.342945
$377$	−5.32949	−0.274483
$378$	−1.65558	−0.0851536
$379$	−13.3989	−0.688254	−0.344127	−	0.938923i	$-0.611825\pi$
$379$	−13.3989	−0.688254	−0.344127	+	0.938923i	$0.611825\pi$
$380$	43.3326	2.22291
$381$	−21.1929	−1.08574
$382$	−0.960488	−0.0491428
$383$	−15.2919	−0.781381	−0.390691	−	0.920522i	$-0.627764\pi$
$383$	−15.2919	−0.781381	−0.390691	+	0.920522i	$0.627764\pi$
$384$	−15.6512	−0.798697
$385$	10.3457	0.527264
$386$	14.7264	0.749557
$387$	10.4091	0.529126
$388$	20.2654	1.02882
$389$	5.88468	0.298365	0.149183	−	0.988810i	$-0.452336\pi$
$389$	5.88468	0.298365	0.149183	+	0.988810i	$0.452336\pi$
$390$	−2.41924	−0.122503
$391$	12.9031	0.652538
$392$	13.5004	0.681875
$393$	28.1826	1.42162
$394$	−14.7582	−0.743505
$395$	45.3040	2.27949
$396$	−11.5096	−0.578382
$397$	31.9655	1.60430	0.802152	−	0.597120i	$-0.203688\pi$
$397$	31.9655	1.60430	0.802152	+	0.597120i	$0.203688\pi$
$398$	−2.93127	−0.146931
$399$	5.68389	0.284550
$400$	13.3337	0.666683
$401$	2.05389	0.102567	0.0512833	−	0.998684i	$-0.483669\pi$
$401$	2.05389	0.102567	0.0512833	+	0.998684i	$0.483669\pi$
$402$	4.28449	0.213691
$403$	1.00000	0.0498135
$404$	0.866367	0.0431034
$405$	13.7651	0.683994
$406$	−1.56342	−0.0775910
$407$	−8.13870	−0.403420
$408$	10.4043	0.515090
$409$	0.974925	0.0482070	0.0241035	−	0.999709i	$-0.492327\pi$
$409$	0.974925	0.0482070	0.0241035	+	0.999709i	$0.492327\pi$
$410$	−20.7078	−1.02268
$411$	−17.0495	−0.840989
$412$	−14.0079	−0.690117
$413$	−5.64870	−0.277955
$414$	2.12777	0.104574
$415$	−13.3092	−0.653325
$416$	5.28339	0.259040
$417$	−17.7269	−0.868090
$418$	−24.5895	−1.20271
$419$	30.0052	1.46585	0.732925	−	0.680310i	$-0.238155\pi$
$419$	30.0052	1.46585	0.732925	+	0.680310i	$0.238155\pi$
$420$	4.10089	0.200103
$421$	21.2469	1.03551	0.517754	−	0.855529i	$-0.326768\pi$
$421$	21.2469	1.03551	0.517754	+	0.855529i	$0.326768\pi$
$422$	0.237360	0.0115545
$423$	−3.81883	−0.185678
$424$	−2.70451	−0.131342
$425$	−21.9098	−1.06278
$426$	−4.78450	−0.231810
$427$	−1.33756	−0.0647290
$428$	−0.163850	−0.00791996
$429$	−7.93275	−0.382997
$430$	16.0448	0.773748
$431$	3.05200	0.147010	0.0735048	−	0.997295i	$-0.476582\pi$
$431$	3.05200	0.147010	0.0735048	+	0.997295i	$0.476582\pi$
$432$	−13.0748	−0.629063
$433$	7.32400	0.351969	0.175984	−	0.984393i	$-0.443689\pi$
$433$	7.32400	0.351969	0.175984	+	0.984393i	$0.443689\pi$
$434$	0.293352	0.0140813
$435$	23.7364	1.13807
$436$	24.0625	1.15238
$437$	−26.2676	−1.25655
$438$	3.38735	0.161854
$439$	−9.17381	−0.437842	−0.218921	−	0.975743i	$-0.570254\pi$
$439$	−9.17381	−0.437842	−0.218921	+	0.975743i	$0.570254\pi$
$440$	−38.5525	−1.83792
$441$	7.75281	0.369181
$442$	−2.06783	−0.0983568
$443$	28.1095	1.33552	0.667762	−	0.744375i	$-0.267253\pi$
$443$	28.1095	1.33552	0.667762	+	0.744375i	$0.267253\pi$
$444$	−3.22607	−0.153103
$445$	−13.1922	−0.625371
$446$	−2.31055	−0.109407
$447$	2.88958	0.136672
$448$	−0.952434	−0.0449983
$449$	14.0708	0.664042	0.332021	−	0.943272i	$-0.392270\pi$
$449$	14.0708	0.664042	0.332021	+	0.943272i	$0.392270\pi$
$450$	−3.61300	−0.170319
$451$	−67.9012	−3.19734
$452$	−25.4256	−1.19592
$453$	−22.4410	−1.05437
$454$	−11.0730	−0.519683
$455$	−1.77113	−0.0830320
$456$	−21.1807	−0.991876
$457$	24.7018	1.15550	0.577752	−	0.816213i	$-0.303930\pi$
$457$	24.7018	1.15550	0.577752	+	0.816213i	$0.303930\pi$
$458$	−6.30670	−0.294693
$459$	21.4845	1.00281
$460$	−18.9519	−0.883638
$461$	5.99150	0.279052	0.139526	−	0.990218i	$-0.455442\pi$
$461$	5.99150	0.279052	0.139526	+	0.990218i	$0.455442\pi$
$462$	−2.32709	−0.108266
$463$	24.7125	1.14849	0.574243	−	0.818685i	$-0.305297\pi$
$463$	24.7125	1.14849	0.574243	+	0.818685i	$0.305297\pi$
$464$	−12.3470	−0.573195
$465$	−4.45378	−0.206539
$466$	−0.563651	−0.0261106
$467$	37.1190	1.71766	0.858831	−	0.512259i	$-0.171191\pi$
$467$	37.1190	1.71766	0.858831	+	0.512259i	$0.171191\pi$
$468$	1.97040	0.0910818
$469$	3.13668	0.144839
$470$	−5.88640	−0.271519
$471$	−7.99422	−0.368354
$472$	21.0496	0.968885
$473$	52.6111	2.41906
$474$	−10.1904	−0.468060
$475$	44.6030	2.04653
$476$	3.50521	0.160661
$477$	−1.55310	−0.0711116
$478$	7.32366	0.334976
$479$	9.82088	0.448727	0.224364	−	0.974505i	$-0.427970\pi$
$479$	9.82088	0.448727	0.224364	+	0.974505i	$0.427970\pi$
$480$	−23.5311	−1.07404
$481$	1.39331	0.0635295
$482$	2.07258	0.0944034
$483$	−2.48590	−0.113113
$484$	−39.4191	−1.79178
$485$	38.9813	1.77005
$486$	6.10046	0.276723
$487$	−17.0244	−0.771450	−0.385725	−	0.922614i	$-0.626049\pi$
$487$	−17.0244	−0.771450	−0.385725	+	0.922614i	$0.626049\pi$
$488$	4.98434	0.225630
$489$	−1.93176	−0.0873570
$490$	11.9503	0.539859
$491$	−9.11752	−0.411468	−0.205734	−	0.978608i	$-0.565958\pi$
$491$	−9.11752	−0.411468	−0.205734	+	0.978608i	$0.565958\pi$
$492$	−26.9151	−1.21343
$493$	20.2885	0.913748
$494$	4.20961	0.189399
$495$	−22.1393	−0.995088
$496$	2.31673	0.104024
$497$	−3.50274	−0.157119
$498$	2.99369	0.134151
$499$	−3.61464	−0.161813	−0.0809067	−	0.996722i	$-0.525782\pi$
$499$	−3.61464	−0.161813	−0.0809067	+	0.996722i	$0.525782\pi$
$500$	4.22360	0.188885
$501$	11.4538	0.511719
$502$	2.22317	0.0992248
$503$	−31.1701	−1.38981	−0.694903	−	0.719103i	$-0.744553\pi$
$503$	−31.1701	−1.38981	−0.694903	+	0.719103i	$0.744553\pi$
$504$	1.25607	0.0559500
$505$	1.66649	0.0741581
$506$	10.7544	0.478093
$507$	1.35805	0.0603132
$508$	26.6063	1.18046
$509$	38.1296	1.69006	0.845032	−	0.534716i	$-0.179581\pi$
$509$	38.1296	1.69006	0.845032	+	0.534716i	$0.179581\pi$
$510$	9.20967	0.407811
$511$	2.47989	0.109704
$512$	21.5650	0.953047
$513$	−43.7372	−1.93105
$514$	4.81416	0.212343
$515$	−26.9447	−1.18733
$516$	20.8544	0.918062
$517$	−19.3016	−0.848884
$518$	0.408730	0.0179586
$519$	−12.5203	−0.549580
$520$	6.60002	0.289430
$521$	−2.39876	−0.105091	−0.0525457	−	0.998619i	$-0.516734\pi$
$521$	−2.39876	−0.105091	−0.0525457	+	0.998619i	$0.516734\pi$
$522$	3.34565	0.146435
$523$	14.3960	0.629494	0.314747	−	0.949176i	$-0.398080\pi$
$523$	14.3960	0.629494	0.314747	+	0.949176i	$0.398080\pi$
$524$	−35.3815	−1.54565
$525$	4.22112	0.184225
$526$	15.4957	0.675646
$527$	−3.80684	−0.165829
$528$	−18.3781	−0.799802
$529$	−11.5116	−0.500504
$530$	−2.39397	−0.103987
$531$	12.0880	0.524575
$532$	−7.13576	−0.309374
$533$	11.6244	0.503508
$534$	2.96737	0.128411
$535$	−0.315172	−0.0136261
$536$	−11.6887	−0.504874
$537$	11.0694	0.477679
$538$	−7.89746	−0.340484
$539$	39.1852	1.68783
$540$	−31.5561	−1.35796
$541$	19.5342	0.839842	0.419921	−	0.907561i	$-0.362058\pi$
$541$	19.5342	0.839842	0.419921	+	0.907561i	$0.362058\pi$
$542$	15.5996	0.670061
$543$	−6.89327	−0.295819
$544$	−20.1130	−0.862339
$545$	46.2852	1.98264
$546$	0.398387	0.0170494
$547$	−40.4710	−1.73041	−0.865207	−	0.501415i	$-0.832813\pi$
$547$	−40.4710	−1.73041	−0.865207	+	0.501415i	$0.832813\pi$
$548$	21.4045	0.914356
$549$	2.86232	0.122161
$550$	−18.2613	−0.778664
$551$	−41.3025	−1.75955
$552$	9.26357	0.394284
$553$	−7.46040	−0.317248
$554$	−6.00188	−0.254995
$555$	−6.20550	−0.263409
$556$	22.2550	0.943822
$557$	−29.0703	−1.23175	−0.615873	−	0.787845i	$-0.711197\pi$
$557$	−29.0703	−1.23175	−0.615873	+	0.787845i	$0.711197\pi$
$558$	−0.627762	−0.0265753
$559$	−9.00680	−0.380947
$560$	−4.10324	−0.173394
$561$	30.1987	1.27499
$562$	−14.1128	−0.595311
$563$	−6.97578	−0.293994	−0.146997	−	0.989137i	$-0.546961\pi$
$563$	−6.97578	−0.293994	−0.146997	+	0.989137i	$0.546961\pi$
$564$	−7.65090	−0.322161
$565$	−48.9073	−2.05755
$566$	−6.61797	−0.278174
$567$	−2.26676	−0.0951949
$568$	13.0528	0.547682
$569$	−2.07193	−0.0868600	−0.0434300	−	0.999056i	$-0.513829\pi$
$569$	−2.07193	−0.0868600	−0.0434300	+	0.999056i	$0.513829\pi$
$570$	−18.7487	−0.785296
$571$	−29.9448	−1.25315	−0.626575	−	0.779361i	$-0.715544\pi$
$571$	−29.9448	−1.25315	−0.626575	+	0.779361i	$0.715544\pi$
$572$	9.95905	0.416409
$573$	−2.40136	−0.100318
$574$	3.41004	0.142332
$575$	−19.5076	−0.813522
$576$	2.03817	0.0849238
$577$	4.39645	0.183026	0.0915132	−	0.995804i	$-0.470830\pi$
$577$	4.39645	0.183026	0.0915132	+	0.995804i	$0.470830\pi$
$578$	−1.36230	−0.0566643
$579$	36.8183	1.53011
$580$	−29.7995	−1.23736
$581$	2.19169	0.0909266
$582$	−8.76821	−0.363454
$583$	−7.84987	−0.325109
$584$	−9.24115	−0.382402
$585$	3.79015	0.156704
$586$	−12.0423	−0.497465
$587$	30.3777	1.25382	0.626912	−	0.779090i	$-0.284319\pi$
$587$	30.3777	1.25382	0.626912	+	0.779090i	$0.284319\pi$
$588$	15.5325	0.640550
$589$	7.74981	0.319325
$590$	18.6326	0.767093
$591$	−36.8975	−1.51776
$592$	3.22793	0.132667
$593$	26.2726	1.07889	0.539443	−	0.842022i	$-0.318635\pi$
$593$	26.2726	1.07889	0.539443	+	0.842022i	$0.318635\pi$
$594$	17.9068	0.734725
$595$	6.74242	0.276412
$596$	−3.62768	−0.148596
$597$	−7.32860	−0.299940
$598$	−1.84111	−0.0752887
$599$	−2.03547	−0.0831672	−0.0415836	−	0.999135i	$-0.513240\pi$
$599$	−2.03547	−0.0831672	−0.0415836	+	0.999135i	$0.513240\pi$
$600$	−15.7298	−0.642165
$601$	12.0857	0.492988	0.246494	−	0.969144i	$-0.420721\pi$
$601$	12.0857	0.492988	0.246494	+	0.969144i	$0.420721\pi$
$602$	−2.64216	−0.107686
$603$	−6.71238	−0.273349
$604$	28.1732	1.14635
$605$	−75.8244	−3.08270
$606$	−0.374851	−0.0152273
$607$	−19.0749	−0.774226	−0.387113	−	0.922032i	$-0.626528\pi$
$607$	−19.0749	−0.774226	−0.387113	+	0.922032i	$0.626528\pi$
$608$	40.9453	1.66055
$609$	−3.90877	−0.158391
$610$	4.41203	0.178638
$611$	3.30435	0.133680
$612$	−7.50100	−0.303210
$613$	−27.3881	−1.10620	−0.553098	−	0.833116i	$-0.686555\pi$
$613$	−27.3881	−1.10620	−0.553098	+	0.833116i	$0.686555\pi$
$614$	7.00454	0.282680
$615$	−51.7724	−2.08767
$616$	6.34861	0.255793
$617$	22.0345	0.887077	0.443538	−	0.896255i	$-0.353723\pi$
$617$	22.0345	0.887077	0.443538	+	0.896255i	$0.353723\pi$
$618$	6.06077	0.243800
$619$	−33.4112	−1.34291	−0.671455	−	0.741045i	$-0.734330\pi$
$619$	−33.4112	−1.34291	−0.671455	+	0.741045i	$0.734330\pi$
$620$	5.59144	0.224557
$621$	19.1289	0.767616
$622$	−2.63965	−0.105840
$623$	2.17242	0.0870360
$624$	3.14624	0.125950
$625$	−20.6526	−0.826103
$626$	−7.34358	−0.293509
$627$	−61.4773	−2.45517
$628$	10.0362	0.400489
$629$	−5.30411	−0.211489
$630$	1.11185	0.0442971
$631$	28.4799	1.13377	0.566883	−	0.823799i	$-0.308149\pi$
$631$	28.4799	1.13377	0.566883	+	0.823799i	$0.308149\pi$
$632$	27.8007	1.10585
$633$	0.593435	0.0235869
$634$	−3.24620	−0.128923
$635$	51.1784	2.03095
$636$	−3.11159	−0.123382
$637$	−6.70834	−0.265794
$638$	16.9100	0.669473
$639$	7.49573	0.296526
$640$	37.7958	1.49401
$641$	−0.439962	−0.0173775	−0.00868873	−	0.999962i	$-0.502766\pi$
$641$	−0.439962	−0.0173775	−0.00868873	+	0.999962i	$0.502766\pi$
$642$	0.0708927	0.00279791
$643$	−38.9985	−1.53795	−0.768976	−	0.639277i	$-0.779234\pi$
$643$	−38.9985	−1.53795	−0.768976	+	0.639277i	$0.779234\pi$
$644$	3.12089	0.122980
$645$	40.1143	1.57950
$646$	−16.0253	−0.630508
$647$	−25.0058	−0.983078	−0.491539	−	0.870856i	$-0.663565\pi$
$647$	−25.0058	−0.983078	−0.491539	+	0.870856i	$0.663565\pi$
$648$	8.44695	0.331827
$649$	61.0967	2.39826
$650$	3.12625	0.122622
$651$	0.733423	0.0287451
$652$	2.42520	0.0949780
$653$	−44.6189	−1.74607	−0.873037	−	0.487654i	$-0.837853\pi$
$653$	−44.6189	−1.74607	−0.873037	+	0.487654i	$0.837853\pi$
$654$	−10.4111	−0.407106
$655$	−68.0578	−2.65924
$656$	26.9306	1.05146
$657$	−5.30686	−0.207040
$658$	0.969338	0.0377887
$659$	−25.2842	−0.984933	−0.492466	−	0.870332i	$-0.663905\pi$
$659$	−25.2842	−0.984933	−0.492466	+	0.870332i	$0.663905\pi$
$660$	−44.3554	−1.72653
$661$	1.37636	0.0535340	0.0267670	−	0.999642i	$-0.491479\pi$
$661$	1.37636	0.0535340	0.0267670	+	0.999642i	$0.491479\pi$
$662$	15.2087	0.591101
$663$	−5.16988	−0.200782
$664$	−8.16720	−0.316949
$665$	−13.7259	−0.532269
$666$	−0.874667	−0.0338926
$667$	18.0641	0.699443
$668$	−14.3795	−0.556361
$669$	−5.77670	−0.223340
$670$	−10.3466	−0.399722
$671$	14.4671	0.558497
$672$	3.87496	0.149480
$673$	0.504133	0.0194329	0.00971645	−	0.999953i	$-0.496907\pi$
$673$	0.504133	0.0194329	0.00971645	+	0.999953i	$0.496907\pi$
$674$	−3.62341	−0.139568
$675$	−32.4813	−1.25021
$676$	−1.70495	−0.0655748
$677$	−34.3533	−1.32030	−0.660152	−	0.751132i	$-0.729508\pi$
$677$	−34.3533	−1.32030	−0.660152	+	0.751132i	$0.729508\pi$
$678$	11.0009	0.422487
$679$	−6.41922	−0.246347
$680$	−25.1252	−0.963508
$681$	−27.6842	−1.06086
$682$	−3.17291	−0.121497
$683$	−51.2973	−1.96284	−0.981418	−	0.191881i	$-0.938541\pi$
$683$	−51.2973	−1.96284	−0.981418	+	0.191881i	$0.938541\pi$
$684$	15.2702	0.583872
$685$	41.1726	1.57312
$686$	−4.02137	−0.153537
$687$	−15.7677	−0.601574
$688$	−20.8663	−0.795522
$689$	1.34386	0.0511971
$690$	8.19992	0.312165
$691$	3.08814	0.117478	0.0587392	−	0.998273i	$-0.481292\pi$
$691$	3.08814	0.117478	0.0587392	+	0.998273i	$0.481292\pi$
$692$	15.7184	0.597525
$693$	3.64577	0.138492
$694$	10.7673	0.408721
$695$	42.8084	1.62382
$696$	14.5658	0.552115
$697$	−44.2522	−1.67617
$698$	−17.9826	−0.680651
$699$	−1.40921	−0.0533012
$700$	−5.29934	−0.200296
$701$	−11.2331	−0.424269	−0.212134	−	0.977241i	$-0.568041\pi$
$701$	−11.2331	−0.424269	−0.212134	+	0.977241i	$0.568041\pi$
$702$	−3.06557	−0.115702
$703$	10.7979	0.407250
$704$	10.3016	0.388255
$705$	−14.7168	−0.554269
$706$	−4.14250	−0.155905
$707$	−0.274429	−0.0103210
$708$	24.2179	0.910166
$709$	−19.7625	−0.742196	−0.371098	−	0.928594i	$-0.621019\pi$
$709$	−19.7625	−0.742196	−0.371098	+	0.928594i	$0.621019\pi$
$710$	11.5540	0.433615
$711$	15.9650	0.598733
$712$	−8.09539	−0.303387
$713$	−3.38945	−0.126936
$714$	−1.51660	−0.0567572
$715$	19.1567	0.716419
$716$	−13.8969	−0.519351
$717$	18.3102	0.683807
$718$	10.7393	0.400787
$719$	−5.87278	−0.219018	−0.109509	−	0.993986i	$-0.534928\pi$
$719$	−5.87278	−0.219018	−0.109509	+	0.993986i	$0.534928\pi$
$720$	8.78077	0.327240
$721$	4.43710	0.165246
$722$	22.3031	0.830036
$723$	5.18175	0.192711
$724$	8.65406	0.321626
$725$	−30.6732	−1.13917
$726$	17.0555	0.632987
$727$	−5.91365	−0.219325	−0.109663	−	0.993969i	$-0.534977\pi$
$727$	−5.91365	−0.219325	−0.109663	+	0.993969i	$0.534977\pi$
$728$	−1.08685	−0.0402815
$729$	27.8439	1.03125
$730$	−8.18007	−0.302758
$731$	34.2874	1.26817
$732$	5.73458	0.211956
$733$	−21.2088	−0.783366	−0.391683	−	0.920100i	$-0.628107\pi$
$733$	−21.2088	−0.783366	−0.391683	+	0.920100i	$0.628107\pi$
$734$	14.7469	0.544318
$735$	29.8775	1.10205
$736$	−17.9078	−0.660091
$737$	−33.9266	−1.24970
$738$	−7.29734	−0.268619
$739$	25.5620	0.940313	0.470157	−	0.882583i	$-0.344197\pi$
$739$	25.5620	0.940313	0.470157	+	0.882583i	$0.344197\pi$
$740$	7.79060	0.286388
$741$	10.5246	0.386632
$742$	0.394225	0.0144725
$743$	11.0249	0.404463	0.202232	−	0.979338i	$-0.435181\pi$
$743$	11.0249	0.404463	0.202232	+	0.979338i	$0.435181\pi$
$744$	−2.73306	−0.100199
$745$	−6.97800	−0.255654
$746$	−9.53365	−0.349052
$747$	−4.69013	−0.171603
$748$	−37.9125	−1.38622
$749$	0.0519007	0.00189641
$750$	−1.82742	−0.0667281
$751$	27.4344	1.00110	0.500548	−	0.865709i	$-0.333132\pi$
$751$	27.4344	1.00110	0.500548	+	0.865709i	$0.333132\pi$
$752$	7.65530	0.279160
$753$	5.55824	0.202554
$754$	−2.89492	−0.105427
$755$	54.1924	1.97226
$756$	5.19648	0.188994
$757$	−37.1729	−1.35107	−0.675536	−	0.737327i	$-0.736087\pi$
$757$	−37.1729	−1.35107	−0.675536	+	0.737327i	$0.736087\pi$
$758$	−7.27812	−0.264353
$759$	26.8877	0.975961
$760$	51.1489	1.85537
$761$	−40.3734	−1.46354	−0.731768	−	0.681554i	$-0.761304\pi$
$761$	−40.3734	−1.46354	−0.731768	+	0.681554i	$0.761304\pi$
$762$	−11.5117	−0.417026
$763$	−7.62198	−0.275934
$764$	3.01475	0.109070
$765$	−14.4285	−0.521664
$766$	−8.30640	−0.300123
$767$	−10.4595	−0.377670
$768$	−3.71148	−0.133926
$769$	−26.2892	−0.948013	−0.474006	−	0.880521i	$-0.657193\pi$
$769$	−26.2892	−0.948013	−0.474006	+	0.880521i	$0.657193\pi$
$770$	5.61965	0.202518
$771$	12.0361	0.433470
$772$	−46.2230	−1.66360
$773$	−42.6272	−1.53319	−0.766597	−	0.642128i	$-0.778052\pi$
$773$	−42.6272	−1.53319	−0.766597	+	0.642128i	$0.778052\pi$
$774$	5.65412	0.203233
$775$	5.75537	0.206739
$776$	23.9209	0.858709
$777$	1.02189	0.0366599
$778$	3.19649	0.114600
$779$	90.0868	3.22769
$780$	7.59345	0.271889
$781$	37.8859	1.35566
$782$	7.00882	0.250635
$783$	30.0778	1.07489
$784$	−15.5414	−0.555051
$785$	19.3051	0.689029
$786$	15.3085	0.546035
$787$	8.64737	0.308245	0.154123	−	0.988052i	$-0.450745\pi$
$787$	8.64737	0.308245	0.154123	+	0.988052i	$0.450745\pi$
$788$	46.3225	1.65017
$789$	38.7416	1.37924
$790$	24.6086	0.875535
$791$	8.05378	0.286359
$792$	−13.5858	−0.482749
$793$	−2.47671	−0.0879505
$794$	17.3633	0.616201
$795$	−5.98527	−0.212276
$796$	9.20058	0.326106
$797$	−4.11899	−0.145902	−0.0729510	−	0.997336i	$-0.523242\pi$
$797$	−4.11899	−0.145902	−0.0729510	+	0.997336i	$0.523242\pi$
$798$	3.08742	0.109294
$799$	−12.5791	−0.445018
$800$	30.4079	1.07508
$801$	−4.64889	−0.164260
$802$	1.11565	0.0393950
$803$	−26.8226	−0.946549
$804$	−13.4480	−0.474276
$805$	6.00317	0.211584
$806$	0.543189	0.0191330
$807$	−19.7448	−0.695050
$808$	1.02264	0.0359765
$809$	28.8488	1.01427	0.507134	−	0.861867i	$-0.330705\pi$
$809$	28.8488	1.01427	0.507134	+	0.861867i	$0.330705\pi$
$810$	7.47706	0.262717
$811$	18.8221	0.660934	0.330467	−	0.943818i	$-0.392794\pi$
$811$	18.8221	0.660934	0.330467	+	0.943818i	$0.392794\pi$
$812$	4.90721	0.172209
$813$	39.0013	1.36784
$814$	−4.42085	−0.154951
$815$	4.66497	0.163407
$816$	−11.9772	−0.419287
$817$	−69.8010	−2.44203
$818$	0.529569	0.0185159
$819$	−0.624140	−0.0218092
$820$	64.9970	2.26979
$821$	20.2893	0.708100	0.354050	−	0.935226i	$-0.384804\pi$
$821$	20.2893	0.708100	0.354050	+	0.935226i	$0.384804\pi$
$822$	−9.26109	−0.323018
$823$	13.4385	0.468435	0.234217	−	0.972184i	$-0.424747\pi$
$823$	13.4385	0.468435	0.234217	+	0.972184i	$0.424747\pi$
$824$	−16.5346	−0.576010
$825$	−45.6559	−1.58953
$826$	−3.06831	−0.106760
$827$	−4.38532	−0.152492	−0.0762462	−	0.997089i	$-0.524293\pi$
$827$	−4.38532	−0.152492	−0.0762462	+	0.997089i	$0.524293\pi$
$828$	−6.67859	−0.232097
$829$	13.7448	0.477376	0.238688	−	0.971096i	$-0.423283\pi$
$829$	13.7448	0.477376	0.238688	+	0.971096i	$0.423283\pi$
$830$	−7.22943	−0.250937
$831$	−15.0056	−0.520537
$832$	−1.76359	−0.0611413
$833$	25.5376	0.884825
$834$	−9.62905	−0.333427
$835$	−27.6597	−0.957202
$836$	77.1808	2.66935
$837$	−5.64365	−0.195073
$838$	16.2985	0.563022
$839$	−25.8769	−0.893370	−0.446685	−	0.894691i	$-0.647395\pi$
$839$	−25.8769	−0.893370	−0.446685	+	0.894691i	$0.647395\pi$
$840$	4.84061	0.167017
$841$	−0.596573	−0.0205715
$842$	11.5411	0.397731
$843$	−35.2840	−1.21524
$844$	−0.745019	−0.0256446
$845$	−3.27954	−0.112820
$846$	−2.07434	−0.0713174
$847$	12.4863	0.429035
$848$	3.11337	0.106914
$849$	−16.5459	−0.567853
$850$	−11.9011	−0.408206
$851$	−4.72256	−0.161887
$852$	15.0175	0.514490
$853$	14.4121	0.493461	0.246731	−	0.969084i	$-0.420644\pi$
$853$	14.4121	0.493461	0.246731	+	0.969084i	$0.420644\pi$
$854$	−0.726547	−0.0248619
$855$	29.3730	1.00453
$856$	−0.193405	−0.00661044
$857$	−10.1711	−0.347439	−0.173719	−	0.984795i	$-0.555579\pi$
$857$	−10.1711	−0.347439	−0.173719	+	0.984795i	$0.555579\pi$
$858$	−4.30898	−0.147106
$859$	−1.25050	−0.0426666	−0.0213333	−	0.999772i	$-0.506791\pi$
$859$	−1.25050	−0.0426666	−0.0213333	+	0.999772i	$0.506791\pi$
$860$	−50.3609	−1.71729
$861$	8.52559	0.290551
$862$	1.65781	0.0564653
$863$	30.9689	1.05419	0.527097	−	0.849805i	$-0.323281\pi$
$863$	30.9689	1.05419	0.527097	+	0.849805i	$0.323281\pi$
$864$	−29.8176	−1.01442
$865$	30.2351	1.02802
$866$	3.97831	0.135189
$867$	−3.40596	−0.115672
$868$	−0.920765	−0.0312528
$869$	80.6921	2.73729
$870$	12.8933	0.437125
$871$	5.80808	0.196799
$872$	28.4029	0.961843
$873$	13.7369	0.464923
$874$	−14.2683	−0.482632
$875$	−1.33786	−0.0452279
$876$	−10.6321	−0.359226
$877$	−19.9036	−0.672096	−0.336048	−	0.941845i	$-0.609090\pi$
$877$	−19.9036	−0.672096	−0.336048	+	0.941845i	$0.609090\pi$
$878$	−4.98311	−0.168172
$879$	−30.1076	−1.01550
$880$	44.3809	1.49608
$881$	8.57293	0.288829	0.144415	−	0.989517i	$-0.453870\pi$
$881$	8.57293	0.288829	0.144415	+	0.989517i	$0.453870\pi$
$882$	4.21124	0.141800
$883$	−3.95554	−0.133114	−0.0665572	−	0.997783i	$-0.521201\pi$
$883$	−3.95554	−0.133114	−0.0665572	+	0.997783i	$0.521201\pi$
$884$	6.49046	0.218298
$885$	46.5843	1.56591
$886$	15.2688	0.512964
$887$	2.11568	0.0710376	0.0355188	−	0.999369i	$-0.488692\pi$
$887$	2.11568	0.0710376	0.0355188	+	0.999369i	$0.488692\pi$
$888$	−3.80800	−0.127788
$889$	−8.42777	−0.282658
$890$	−7.16586	−0.240200
$891$	24.5174	0.821364
$892$	7.25228	0.242824
$893$	25.6081	0.856942
$894$	1.56959	0.0524948
$895$	−26.7313	−0.893528
$896$	−6.22400	−0.207929
$897$	−4.60305	−0.153691
$898$	7.64310	0.255053
$899$	−5.32949	−0.177748
$900$	11.3404	0.378013
$901$	−5.11587	−0.170434
$902$	−36.8832	−1.22807
$903$	−6.60579	−0.219827
$904$	−30.0119	−0.998182
$905$	16.6465	0.553347
$906$	−12.1897	−0.404975
$907$	21.3279	0.708182	0.354091	−	0.935211i	$-0.384790\pi$
$907$	21.3279	0.708182	0.354091	+	0.935211i	$0.384790\pi$
$908$	34.7557	1.15341
$909$	0.587267	0.0194784
$910$	−0.962059	−0.0318919
$911$	−2.48215	−0.0822372	−0.0411186	−	0.999154i	$-0.513092\pi$
$911$	−2.48215	−0.0822372	−0.0411186	+	0.999154i	$0.513092\pi$
$912$	24.3828	0.807395
$913$	−23.7054	−0.784535
$914$	13.4178	0.443820
$915$	11.0307	0.364664
$916$	19.7953	0.654055
$917$	11.2074	0.370100
$918$	11.6701	0.385171
$919$	−10.9587	−0.361493	−0.180747	−	0.983530i	$-0.557851\pi$
$919$	−10.9587	−0.361493	−0.180747	+	0.983530i	$0.557851\pi$
$920$	−22.3705	−0.737533
$921$	17.5124	0.577052
$922$	3.25451	0.107182
$923$	−6.48589	−0.213486
$924$	7.30420	0.240290
$925$	8.01901	0.263664
$926$	13.4235	0.441125
$927$	−9.49522	−0.311864
$928$	−28.1578	−0.924324
$929$	−43.9786	−1.44289	−0.721445	−	0.692471i	$-0.756522\pi$
$929$	−43.9786	−1.44289	−0.721445	+	0.692471i	$0.756522\pi$
$930$	−2.41924	−0.0793301
$931$	−51.9884	−1.70385
$932$	1.76917	0.0579511
$933$	−6.59951	−0.216058
$934$	20.1626	0.659741
$935$	−72.9264	−2.38495
$936$	2.32582	0.0760219
$937$	22.9720	0.750461	0.375230	−	0.926932i	$-0.377564\pi$
$937$	22.9720	0.750461	0.375230	+	0.926932i	$0.377564\pi$
$938$	1.70381	0.0556314
$939$	−18.3600	−0.599157
$940$	18.4761	0.602623
$941$	−32.9704	−1.07481	−0.537403	−	0.843326i	$-0.680595\pi$
$941$	−32.9704	−1.07481	−0.537403	+	0.843326i	$0.680595\pi$
$942$	−4.34237	−0.141482
$943$	−39.4003	−1.28305
$944$	−24.2318	−0.788679
$945$	9.99565	0.325158
$946$	28.5778	0.929144
$947$	−16.1115	−0.523554	−0.261777	−	0.965128i	$-0.584308\pi$
$947$	−16.1115	−0.523554	−0.261777	+	0.965128i	$0.584308\pi$
$948$	31.9853	1.03883
$949$	4.59191	0.149060
$950$	24.2279	0.786056
$951$	−8.11597	−0.263178
$952$	4.13748	0.134096
$953$	−53.6708	−1.73857	−0.869283	−	0.494314i	$-0.835419\pi$
$953$	−53.6708	−1.73857	−0.869283	+	0.494314i	$0.835419\pi$
$954$	−0.843626	−0.0273134
$955$	5.79901	0.187652
$956$	−22.9873	−0.743462
$957$	42.2775	1.36664
$958$	5.33459	0.172353
$959$	−6.78006	−0.218940
$960$	7.85462	0.253507
$961$	1.00000	0.0322581
$962$	0.756830	0.0244012
$963$	−0.111065	−0.00357903
$964$	−6.50535	−0.209523
$965$	−88.9119	−2.86218
$966$	−1.35031	−0.0434457
$967$	−29.2341	−0.940106	−0.470053	−	0.882638i	$-0.655765\pi$
$967$	−29.2341	−0.940106	−0.470053	+	0.882638i	$0.655765\pi$
$968$	−46.5296	−1.49552
$969$	−40.0656	−1.28709
$970$	21.1742	0.679863
$971$	−53.5625	−1.71890	−0.859452	−	0.511217i	$-0.829195\pi$
$971$	−53.5625	−1.71890	−0.859452	+	0.511217i	$0.829195\pi$
$972$	−19.1480	−0.614171
$973$	−7.04945	−0.225995
$974$	−9.24747	−0.296308
$975$	7.81609	0.250315
$976$	−5.73787	−0.183665
$977$	39.1870	1.25370	0.626852	−	0.779139i	$-0.284343\pi$
$977$	39.1870	1.25370	0.626852	+	0.779139i	$0.284343\pi$
$978$	−1.04931	−0.0335532
$979$	−23.4970	−0.750967
$980$	−37.5092	−1.19819
$981$	16.3107	0.520762
$982$	−4.95253	−0.158042
$983$	29.0728	0.927277	0.463638	−	0.886024i	$-0.346544\pi$
$983$	29.0728	0.927277	0.463638	+	0.886024i	$0.346544\pi$
$984$	−31.7701	−1.01279
$985$	89.1033	2.83907
$986$	11.0205	0.350964
$987$	2.42349	0.0771404
$988$	−13.2130	−0.420362
$989$	30.5281	0.970738
$990$	−12.0258	−0.382206
$991$	−35.8936	−1.14020	−0.570099	−	0.821576i	$-0.693095\pi$
$991$	−35.8936	−1.14020	−0.570099	+	0.821576i	$0.693095\pi$
$992$	5.28339	0.167748
$993$	38.0238	1.20665
$994$	−1.90265	−0.0603484
$995$	17.6977	0.561056
$996$	−9.39652	−0.297740
$997$	−32.0404	−1.01473	−0.507366	−	0.861731i	$-0.669381\pi$
$997$	−32.0404	−1.01473	−0.507366	+	0.861731i	$0.669381\pi$
$998$	−1.96343	−0.0621513
$999$	−7.86335	−0.248785

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.a.b.1.5	✓	6
3.2	odd	2		3627.2.a.m.1.2		6
4.3	odd	2		6448.2.a.y.1.1		6
13.12	even	2		5239.2.a.g.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.a.b.1.5	✓	6	1.1	even	1	trivial
3627.2.a.m.1.2		6	3.2	odd	2
5239.2.a.g.1.2		6	13.12	even	2
6448.2.a.y.1.1		6	4.3	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:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	403.a (trivial)

\(f(q)\)	\(=\)	\(q+0.543189 q^{2} +1.35805 q^{3} -1.70495 q^{4} -3.27954 q^{5} +0.737678 q^{6} +0.540055 q^{7} -2.01249 q^{8} -1.15570 q^{9} +O(q^{10})\)	\(q+0.543189 q^{2} +1.35805 q^{3} -1.70495 q^{4} -3.27954 q^{5} +0.737678 q^{6} +0.540055 q^{7} -2.01249 q^{8} -1.15570 q^{9} -1.78141 q^{10} -5.84127 q^{11} -2.31540 q^{12} +1.00000 q^{13} +0.293352 q^{14} -4.45378 q^{15} +2.31673 q^{16} -3.80684 q^{17} -0.627762 q^{18} +7.74981 q^{19} +5.59144 q^{20} +0.733423 q^{21} -3.17291 q^{22} -3.38945 q^{23} -2.73306 q^{24} +5.75537 q^{25} +0.543189 q^{26} -5.64365 q^{27} -0.920765 q^{28} -5.32949 q^{29} -2.41924 q^{30} +1.00000 q^{31} +5.28339 q^{32} -7.93275 q^{33} -2.06783 q^{34} -1.77113 q^{35} +1.97040 q^{36} +1.39331 q^{37} +4.20961 q^{38} +1.35805 q^{39} +6.60002 q^{40} +11.6244 q^{41} +0.398387 q^{42} -9.00680 q^{43} +9.95905 q^{44} +3.79015 q^{45} -1.84111 q^{46} +3.30435 q^{47} +3.14624 q^{48} -6.70834 q^{49} +3.12625 q^{50} -5.16988 q^{51} -1.70495 q^{52} +1.34386 q^{53} -3.06557 q^{54} +19.1567 q^{55} -1.08685 q^{56} +10.5246 q^{57} -2.89492 q^{58} -10.4595 q^{59} +7.59345 q^{60} -2.47671 q^{61} +0.543189 q^{62} -0.624140 q^{63} -1.76359 q^{64} -3.27954 q^{65} -4.30898 q^{66} +5.80808 q^{67} +6.49046 q^{68} -4.60305 q^{69} -0.962059 q^{70} -6.48589 q^{71} +2.32582 q^{72} +4.59191 q^{73} +0.756830 q^{74} +7.81609 q^{75} -13.2130 q^{76} -3.15461 q^{77} +0.737678 q^{78} -13.8141 q^{79} -7.59781 q^{80} -4.19727 q^{81} +6.31423 q^{82} +4.05827 q^{83} -1.25045 q^{84} +12.4847 q^{85} -4.89239 q^{86} -7.23772 q^{87} +11.7555 q^{88} +4.02258 q^{89} +2.05877 q^{90} +0.540055 q^{91} +5.77884 q^{92} +1.35805 q^{93} +1.79489 q^{94} -25.4158 q^{95} +7.17512 q^{96} -11.8862 q^{97} -3.64390 q^{98} +6.75074 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{2} - 5 q^{3} + 6 q^{4} - 9 q^{5} - 3 q^{8} - q^{9}+O(q^{10}) \) 6 * q - 2 * q^2 - 5 * q^3 + 6 * q^4 - 9 * q^5 - 3 * q^8 - q^9	\( 6 q - 2 q^{2} - 5 q^{3} + 6 q^{4} - 9 q^{5} - 3 q^{8} - q^{9} - 8 q^{10} - 5 q^{11} - 13 q^{12} + 6 q^{13} - 17 q^{14} + 4 q^{15} + 14 q^{16} - 23 q^{17} + 9 q^{18} + 7 q^{19} - 10 q^{20} + 2 q^{21} + 2 q^{22} - 18 q^{23} - 13 q^{24} + 11 q^{25} - 2 q^{26} - 5 q^{27} - 25 q^{28} - 18 q^{29} + 25 q^{30} + 6 q^{31} + 2 q^{32} - 2 q^{33} - 16 q^{34} - q^{35} - 2 q^{36} - 13 q^{37} - 8 q^{38} - 5 q^{39} - 29 q^{40} - 5 q^{41} + 31 q^{42} - 7 q^{43} + 30 q^{44} - 5 q^{45} + 19 q^{46} - 9 q^{47} - 19 q^{48} + 16 q^{49} + 29 q^{50} + 26 q^{51} + 6 q^{52} - 31 q^{53} - 4 q^{54} + 7 q^{55} + 8 q^{56} - 5 q^{57} + 35 q^{58} - q^{59} + 33 q^{60} - 15 q^{61} - 2 q^{62} + 11 q^{63} - 5 q^{64} - 9 q^{65} - 29 q^{66} - 28 q^{67} - 12 q^{68} + 5 q^{69} + 73 q^{70} + q^{71} + 45 q^{72} - 20 q^{73} + 4 q^{74} + q^{75} + 38 q^{76} - 29 q^{77} - 15 q^{79} + 7 q^{80} + 2 q^{81} + 36 q^{82} + q^{83} + 68 q^{84} + 29 q^{85} + 3 q^{86} + 10 q^{87} + 9 q^{88} - q^{89} - 32 q^{90} - 60 q^{92} - 5 q^{93} + 54 q^{94} - 13 q^{95} + 36 q^{96} - 5 q^{97} + 20 q^{98} - 15 q^{99}+O(q^{100}) \) 6 * q - 2 * q^2 - 5 * q^3 + 6 * q^4 - 9 * q^5 - 3 * q^8 - q^9 - 8 * q^10 - 5 * q^11 - 13 * q^12 + 6 * q^13 - 17 * q^14 + 4 * q^15 + 14 * q^16 - 23 * q^17 + 9 * q^18 + 7 * q^19 - 10 * q^20 + 2 * q^21 + 2 * q^22 - 18 * q^23 - 13 * q^24 + 11 * q^25 - 2 * q^26 - 5 * q^27 - 25 * q^28 - 18 * q^29 + 25 * q^30 + 6 * q^31 + 2 * q^32 - 2 * q^33 - 16 * q^34 - q^35 - 2 * q^36 - 13 * q^37 - 8 * q^38 - 5 * q^39 - 29 * q^40 - 5 * q^41 + 31 * q^42 - 7 * q^43 + 30 * q^44 - 5 * q^45 + 19 * q^46 - 9 * q^47 - 19 * q^48 + 16 * q^49 + 29 * q^50 + 26 * q^51 + 6 * q^52 - 31 * q^53 - 4 * q^54 + 7 * q^55 + 8 * q^56 - 5 * q^57 + 35 * q^58 - q^59 + 33 * q^60 - 15 * q^61 - 2 * q^62 + 11 * q^63 - 5 * q^64 - 9 * q^65 - 29 * q^66 - 28 * q^67 - 12 * q^68 + 5 * q^69 + 73 * q^70 + q^71 + 45 * q^72 - 20 * q^73 + 4 * q^74 + q^75 + 38 * q^76 - 29 * q^77 - 15 * q^79 + 7 * q^80 + 2 * q^81 + 36 * q^82 + q^83 + 68 * q^84 + 29 * q^85 + 3 * q^86 + 10 * q^87 + 9 * q^88 - q^89 - 32 * q^90 - 60 * q^92 - 5 * q^93 + 54 * q^94 - 13 * q^95 + 36 * q^96 - 5 * q^97 + 20 * q^98 - 15 * q^99

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