LMFDB - Embedded newform 401.2.a.a.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [401,2,Mod(1,401)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(401, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("401.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 401 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	401.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:	$3.20200112105$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} - 10 x^{10} + 34 x^{9} + 29 x^{8} - 129 x^{7} - 24 x^{6} + 203 x^{5} + x^{4} + \cdots + 4 $ x^12 - 3x^11 - 10x^10 + 34x^9 + 29x^8 - 129x^7 - 24x^6 + 203x^5 + x^4 - 130x^3 - 5x^2 + 22x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.44790$ of defining polynomial
Character	$\chi$	$=$	401.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.44790 q^{2} -2.35797 q^{3} +0.0964152 q^{4} -1.54450 q^{5} +3.41411 q^{6} +2.00193 q^{7} +2.75620 q^{8} +2.56004 q^{9} +O(q^{10})$	\(q-1.44790 q^{2} -2.35797 q^{3} +0.0964152 q^{4} -1.54450 q^{5} +3.41411 q^{6} +2.00193 q^{7} +2.75620 q^{8} +2.56004 q^{9} +2.23628 q^{10} +4.24832 q^{11} -0.227344 q^{12} +0.309927 q^{13} -2.89860 q^{14} +3.64189 q^{15} -4.18353 q^{16} -2.86865 q^{17} -3.70669 q^{18} -7.71219 q^{19} -0.148913 q^{20} -4.72050 q^{21} -6.15114 q^{22} +1.87651 q^{23} -6.49905 q^{24} -2.61452 q^{25} -0.448744 q^{26} +1.03741 q^{27} +0.193017 q^{28} +8.85455 q^{29} -5.27310 q^{30} -8.84171 q^{31} +0.544939 q^{32} -10.0174 q^{33} +4.15351 q^{34} -3.09199 q^{35} +0.246827 q^{36} +10.5255 q^{37} +11.1665 q^{38} -0.730800 q^{39} -4.25695 q^{40} -12.7017 q^{41} +6.83482 q^{42} -9.08404 q^{43} +0.409602 q^{44} -3.95399 q^{45} -2.71699 q^{46} +5.36479 q^{47} +9.86467 q^{48} -2.99227 q^{49} +3.78556 q^{50} +6.76420 q^{51} +0.0298817 q^{52} -2.25895 q^{53} -1.50206 q^{54} -6.56153 q^{55} +5.51773 q^{56} +18.1852 q^{57} -12.8205 q^{58} -5.34882 q^{59} +0.351134 q^{60} -1.34687 q^{61} +12.8019 q^{62} +5.12503 q^{63} +7.57805 q^{64} -0.478683 q^{65} +14.5042 q^{66} -6.27088 q^{67} -0.276581 q^{68} -4.42475 q^{69} +4.47689 q^{70} +5.10513 q^{71} +7.05599 q^{72} -3.61987 q^{73} -15.2399 q^{74} +6.16497 q^{75} -0.743572 q^{76} +8.50485 q^{77} +1.05813 q^{78} -9.16870 q^{79} +6.46147 q^{80} -10.1263 q^{81} +18.3907 q^{82} -10.9517 q^{83} -0.455128 q^{84} +4.43063 q^{85} +13.1528 q^{86} -20.8788 q^{87} +11.7092 q^{88} -11.9514 q^{89} +5.72498 q^{90} +0.620453 q^{91} +0.180924 q^{92} +20.8485 q^{93} -7.76767 q^{94} +11.9115 q^{95} -1.28495 q^{96} +17.5324 q^{97} +4.33250 q^{98} +10.8759 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 3 q^{2} - 5 q^{3} + 5 q^{4} - 7 q^{5} - 9 q^{6} - 20 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) $ 12 * q - 3 * q^2 - 5 * q^3 + 5 * q^4 - 7 * q^5 - 9 * q^6 - 20 * q^7 - 3 * q^8 + 3 * q^9	\( 12 q - 3 q^{2} - 5 q^{3} + 5 q^{4} - 7 q^{5} - 9 q^{6} - 20 q^{7} - 3 q^{8} + 3 q^{9} - 11 q^{10} - 11 q^{11} - 9 q^{12} - 11 q^{13} - 3 q^{14} - 11 q^{15} - 9 q^{16} + q^{17} - q^{18} - 34 q^{19} - 5 q^{20} - 3 q^{21} + 3 q^{22} - 7 q^{23} - 9 q^{24} + 7 q^{25} + 6 q^{26} - 2 q^{27} - 23 q^{28} - 6 q^{29} + 23 q^{30} - 52 q^{31} + 11 q^{32} + 4 q^{33} - 5 q^{34} + 12 q^{35} + 16 q^{36} + 3 q^{37} + 25 q^{38} - 24 q^{39} - 25 q^{40} - 16 q^{41} + 47 q^{42} - 2 q^{43} - 2 q^{44} - 23 q^{45} - 16 q^{46} - 3 q^{47} + 24 q^{48} + 6 q^{49} + 27 q^{50} - 16 q^{51} - 5 q^{52} + 19 q^{53} + 5 q^{54} - 43 q^{55} + 7 q^{56} + 11 q^{57} + 11 q^{58} - q^{59} + 30 q^{60} - 24 q^{61} + 39 q^{62} - 11 q^{63} - q^{64} + 13 q^{65} + 14 q^{66} + 6 q^{67} + 32 q^{68} + 29 q^{69} + 47 q^{70} - 15 q^{71} + 32 q^{72} - 20 q^{73} + 25 q^{74} + 31 q^{75} - 42 q^{76} + 38 q^{77} + 52 q^{78} - 53 q^{79} + 23 q^{80} - 8 q^{81} + 4 q^{82} + 17 q^{83} + 35 q^{84} + 7 q^{85} + 28 q^{86} - 5 q^{87} + 38 q^{88} - q^{89} + 58 q^{90} - 6 q^{91} + 46 q^{92} + 44 q^{93} - 4 q^{94} + 34 q^{95} + 28 q^{96} + 12 q^{97} + 27 q^{98}+O(q^{100}) \) 12 * q - 3 * q^2 - 5 * q^3 + 5 * q^4 - 7 * q^5 - 9 * q^6 - 20 * q^7 - 3 * q^8 + 3 * q^9 - 11 * q^10 - 11 * q^11 - 9 * q^12 - 11 * q^13 - 3 * q^14 - 11 * q^15 - 9 * q^16 + q^17 - q^18 - 34 * q^19 - 5 * q^20 - 3 * q^21 + 3 * q^22 - 7 * q^23 - 9 * q^24 + 7 * q^25 + 6 * q^26 - 2 * q^27 - 23 * q^28 - 6 * q^29 + 23 * q^30 - 52 * q^31 + 11 * q^32 + 4 * q^33 - 5 * q^34 + 12 * q^35 + 16 * q^36 + 3 * q^37 + 25 * q^38 - 24 * q^39 - 25 * q^40 - 16 * q^41 + 47 * q^42 - 2 * q^43 - 2 * q^44 - 23 * q^45 - 16 * q^46 - 3 * q^47 + 24 * q^48 + 6 * q^49 + 27 * q^50 - 16 * q^51 - 5 * q^52 + 19 * q^53 + 5 * q^54 - 43 * q^55 + 7 * q^56 + 11 * q^57 + 11 * q^58 - q^59 + 30 * q^60 - 24 * q^61 + 39 * q^62 - 11 * q^63 - q^64 + 13 * q^65 + 14 * q^66 + 6 * q^67 + 32 * q^68 + 29 * q^69 + 47 * q^70 - 15 * q^71 + 32 * q^72 - 20 * q^73 + 25 * q^74 + 31 * q^75 - 42 * q^76 + 38 * q^77 + 52 * q^78 - 53 * q^79 + 23 * q^80 - 8 * q^81 + 4 * q^82 + 17 * q^83 + 35 * q^84 + 7 * q^85 + 28 * q^86 - 5 * q^87 + 38 * q^88 - q^89 + 58 * q^90 - 6 * q^91 + 46 * q^92 + 44 * q^93 - 4 * q^94 + 34 * q^95 + 28 * q^96 + 12 * q^97 + 27 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.44790	−1.02382	−0.511910	−	0.859039i	$-0.671062\pi$
$2$	−1.44790	−1.02382	−0.511910	+	0.859039i	$0.671062\pi$
$3$	−2.35797	−1.36138	−0.680689	−	0.732573i	$-0.738319\pi$
$3$	−2.35797	−1.36138	−0.680689	+	0.732573i	$0.738319\pi$
$4$	0.0964152	0.0482076
$5$	−1.54450	−0.690722	−0.345361	−	0.938470i	$-0.612243\pi$
$5$	−1.54450	−0.690722	−0.345361	+	0.938470i	$0.612243\pi$
$6$	3.41411	1.39381
$7$	2.00193	0.756659	0.378330	−	0.925671i	$-0.376499\pi$
$7$	2.00193	0.756659	0.378330	+	0.925671i	$0.376499\pi$
$8$	2.75620	0.974464
$9$	2.56004	0.853348
$10$	2.23628	0.707175
$11$	4.24832	1.28092	0.640458	−	0.767993i	$-0.278744\pi$
$11$	4.24832	1.28092	0.640458	+	0.767993i	$0.278744\pi$
$12$	−0.227344	−0.0656287
$13$	0.309927	0.0859583	0.0429792	−	0.999076i	$-0.486315\pi$
$13$	0.309927	0.0859583	0.0429792	+	0.999076i	$0.486315\pi$
$14$	−2.89860	−0.774683
$15$	3.64189	0.940333
$16$	−4.18353	−1.04588
$17$	−2.86865	−0.695749	−0.347875	−	0.937541i	$-0.613096\pi$
$17$	−2.86865	−0.695749	−0.347875	+	0.937541i	$0.613096\pi$
$18$	−3.70669	−0.873674
$19$	−7.71219	−1.76930	−0.884649	−	0.466257i	$-0.845602\pi$
$19$	−7.71219	−1.76930	−0.884649	+	0.466257i	$0.845602\pi$
$20$	−0.148913	−0.0332980
$21$	−4.72050	−1.03010
$22$	−6.15114	−1.31143
$23$	1.87651	0.391279	0.195639	−	0.980676i	$-0.437322\pi$
$23$	1.87651	0.391279	0.195639	+	0.980676i	$0.437322\pi$
$24$	−6.49905	−1.32661
$25$	−2.61452	−0.522904
$26$	−0.448744	−0.0880059
$27$	1.03741	0.199649
$28$	0.193017	0.0364767
$29$	8.85455	1.64425	0.822124	−	0.569308i	$-0.192789\pi$
$29$	8.85455	1.64425	0.822124	+	0.569308i	$0.192789\pi$
$30$	−5.27310	−0.962731
$31$	−8.84171	−1.58802	−0.794009	−	0.607906i	$-0.792010\pi$
$31$	−8.84171	−1.58802	−0.794009	+	0.607906i	$0.792010\pi$
$32$	0.544939	0.0963325
$33$	−10.0174	−1.74381
$34$	4.15351	0.712322
$35$	−3.09199	−0.522641
$36$	0.246827	0.0411378
$37$	10.5255	1.73038	0.865190	−	0.501445i	$-0.167198\pi$
$37$	10.5255	1.73038	0.865190	+	0.501445i	$0.167198\pi$
$38$	11.1665	1.81144
$39$	−0.730800	−0.117022
$40$	−4.25695	−0.673084
$41$	−12.7017	−1.98367	−0.991833	−	0.127541i	$-0.959292\pi$
$41$	−12.7017	−1.98367	−0.991833	+	0.127541i	$0.959292\pi$
$42$	6.83482	1.05464
$43$	−9.08404	−1.38530	−0.692652	−	0.721272i	$-0.743558\pi$
$43$	−9.08404	−1.38530	−0.692652	+	0.721272i	$0.743558\pi$
$44$	0.409602	0.0617499
$45$	−3.95399	−0.589426
$46$	−2.71699	−0.400599
$47$	5.36479	0.782534	0.391267	−	0.920277i	$-0.372037\pi$
$47$	5.36479	0.782534	0.391267	+	0.920277i	$0.372037\pi$
$48$	9.86467	1.42384
$49$	−2.99227	−0.427467
$50$	3.78556	0.535359
$51$	6.76420	0.947177
$52$	0.0298817	0.00414384
$53$	−2.25895	−0.310291	−0.155146	−	0.987892i	$-0.549585\pi$
$53$	−2.25895	−0.310291	−0.155146	+	0.987892i	$0.549585\pi$
$54$	−1.50206	−0.204405
$55$	−6.56153	−0.884757
$56$	5.51773	0.737337
$57$	18.1852	2.40868
$58$	−12.8205	−1.68341
$59$	−5.34882	−0.696357	−0.348178	−	0.937428i	$-0.613200\pi$
$59$	−5.34882	−0.696357	−0.348178	+	0.937428i	$0.613200\pi$
$60$	0.351134	0.0453312
$61$	−1.34687	−0.172449	−0.0862247	−	0.996276i	$-0.527480\pi$
$61$	−1.34687	−0.172449	−0.0862247	+	0.996276i	$0.527480\pi$
$62$	12.8019	1.62585
$63$	5.12503	0.645693
$64$	7.57805	0.947257
$65$	−0.478683	−0.0593733
$66$	14.5042	1.78535
$67$	−6.27088	−0.766110	−0.383055	−	0.923726i	$-0.625128\pi$
$67$	−6.27088	−0.766110	−0.383055	+	0.923726i	$0.625128\pi$
$68$	−0.276581	−0.0335404
$69$	−4.42475	−0.532678
$70$	4.47689	0.535090
$71$	5.10513	0.605867	0.302933	−	0.953012i	$-0.402034\pi$
$71$	5.10513	0.605867	0.302933	+	0.953012i	$0.402034\pi$
$72$	7.05599	0.831557
$73$	−3.61987	−0.423674	−0.211837	−	0.977305i	$-0.567945\pi$
$73$	−3.61987	−0.423674	−0.211837	+	0.977305i	$0.567945\pi$
$74$	−15.2399	−1.77160
$75$	6.16497	0.711869
$76$	−0.743572	−0.0852936
$77$	8.50485	0.969217
$78$	1.05813	0.119809
$79$	−9.16870	−1.03156	−0.515780	−	0.856721i	$-0.672498\pi$
$79$	−9.16870	−1.03156	−0.515780	+	0.856721i	$0.672498\pi$
$80$	6.46147	0.722414
$81$	−10.1263	−1.12515
$82$	18.3907	2.03092
$83$	−10.9517	−1.20211	−0.601053	−	0.799209i	$-0.705252\pi$
$83$	−10.9517	−1.20211	−0.601053	+	0.799209i	$0.705252\pi$
$84$	−0.455128	−0.0496586
$85$	4.43063	0.480569
$86$	13.1528	1.41830
$87$	−20.8788	−2.23844
$88$	11.7092	1.24821
$89$	−11.9514	−1.26684	−0.633422	−	0.773807i	$-0.718350\pi$
$89$	−11.9514	−1.26684	−0.633422	+	0.773807i	$0.718350\pi$
$90$	5.72498	0.603466
$91$	0.620453	0.0650412
$92$	0.180924	0.0188626
$93$	20.8485	2.16189
$94$	−7.76767	−0.801174
$95$	11.9115	1.22209
$96$	−1.28495	−0.131145
$97$	17.5324	1.78015	0.890074	−	0.455815i	$-0.150652\pi$
$97$	17.5324	1.78015	0.890074	+	0.455815i	$0.150652\pi$
$98$	4.33250	0.437649
$99$	10.8759	1.09307
$100$	−0.252079	−0.0252079
$101$	4.51166	0.448927	0.224463	−	0.974483i	$-0.427937\pi$
$101$	4.51166	0.448927	0.224463	+	0.974483i	$0.427937\pi$
$102$	−9.79388	−0.969739
$103$	−11.8260	−1.16525	−0.582624	−	0.812742i	$-0.697974\pi$
$103$	−11.8260	−1.16525	−0.582624	+	0.812742i	$0.697974\pi$
$104$	0.854222	0.0837633
$105$	7.29082	0.711511
$106$	3.27074	0.317682
$107$	−7.80214	−0.754262	−0.377131	−	0.926160i	$-0.623089\pi$
$107$	−7.80214	−0.754262	−0.377131	+	0.926160i	$0.623089\pi$
$108$	0.100022	0.00962461
$109$	−3.84779	−0.368552	−0.184276	−	0.982875i	$-0.558994\pi$
$109$	−3.84779	−0.368552	−0.184276	+	0.982875i	$0.558994\pi$
$110$	9.50044	0.905832
$111$	−24.8188	−2.35570
$112$	−8.37515	−0.791378
$113$	6.69321	0.629644	0.314822	−	0.949151i	$-0.398055\pi$
$113$	6.69321	0.629644	0.314822	+	0.949151i	$0.398055\pi$
$114$	−26.3303	−2.46606
$115$	−2.89827	−0.270265
$116$	0.853713	0.0792652
$117$	0.793427	0.0733523
$118$	7.74455	0.712944
$119$	−5.74284	−0.526445
$120$	10.0378	0.916320
$121$	7.04821	0.640747
$122$	1.95014	0.176557
$123$	29.9502	2.70052
$124$	−0.852475	−0.0765545
$125$	11.7606	1.05190
$126$	−7.42053	−0.661074
$127$	−7.62671	−0.676761	−0.338381	−	0.941009i	$-0.609879\pi$
$127$	−7.62671	−0.676761	−0.338381	+	0.941009i	$0.609879\pi$
$128$	−12.0621	−1.06615
$129$	21.4199	1.88592
$130$	0.693085	0.0607876
$131$	5.84345	0.510545	0.255272	−	0.966869i	$-0.417835\pi$
$131$	5.84345	0.510545	0.255272	+	0.966869i	$0.417835\pi$
$132$	−0.965832	−0.0840649
$133$	−15.4393	−1.33876
$134$	9.07960	0.784358
$135$	−1.60228	−0.137902
$136$	−7.90657	−0.677983
$137$	6.21454	0.530944	0.265472	−	0.964119i	$-0.414472\pi$
$137$	6.21454	0.530944	0.265472	+	0.964119i	$0.414472\pi$
$138$	6.40660	0.545366
$139$	5.51223	0.467541	0.233771	−	0.972292i	$-0.424894\pi$
$139$	5.51223	0.467541	0.233771	+	0.972292i	$0.424894\pi$
$140$	−0.298114	−0.0251953
$141$	−12.6500	−1.06532
$142$	−7.39171	−0.620299
$143$	1.31667	0.110105
$144$	−10.7100	−0.892502
$145$	−13.6759	−1.13572
$146$	5.24121	0.433766
$147$	7.05569	0.581943
$148$	1.01482	0.0834174
$149$	−6.97164	−0.571139	−0.285570	−	0.958358i	$-0.592183\pi$
$149$	−6.97164	−0.571139	−0.285570	+	0.958358i	$0.592183\pi$
$150$	−8.92626	−0.728826
$151$	−18.1730	−1.47890	−0.739450	−	0.673211i	$-0.764915\pi$
$151$	−18.1730	−1.47890	−0.739450	+	0.673211i	$0.764915\pi$
$152$	−21.2564	−1.72412
$153$	−7.34386	−0.593716
$154$	−12.3142	−0.992304
$155$	13.6560	1.09688
$156$	−0.0704602	−0.00564133
$157$	15.2157	1.21435	0.607174	−	0.794569i	$-0.292303\pi$
$157$	15.2157	1.21435	0.607174	+	0.794569i	$0.292303\pi$
$158$	13.2754	1.05613
$159$	5.32656	0.422423
$160$	−0.841658	−0.0665389
$161$	3.75664	0.296065
$162$	14.6619	1.15195
$163$	−14.9873	−1.17390	−0.586950	−	0.809623i	$-0.699671\pi$
$163$	−14.9873	−1.17390	−0.586950	+	0.809623i	$0.699671\pi$
$164$	−1.22463	−0.0956278
$165$	15.4719	1.20449
$166$	15.8570	1.23074
$167$	−11.6367	−0.900476	−0.450238	−	0.892909i	$-0.648661\pi$
$167$	−11.6367	−0.900476	−0.450238	+	0.892909i	$0.648661\pi$
$168$	−13.0107	−1.00379
$169$	−12.9039	−0.992611
$170$	−6.41511	−0.492016
$171$	−19.7435	−1.50983
$172$	−0.875839	−0.0667821
$173$	7.26706	0.552504	0.276252	−	0.961085i	$-0.410908\pi$
$173$	7.26706	0.552504	0.276252	+	0.961085i	$0.410908\pi$
$174$	30.2304	2.29176
$175$	−5.23409	−0.395660
$176$	−17.7730	−1.33969
$177$	12.6124	0.948004
$178$	17.3044	1.29702
$179$	−13.5764	−1.01475	−0.507374	−	0.861726i	$-0.669384\pi$
$179$	−13.5764	−1.01475	−0.507374	+	0.861726i	$0.669384\pi$
$180$	−0.381224	−0.0284148
$181$	8.16356	0.606792	0.303396	−	0.952864i	$-0.401879\pi$
$181$	8.16356	0.606792	0.303396	+	0.952864i	$0.401879\pi$
$182$	−0.898354	−0.0665905
$183$	3.17589	0.234769
$184$	5.17203	0.381287
$185$	−16.2566	−1.19521
$186$	−30.1866	−2.21339
$187$	−12.1869	−0.891196
$188$	0.517247	0.0377241
$189$	2.07682	0.151067
$190$	−17.2466	−1.25120
$191$	−5.44986	−0.394338	−0.197169	−	0.980370i	$-0.563175\pi$
$191$	−5.44986	−0.394338	−0.197169	+	0.980370i	$0.563175\pi$
$192$	−17.8689	−1.28957
$193$	−3.23770	−0.233055	−0.116527	−	0.993187i	$-0.537176\pi$
$193$	−3.23770	−0.233055	−0.116527	+	0.993187i	$0.537176\pi$
$194$	−25.3852	−1.82255
$195$	1.12872	0.0808294
$196$	−0.288500	−0.0206071
$197$	13.9776	0.995861	0.497930	−	0.867217i	$-0.334094\pi$
$197$	13.9776	0.995861	0.497930	+	0.867217i	$0.334094\pi$
$198$	−15.7472	−1.11910
$199$	−21.6026	−1.53137	−0.765683	−	0.643218i	$-0.777599\pi$
$199$	−21.6026	−1.53137	−0.765683	+	0.643218i	$0.777599\pi$
$200$	−7.20614	−0.509551
$201$	14.7866	1.04296
$202$	−6.53243	−0.459620
$203$	17.7262	1.24414
$204$	0.652171	0.0456611
$205$	19.6177	1.37016
$206$	17.1228	1.19300
$207$	4.80394	0.333897
$208$	−1.29659	−0.0899024
$209$	−32.7639	−2.26632
$210$	−10.5564	−0.728460
$211$	20.2906	1.39686	0.698430	−	0.715678i	$-0.253882\pi$
$211$	20.2906	1.39686	0.698430	+	0.715678i	$0.253882\pi$
$212$	−0.217797	−0.0149584
$213$	−12.0378	−0.824813
$214$	11.2967	0.772228
$215$	14.0303	0.956859
$216$	2.85931	0.194551
$217$	−17.7005	−1.20159
$218$	5.57122	0.377331
$219$	8.53556	0.576780
$220$	−0.632631	−0.0426520
$221$	−0.889072	−0.0598054
$222$	35.9352	2.41181
$223$	10.5997	0.709810	0.354905	−	0.934902i	$-0.384513\pi$
$223$	10.5997	0.709810	0.354905	+	0.934902i	$0.384513\pi$
$224$	1.09093	0.0728908
$225$	−6.69328	−0.446218
$226$	−9.69110	−0.644643
$227$	14.6249	0.970687	0.485344	−	0.874324i	$-0.338695\pi$
$227$	14.6249	0.970687	0.485344	+	0.874324i	$0.338695\pi$
$228$	1.75332	0.116117
$229$	−12.3801	−0.818102	−0.409051	−	0.912511i	$-0.634140\pi$
$229$	−12.3801	−0.818102	−0.409051	+	0.912511i	$0.634140\pi$
$230$	4.19640	0.276702
$231$	−20.0542	−1.31947
$232$	24.4049	1.60226
$233$	−12.1067	−0.793139	−0.396570	−	0.918005i	$-0.629799\pi$
$233$	−12.1067	−0.793139	−0.396570	+	0.918005i	$0.629799\pi$
$234$	−1.14880	−0.0750996
$235$	−8.28591	−0.540513
$236$	−0.515707	−0.0335697
$237$	21.6196	1.40434
$238$	8.31506	0.538985
$239$	3.39942	0.219890	0.109945	−	0.993938i	$-0.464933\pi$
$239$	3.39942	0.219890	0.109945	+	0.993938i	$0.464933\pi$
$240$	−15.2360	−0.983478
$241$	12.8618	0.828505	0.414252	−	0.910162i	$-0.364043\pi$
$241$	12.8618	0.828505	0.414252	+	0.910162i	$0.364043\pi$
$242$	−10.2051	−0.656009
$243$	20.7654	1.33210
$244$	−0.129859	−0.00831337
$245$	4.62156	0.295261
$246$	−43.3649	−2.76484
$247$	−2.39022	−0.152086
$248$	−24.3695	−1.54747
$249$	25.8238	1.63652
$250$	−17.0282	−1.07696
$251$	−9.40210	−0.593456	−0.296728	−	0.954962i	$-0.595895\pi$
$251$	−9.40210	−0.593456	−0.296728	+	0.954962i	$0.595895\pi$
$252$	0.494131	0.0311273
$253$	7.97200	0.501195
$254$	11.0427	0.692882
$255$	−10.4473	−0.654236
$256$	2.30867	0.144292
$257$	16.5625	1.03314	0.516571	−	0.856244i	$-0.327208\pi$
$257$	16.5625	1.03314	0.516571	+	0.856244i	$0.327208\pi$
$258$	−31.0139	−1.93084
$259$	21.0713	1.30931
$260$	−0.0461523	−0.00286224
$261$	22.6680	1.40312
$262$	−8.46074	−0.522706
$263$	−14.6415	−0.902834	−0.451417	−	0.892313i	$-0.649081\pi$
$263$	−14.6415	−0.902834	−0.451417	+	0.892313i	$0.649081\pi$
$264$	−27.6100	−1.69928
$265$	3.48896	0.214325
$266$	22.3545	1.37065
$267$	28.1810	1.72465
$268$	−0.604608	−0.0369323
$269$	5.62665	0.343063	0.171531	−	0.985179i	$-0.445128\pi$
$269$	5.62665	0.343063	0.171531	+	0.985179i	$0.445128\pi$
$270$	2.31994	0.141187
$271$	1.17281	0.0712430	0.0356215	−	0.999365i	$-0.488659\pi$
$271$	1.17281	0.0712430	0.0356215	+	0.999365i	$0.488659\pi$
$272$	12.0011	0.727673
$273$	−1.46301	−0.0885456
$274$	−8.99803	−0.543591
$275$	−11.1073	−0.669796
$276$	−0.426614	−0.0256791
$277$	−5.50071	−0.330505	−0.165253	−	0.986251i	$-0.552844\pi$
$277$	−5.50071	−0.330505	−0.165253	+	0.986251i	$0.552844\pi$
$278$	−7.98116	−0.478678
$279$	−22.6352	−1.35513
$280$	−8.52213	−0.509295
$281$	14.2514	0.850170	0.425085	−	0.905153i	$-0.360244\pi$
$281$	14.2514	0.850170	0.425085	+	0.905153i	$0.360244\pi$
$282$	18.3160	1.09070
$283$	20.0137	1.18969	0.594845	−	0.803840i	$-0.297213\pi$
$283$	20.0137	1.18969	0.594845	+	0.803840i	$0.297213\pi$
$284$	0.492212	0.0292074
$285$	−28.0870	−1.66373
$286$	−1.90641	−0.112728
$287$	−25.4279	−1.50096
$288$	1.39507	0.0822051
$289$	−8.77086	−0.515933
$290$	19.8013	1.16277
$291$	−41.3410	−2.42345
$292$	−0.349010	−0.0204243
$293$	11.6522	0.680726	0.340363	−	0.940294i	$-0.389450\pi$
$293$	11.6522	0.680726	0.340363	+	0.940294i	$0.389450\pi$
$294$	−10.2159	−0.595805
$295$	8.26125	0.480989
$296$	29.0104	1.68619
$297$	4.40724	0.255734
$298$	10.0942	0.584744
$299$	0.581581	0.0336337
$300$	0.594396	0.0343175
$301$	−18.1856	−1.04820
$302$	26.3127	1.51413
$303$	−10.6384	−0.611158
$304$	32.2642	1.85048
$305$	2.08025	0.119115
$306$	10.6332	0.607858
$307$	−6.45201	−0.368236	−0.184118	−	0.982904i	$-0.558943\pi$
$307$	−6.45201	−0.368236	−0.184118	+	0.982904i	$0.558943\pi$
$308$	0.819996	0.0467236
$309$	27.8854	1.58634
$310$	−19.7726	−1.12301
$311$	−17.4462	−0.989286	−0.494643	−	0.869096i	$-0.664701\pi$
$311$	−17.4462	−0.989286	−0.494643	+	0.869096i	$0.664701\pi$
$312$	−2.01423	−0.114033
$313$	−17.6694	−0.998733	−0.499366	−	0.866391i	$-0.666434\pi$
$313$	−17.6694	−0.998733	−0.499366	+	0.866391i	$0.666434\pi$
$314$	−22.0309	−1.24327
$315$	−7.91561	−0.445994
$316$	−0.884002	−0.0497290
$317$	−23.2836	−1.30774	−0.653869	−	0.756608i	$-0.726855\pi$
$317$	−23.2836	−1.30774	−0.653869	+	0.756608i	$0.726855\pi$
$318$	−7.71232	−0.432485
$319$	37.6169	2.10614
$320$	−11.7043	−0.654291
$321$	18.3973	1.02683
$322$	−5.43924	−0.303117
$323$	22.1236	1.23099
$324$	−0.976330	−0.0542406
$325$	−0.810310	−0.0449479
$326$	21.7002	1.20186
$327$	9.07300	0.501738
$328$	−35.0083	−1.93301
$329$	10.7399	0.592112
$330$	−22.4018	−1.23318
$331$	9.40261	0.516814	0.258407	−	0.966036i	$-0.416802\pi$
$331$	9.40261	0.516814	0.258407	+	0.966036i	$0.416802\pi$
$332$	−1.05591	−0.0579506
$333$	26.9457	1.47661
$334$	16.8488	0.921925
$335$	9.68537	0.529168
$336$	19.7484	1.07736
$337$	0.535293	0.0291592	0.0145796	−	0.999894i	$-0.495359\pi$
$337$	0.535293	0.0291592	0.0145796	+	0.999894i	$0.495359\pi$
$338$	18.6836	1.01626
$339$	−15.7824	−0.857184
$340$	0.427180	0.0231671
$341$	−37.5624	−2.03412
$342$	28.5867	1.54579
$343$	−20.0038	−1.08011
$344$	−25.0374	−1.34993
$345$	6.83404	0.367932
$346$	−10.5220	−0.565665
$347$	11.8396	0.635583	0.317792	−	0.948161i	$-0.397059\pi$
$347$	11.8396	0.635583	0.317792	+	0.948161i	$0.397059\pi$
$348$	−2.01303	−0.107910
$349$	25.1735	1.34751	0.673753	−	0.738957i	$-0.264681\pi$
$349$	25.1735	1.34751	0.673753	+	0.738957i	$0.264681\pi$
$350$	7.57844	0.405084
$351$	0.321521	0.0171615
$352$	2.31507	0.123394
$353$	16.7159	0.889699	0.444850	−	0.895605i	$-0.353257\pi$
$353$	16.7159	0.889699	0.444850	+	0.895605i	$0.353257\pi$
$354$	−18.2615	−0.970586
$355$	−7.88487	−0.418485
$356$	−1.15229	−0.0610715
$357$	13.5415	0.716690
$358$	19.6573	1.03892
$359$	8.47215	0.447143	0.223572	−	0.974688i	$-0.428228\pi$
$359$	8.47215	0.447143	0.223572	+	0.974688i	$0.428228\pi$
$360$	−10.8980	−0.574374
$361$	40.4779	2.13042
$362$	−11.8200	−0.621246
$363$	−16.6195	−0.872298
$364$	0.0598211	0.00313548
$365$	5.59089	0.292641
$366$	−4.59837	−0.240361
$367$	18.9557	0.989480	0.494740	−	0.869041i	$-0.335263\pi$
$367$	18.9557	0.989480	0.494740	+	0.869041i	$0.335263\pi$
$368$	−7.85043	−0.409232
$369$	−32.5168	−1.69276
$370$	23.5380	1.22368
$371$	−4.52227	−0.234785
$372$	2.01011	0.104220
$373$	−35.8813	−1.85787	−0.928933	−	0.370249i	$-0.879272\pi$
$373$	−35.8813	−1.85787	−0.928933	+	0.370249i	$0.879272\pi$
$374$	17.6455	0.912425
$375$	−27.7313	−1.43204
$376$	14.7864	0.762552
$377$	2.74427	0.141337
$378$	−3.00703	−0.154665
$379$	16.1721	0.830706	0.415353	−	0.909660i	$-0.363658\pi$
$379$	16.1721	0.830706	0.415353	+	0.909660i	$0.363658\pi$
$380$	1.14845	0.0589141
$381$	17.9836	0.921327
$382$	7.89086	0.403731
$383$	30.1053	1.53831	0.769155	−	0.639062i	$-0.220677\pi$
$383$	30.1053	1.53831	0.769155	+	0.639062i	$0.220677\pi$
$384$	28.4422	1.45144
$385$	−13.1357	−0.669459
$386$	4.68787	0.238606
$387$	−23.2555	−1.18215
$388$	1.69039	0.0858167
$389$	−6.54786	−0.331989	−0.165995	−	0.986127i	$-0.553083\pi$
$389$	−6.54786	−0.331989	−0.165995	+	0.986127i	$0.553083\pi$
$390$	−1.63428	−0.0827548
$391$	−5.38304	−0.272232
$392$	−8.24729	−0.416551
$393$	−13.7787	−0.695044
$394$	−20.2381	−1.01958
$395$	14.1611	0.712521
$396$	1.04860	0.0526941
$397$	35.7299	1.79323	0.896617	−	0.442806i	$-0.146017\pi$
$397$	35.7299	1.79323	0.896617	+	0.442806i	$0.146017\pi$
$398$	31.2784	1.56784
$399$	36.4054	1.82255
$400$	10.9379	0.546896
$401$	−1.00000	−0.0499376
$401$	−1.00000	−0.0499376
$402$	−21.4095	−1.06781
$403$	−2.74029	−0.136503
$404$	0.434992	0.0216417
$405$	15.6401	0.777162
$406$	−25.6658	−1.27377
$407$	44.7156	2.21647
$408$	18.6435	0.922990
$409$	25.3678	1.25436	0.627178	−	0.778876i	$-0.284210\pi$
$409$	25.3678	1.25436	0.627178	+	0.778876i	$0.284210\pi$
$410$	−28.4045	−1.40280
$411$	−14.6537	−0.722815
$412$	−1.14020	−0.0561738
$413$	−10.7080	−0.526905
$414$	−6.95562	−0.341850
$415$	16.9149	0.830320
$416$	0.168891	0.00828058
$417$	−12.9977	−0.636500
$418$	47.4388	2.32031
$419$	−27.1917	−1.32840	−0.664201	−	0.747554i	$-0.731228\pi$
$419$	−27.1917	−1.32840	−0.664201	+	0.747554i	$0.731228\pi$
$420$	0.702946	0.0343002
$421$	−8.01328	−0.390543	−0.195272	−	0.980749i	$-0.562559\pi$
$421$	−8.01328	−0.390543	−0.195272	+	0.980749i	$0.562559\pi$
$422$	−29.3787	−1.43013
$423$	13.7341	0.667774
$424$	−6.22613	−0.302368
$425$	7.50013	0.363810
$426$	17.4295	0.844460
$427$	−2.69635	−0.130485
$428$	−0.752245	−0.0363611
$429$	−3.10467	−0.149895
$430$	−20.3145	−0.979652
$431$	32.7063	1.57541	0.787704	−	0.616054i	$-0.211270\pi$
$431$	32.7063	1.57541	0.787704	+	0.616054i	$0.211270\pi$
$432$	−4.34003	−0.208810
$433$	−33.3264	−1.60156	−0.800782	−	0.598957i	$-0.795582\pi$
$433$	−33.3264	−1.60156	−0.800782	+	0.598957i	$0.795582\pi$
$434$	25.6286	1.23021
$435$	32.2473	1.54614
$436$	−0.370986	−0.0177670
$437$	−14.4720	−0.692289
$438$	−12.3586	−0.590519
$439$	6.80119	0.324603	0.162302	−	0.986741i	$-0.448108\pi$
$439$	6.80119	0.324603	0.162302	+	0.986741i	$0.448108\pi$
$440$	−18.0849	−0.862164
$441$	−7.66033	−0.364778
$442$	1.28729	0.0612300
$443$	5.25361	0.249606	0.124803	−	0.992182i	$-0.460170\pi$
$443$	5.25361	0.249606	0.124803	+	0.992182i	$0.460170\pi$
$444$	−2.39291	−0.113563
$445$	18.4589	0.875036
$446$	−15.3473	−0.726718
$447$	16.4389	0.777536
$448$	15.1707	0.716750
$449$	6.13520	0.289538	0.144769	−	0.989465i	$-0.453756\pi$
$449$	6.13520	0.289538	0.144769	+	0.989465i	$0.453756\pi$
$450$	9.69120	0.456847
$451$	−53.9607	−2.54091
$452$	0.645327	0.0303536
$453$	42.8515	2.01334
$454$	−21.1754	−0.993809
$455$	−0.958290	−0.0449254
$456$	50.1219	2.34717
$457$	−31.9738	−1.49567	−0.747835	−	0.663885i	$-0.768906\pi$
$457$	−31.9738	−1.49567	−0.747835	+	0.663885i	$0.768906\pi$
$458$	17.9252	0.837590
$459$	−2.97596	−0.138906
$460$	−0.279437	−0.0130288
$461$	−29.2548	−1.36253	−0.681267	−	0.732035i	$-0.738571\pi$
$461$	−29.2548	−1.36253	−0.681267	+	0.732035i	$0.738571\pi$
$462$	29.0365	1.35090
$463$	0.482750	0.0224353	0.0112176	−	0.999937i	$-0.496429\pi$
$463$	0.482750	0.0224353	0.0112176	+	0.999937i	$0.496429\pi$
$464$	−37.0433	−1.71969
$465$	−32.2006	−1.49327
$466$	17.5294	0.812032
$467$	−2.40144	−0.111125	−0.0555627	−	0.998455i	$-0.517695\pi$
$467$	−2.40144	−0.111125	−0.0555627	+	0.998455i	$0.517695\pi$
$468$	0.0764984	0.00353614
$469$	−12.5539	−0.579684
$470$	11.9972	0.553389
$471$	−35.8783	−1.65319
$472$	−14.7424	−0.678575
$473$	−38.5919	−1.77446
$474$	−31.3030	−1.43779
$475$	20.1637	0.925172
$476$	−0.553697	−0.0253786
$477$	−5.78302	−0.264786
$478$	−4.92202	−0.225128
$479$	−17.2392	−0.787678	−0.393839	−	0.919179i	$-0.628853\pi$
$479$	−17.2392	−0.787678	−0.393839	+	0.919179i	$0.628853\pi$
$480$	1.98461	0.0905846
$481$	3.26213	0.148741
$482$	−18.6227	−0.848240
$483$	−8.85806	−0.403056
$484$	0.679555	0.0308889
$485$	−27.0789	−1.22959
$486$	−30.0662	−1.36383
$487$	5.26459	0.238561	0.119281	−	0.992861i	$-0.461941\pi$
$487$	5.26459	0.238561	0.119281	+	0.992861i	$0.461941\pi$
$488$	−3.71225	−0.168046
$489$	35.3398	1.59812
$490$	−6.69156	−0.302294
$491$	−36.7189	−1.65710	−0.828551	−	0.559913i	$-0.810835\pi$
$491$	−36.7189	−1.65710	−0.828551	+	0.559913i	$0.810835\pi$
$492$	2.88765	0.130185
$493$	−25.4006	−1.14398
$494$	3.46080	0.155709
$495$	−16.7978	−0.755005
$496$	36.9896	1.66088
$497$	10.2201	0.458435
$498$	−37.3903	−1.67550
$499$	−24.2996	−1.08780	−0.543899	−	0.839151i	$-0.683052\pi$
$499$	−24.2996	−1.08780	−0.543899	+	0.839151i	$0.683052\pi$
$500$	1.13390	0.0507097
$501$	27.4391	1.22589
$502$	13.6133	0.607592
$503$	−2.28190	−0.101745	−0.0508725	−	0.998705i	$-0.516200\pi$
$503$	−2.28190	−0.101745	−0.0508725	+	0.998705i	$0.516200\pi$
$504$	14.1256	0.629205
$505$	−6.96826	−0.310083
$506$	−11.5427	−0.513134
$507$	30.4272	1.35132
$508$	−0.735331	−0.0326250
$509$	1.28852	0.0571128	0.0285564	−	0.999592i	$-0.490909\pi$
$509$	1.28852	0.0571128	0.0285564	+	0.999592i	$0.490909\pi$
$510$	15.1267	0.669820
$511$	−7.24673	−0.320577
$512$	20.7816	0.918424
$513$	−8.00069	−0.353239
$514$	−23.9809	−1.05775
$515$	18.2652	0.804862
$516$	2.06521	0.0909157
$517$	22.7913	1.00236
$518$	−30.5092	−1.34050
$519$	−17.1355	−0.752166
$520$	−1.31935	−0.0578571
$521$	−33.4172	−1.46403	−0.732017	−	0.681287i	$-0.761421\pi$
$521$	−33.4172	−1.46403	−0.732017	+	0.681287i	$0.761421\pi$
$522$	−32.8210	−1.43654
$523$	17.4914	0.764848	0.382424	−	0.923987i	$-0.375089\pi$
$523$	17.4914	0.764848	0.382424	+	0.923987i	$0.375089\pi$
$524$	0.563398	0.0246121
$525$	12.3418	0.538642
$526$	21.1994	0.924339
$527$	25.3638	1.10486
$528$	41.9083	1.82382
$529$	−19.4787	−0.846901
$530$	−5.05166	−0.219430
$531$	−13.6932	−0.594234
$532$	−1.48858	−0.0645382
$533$	−3.93659	−0.170513
$534$	−40.8033	−1.76573
$535$	12.0504	0.520985
$536$	−17.2838	−0.746546
$537$	32.0128	1.38145
$538$	−8.14683	−0.351235
$539$	−12.7121	−0.547549
$540$	−0.154484	−0.00664793
$541$	−16.7427	−0.719824	−0.359912	−	0.932986i	$-0.617193\pi$
$541$	−16.7427	−0.719824	−0.359912	+	0.932986i	$0.617193\pi$
$542$	−1.69811	−0.0729400
$543$	−19.2495	−0.826073
$544$	−1.56324	−0.0670232
$545$	5.94292	0.254567
$546$	2.11830	0.0906547
$547$	−23.1127	−0.988228	−0.494114	−	0.869397i	$-0.664507\pi$
$547$	−23.1127	−0.988228	−0.494114	+	0.869397i	$0.664507\pi$
$548$	0.599176	0.0255955
$549$	−3.44805	−0.147159
$550$	16.0823	0.685750
$551$	−68.2880	−2.90917
$552$	−12.1955	−0.519076
$553$	−18.3551	−0.780539
$554$	7.96447	0.338378
$555$	38.3327	1.62713
$556$	0.531463	0.0225390
$557$	−35.9747	−1.52430	−0.762149	−	0.647401i	$-0.775856\pi$
$557$	−35.9747	−1.52430	−0.762149	+	0.647401i	$0.775856\pi$
$558$	32.7735	1.38741
$559$	−2.81539	−0.119078
$560$	12.9354	0.546622
$561$	28.7365	1.21325
$562$	−20.6347	−0.870421
$563$	9.07812	0.382597	0.191299	−	0.981532i	$-0.438730\pi$
$563$	9.07812	0.382597	0.191299	+	0.981532i	$0.438730\pi$
$564$	−1.21965	−0.0513567
$565$	−10.3377	−0.434909
$566$	−28.9778	−1.21803
$567$	−20.2722	−0.851352
$568$	14.0708	0.590396
$569$	40.8911	1.71425	0.857123	−	0.515112i	$-0.172250\pi$
$569$	40.8911	1.71425	0.857123	+	0.515112i	$0.172250\pi$
$570$	40.6671	1.70336
$571$	17.7500	0.742815	0.371408	−	0.928470i	$-0.378875\pi$
$571$	17.7500	0.742815	0.371408	+	0.928470i	$0.378875\pi$
$572$	0.126947	0.00530792
$573$	12.8506	0.536843
$574$	36.8170	1.53671
$575$	−4.90616	−0.204601
$576$	19.4001	0.808339
$577$	−12.0135	−0.500130	−0.250065	−	0.968229i	$-0.580452\pi$
$577$	−12.0135	−0.500130	−0.250065	+	0.968229i	$0.580452\pi$
$578$	12.6993	0.528223
$579$	7.63442	0.317276
$580$	−1.31856	−0.0547502
$581$	−21.9246	−0.909584
$582$	59.8577	2.48118
$583$	−9.59676	−0.397457
$584$	−9.97709	−0.412855
$585$	−1.22545	−0.0506661
$586$	−16.8712	−0.696941
$587$	−27.0026	−1.11452	−0.557258	−	0.830339i	$-0.688147\pi$
$587$	−27.0026	−1.11452	−0.557258	+	0.830339i	$0.688147\pi$
$588$	0.680276	0.0280541
$589$	68.1890	2.80968
$590$	−11.9615	−0.492446
$591$	−32.9588	−1.35574
$592$	−44.0337	−1.80978
$593$	−9.97414	−0.409589	−0.204794	−	0.978805i	$-0.565653\pi$
$593$	−9.97414	−0.409589	−0.204794	+	0.978805i	$0.565653\pi$
$594$	−6.38125	−0.261826
$595$	8.86982	0.363627
$596$	−0.672172	−0.0275332
$597$	50.9383	2.08477
$598$	−0.842071	−0.0344348
$599$	21.8216	0.891607	0.445803	−	0.895131i	$-0.352918\pi$
$599$	21.8216	0.891607	0.445803	+	0.895131i	$0.352918\pi$
$600$	16.9919	0.693691
$601$	18.6041	0.758879	0.379439	−	0.925217i	$-0.376117\pi$
$601$	18.6041	0.758879	0.379439	+	0.925217i	$0.376117\pi$
$602$	26.3310	1.07317
$603$	−16.0537	−0.653758
$604$	−1.75216	−0.0712942
$605$	−10.8860	−0.442578
$606$	15.4033	0.625716
$607$	27.4663	1.11482	0.557411	−	0.830236i	$-0.311795\pi$
$607$	27.4663	1.11482	0.557411	+	0.830236i	$0.311795\pi$
$608$	−4.20267	−0.170441
$609$	−41.7979	−1.69374
$610$	−3.01199	−0.121952
$611$	1.66269	0.0672654
$612$	−0.708059	−0.0286216
$613$	0.616794	0.0249121	0.0124560	−	0.999922i	$-0.496035\pi$
$613$	0.616794	0.0249121	0.0124560	+	0.999922i	$0.496035\pi$
$614$	9.34187	0.377007
$615$	−46.2581	−1.86531
$616$	23.4411	0.944467
$617$	−22.7778	−0.916999	−0.458499	−	0.888695i	$-0.651613\pi$
$617$	−22.7778	−0.916999	−0.458499	+	0.888695i	$0.651613\pi$
$618$	−40.3752	−1.62413
$619$	42.7584	1.71860	0.859302	−	0.511469i	$-0.170898\pi$
$619$	42.7584	1.71860	0.859302	+	0.511469i	$0.170898\pi$
$620$	1.31665	0.0528779
$621$	1.94670	0.0781185
$622$	25.2604	1.01285
$623$	−23.9258	−0.958569
$624$	3.05733	0.122391
$625$	−5.09171	−0.203668
$626$	25.5835	1.02252
$627$	77.2563	3.08532
$628$	1.46703	0.0585408
$629$	−30.1939	−1.20391
$630$	11.4610	0.456618
$631$	19.5623	0.778761	0.389381	−	0.921077i	$-0.372689\pi$
$631$	19.5623	0.778761	0.389381	+	0.921077i	$0.372689\pi$
$632$	−25.2708	−1.00522
$633$	−47.8446	−1.90165
$634$	33.7124	1.33889
$635$	11.7795	0.467454
$636$	0.513561	0.0203640
$637$	−0.927385	−0.0367443
$638$	−54.4656	−2.15631
$639$	13.0693	0.517015
$640$	18.6300	0.736415
$641$	38.5585	1.52297	0.761485	−	0.648183i	$-0.224471\pi$
$641$	38.5585	1.52297	0.761485	+	0.648183i	$0.224471\pi$
$642$	−26.6374	−1.05129
$643$	3.11044	0.122664	0.0613318	−	0.998117i	$-0.480465\pi$
$643$	3.11044	0.122664	0.0613318	+	0.998117i	$0.480465\pi$
$644$	0.362197	0.0142726
$645$	−33.0831	−1.30265
$646$	−32.0327	−1.26031
$647$	−28.9494	−1.13812	−0.569060	−	0.822296i	$-0.692693\pi$
$647$	−28.9494	−1.13812	−0.569060	+	0.822296i	$0.692693\pi$
$648$	−27.9101	−1.09641
$649$	−22.7235	−0.891975
$650$	1.17325	0.0460186
$651$	41.7373	1.63582
$652$	−1.44501	−0.0565909
$653$	41.7333	1.63315	0.816575	−	0.577240i	$-0.195870\pi$
$653$	41.7333	1.63315	0.816575	+	0.577240i	$0.195870\pi$
$654$	−13.1368	−0.513689
$655$	−9.02522	−0.352644
$656$	53.1378	2.07468
$657$	−9.26702	−0.361541
$658$	−15.5504	−0.606216
$659$	30.5739	1.19099	0.595495	−	0.803359i	$-0.296956\pi$
$659$	30.5739	1.19099	0.595495	+	0.803359i	$0.296956\pi$
$660$	1.49173	0.0580654
$661$	7.57936	0.294803	0.147402	−	0.989077i	$-0.452909\pi$
$661$	7.57936	0.294803	0.147402	+	0.989077i	$0.452909\pi$
$662$	−13.6140	−0.529125
$663$	2.09641	0.0814178
$664$	−30.1851	−1.17141
$665$	23.8460	0.924708
$666$	−39.0147	−1.51179
$667$	16.6156	0.643359
$668$	−1.12196	−0.0434098
$669$	−24.9939	−0.966319
$670$	−14.0235	−0.541773
$671$	−5.72194	−0.220893
$672$	−2.57239	−0.0992319
$673$	−44.7555	−1.72520	−0.862599	−	0.505889i	$-0.831165\pi$
$673$	−44.7555	−1.72520	−0.862599	+	0.505889i	$0.831165\pi$
$674$	−0.775050	−0.0298538
$675$	−2.71232	−0.104397
$676$	−1.24414	−0.0478514
$677$	18.7731	0.721508	0.360754	−	0.932661i	$-0.382519\pi$
$677$	18.7731	0.721508	0.360754	+	0.932661i	$0.382519\pi$
$678$	22.8514	0.877602
$679$	35.0987	1.34697
$680$	12.2117	0.468297
$681$	−34.4851	−1.32147
$682$	54.3866	2.08257
$683$	−5.11991	−0.195908	−0.0979540	−	0.995191i	$-0.531230\pi$
$683$	−5.11991	−0.195908	−0.0979540	+	0.995191i	$0.531230\pi$
$684$	−1.90358	−0.0727851
$685$	−9.59836	−0.366735
$686$	28.9636	1.10583
$687$	29.1920	1.11375
$688$	38.0034	1.44887
$689$	−0.700111	−0.0266721
$690$	−9.89500	−0.376696
$691$	32.8364	1.24916	0.624578	−	0.780963i	$-0.285271\pi$
$691$	32.8364	1.24916	0.624578	+	0.780963i	$0.285271\pi$
$692$	0.700654	0.0266349
$693$	21.7728	0.827079
$694$	−17.1426	−0.650723
$695$	−8.51364	−0.322941
$696$	−57.5462	−2.18128
$697$	36.4366	1.38013
$698$	−36.4487	−1.37960
$699$	28.5474	1.07976
$700$	−0.504645	−0.0190738
$701$	−17.9843	−0.679258	−0.339629	−	0.940559i	$-0.610302\pi$
$701$	−17.9843	−0.679258	−0.339629	+	0.940559i	$0.610302\pi$
$702$	−0.465531	−0.0175703
$703$	−81.1746	−3.06156
$704$	32.1940	1.21336
$705$	19.5380	0.735843
$706$	−24.2030	−0.910892
$707$	9.03203	0.339684
$708$	1.21602	0.0457010
$709$	4.29461	0.161288	0.0806438	−	0.996743i	$-0.474302\pi$
$709$	4.29461	0.161288	0.0806438	+	0.996743i	$0.474302\pi$
$710$	11.4165	0.428454
$711$	−23.4723	−0.880279
$712$	−32.9404	−1.23449
$713$	−16.5915	−0.621358
$714$	−19.6067	−0.733762
$715$	−2.03360	−0.0760522
$716$	−1.30897	−0.0489186
$717$	−8.01574	−0.299353
$718$	−12.2668	−0.457794
$719$	8.65712	0.322856	0.161428	−	0.986884i	$-0.448390\pi$
$719$	8.65712	0.322856	0.161428	+	0.986884i	$0.448390\pi$
$720$	16.5416	0.616471
$721$	−23.6748	−0.881696
$722$	−58.6080	−2.18116
$723$	−30.3279	−1.12791
$724$	0.787091	0.0292520
$725$	−23.1504	−0.859783
$726$	24.0634	0.893076
$727$	11.1806	0.414666	0.207333	−	0.978270i	$-0.433522\pi$
$727$	11.1806	0.414666	0.207333	+	0.978270i	$0.433522\pi$
$728$	1.71009	0.0633803
$729$	−18.5852	−0.688342
$730$	−8.09505	−0.299611
$731$	26.0589	0.963824
$732$	0.306204	0.0113176
$733$	22.7792	0.841368	0.420684	−	0.907207i	$-0.361790\pi$
$733$	22.7792	0.841368	0.420684	+	0.907207i	$0.361790\pi$
$734$	−27.4460	−1.01305
$735$	−10.8975	−0.401961
$736$	1.02258	0.0376928
$737$	−26.6407	−0.981322
$738$	47.0811	1.73308
$739$	−34.7056	−1.27667	−0.638333	−	0.769760i	$-0.720376\pi$
$739$	−34.7056	−1.27667	−0.638333	+	0.769760i	$0.720376\pi$
$740$	−1.56738	−0.0576182
$741$	5.63607	0.207046
$742$	6.54780	0.240377
$743$	18.9180	0.694033	0.347016	−	0.937859i	$-0.387195\pi$
$743$	18.9180	0.694033	0.347016	+	0.937859i	$0.387195\pi$
$744$	57.4627	2.10669
$745$	10.7677	0.394498
$746$	51.9526	1.90212
$747$	−28.0368	−1.02581
$748$	−1.17500	−0.0429624
$749$	−15.6194	−0.570719
$750$	40.1521	1.46615
$751$	−24.3213	−0.887496	−0.443748	−	0.896152i	$-0.646352\pi$
$751$	−24.3213	−0.887496	−0.443748	+	0.896152i	$0.646352\pi$
$752$	−22.4438	−0.818440
$753$	22.1699	0.807917
$754$	−3.97342	−0.144704
$755$	28.0683	1.02151
$756$	0.200237	0.00728255
$757$	−46.1933	−1.67892	−0.839462	−	0.543419i	$-0.817130\pi$
$757$	−46.1933	−1.67892	−0.839462	+	0.543419i	$0.817130\pi$
$758$	−23.4156	−0.850493
$759$	−18.7978	−0.682316
$760$	32.8304	1.19089
$761$	−11.8756	−0.430492	−0.215246	−	0.976560i	$-0.569055\pi$
$761$	−11.8756	−0.430492	−0.215246	+	0.976560i	$0.569055\pi$
$762$	−26.0384	−0.943273
$763$	−7.70302	−0.278868
$764$	−0.525449	−0.0190101
$765$	11.3426	0.410092
$766$	−43.5895	−1.57495
$767$	−1.65774	−0.0598577
$768$	−5.44379	−0.196436
$769$	−18.4351	−0.664786	−0.332393	−	0.943141i	$-0.607856\pi$
$769$	−18.4351	−0.664786	−0.332393	+	0.943141i	$0.607856\pi$
$770$	19.0192	0.685406
$771$	−39.0540	−1.40650
$772$	−0.312164	−0.0112350
$773$	−5.46060	−0.196404	−0.0982021	−	0.995166i	$-0.531309\pi$
$773$	−5.46060	−0.196404	−0.0982021	+	0.995166i	$0.531309\pi$
$774$	33.6717	1.21030
$775$	23.1168	0.830381
$776$	48.3229	1.73469
$777$	−49.6856	−1.78246
$778$	9.48064	0.339897
$779$	97.9577	3.50970
$780$	0.108826	0.00389659
$781$	21.6882	0.776065
$782$	7.79410	0.278716
$783$	9.18578	0.328273
$784$	12.5183	0.447080
$785$	−23.5007	−0.838777
$786$	19.9502	0.711600
$787$	11.2338	0.400441	0.200220	−	0.979751i	$-0.435834\pi$
$787$	11.2338	0.400441	0.200220	+	0.979751i	$0.435834\pi$
$788$	1.34765	0.0480080
$789$	34.5243	1.22910
$790$	−20.5038	−0.729493
$791$	13.3994	0.476426
$792$	29.9761	1.06515
$793$	−0.417432	−0.0148235
$794$	−51.7334	−1.83595
$795$	−8.22687	−0.291777
$796$	−2.08282	−0.0738235
$797$	−5.66214	−0.200563	−0.100282	−	0.994959i	$-0.531974\pi$
$797$	−5.66214	−0.200563	−0.100282	+	0.994959i	$0.531974\pi$
$798$	−52.7114	−1.86596
$799$	−15.3897	−0.544448
$800$	−1.42475	−0.0503726
$801$	−30.5960	−1.08106
$802$	1.44790	0.0511271
$803$	−15.3784	−0.542691
$804$	1.42565	0.0502788
$805$	−5.80213	−0.204498
$806$	3.96766	0.139755
$807$	−13.2675	−0.467038
$808$	12.4350	0.437463
$809$	6.00571	0.211150	0.105575	−	0.994411i	$-0.466332\pi$
$809$	6.00571	0.211150	0.105575	+	0.994411i	$0.466332\pi$
$810$	−22.6453	−0.795674
$811$	−23.2409	−0.816100	−0.408050	−	0.912960i	$-0.633791\pi$
$811$	−23.2409	−0.816100	−0.408050	+	0.912960i	$0.633791\pi$
$812$	1.70908	0.0599768
$813$	−2.76545	−0.0969886
$814$	−64.7438	−2.26927
$815$	23.1480	0.810838
$816$	−28.2982	−0.990637
$817$	70.0579	2.45101
$818$	−36.7300	−1.28423
$819$	1.58839	0.0555027
$820$	1.89145	0.0660522
$821$	30.3896	1.06060	0.530302	−	0.847809i	$-0.322079\pi$
$821$	30.3896	1.06060	0.530302	+	0.847809i	$0.322079\pi$
$822$	21.2171	0.740032
$823$	13.3662	0.465915	0.232957	−	0.972487i	$-0.425160\pi$
$823$	13.3662	0.465915	0.232957	+	0.972487i	$0.425160\pi$
$824$	−32.5948	−1.13549
$825$	26.1907	0.911845
$826$	15.5041	0.539456
$827$	−39.1913	−1.36282	−0.681408	−	0.731904i	$-0.738632\pi$
$827$	−39.1913	−1.36282	−0.681408	+	0.731904i	$0.738632\pi$
$828$	0.463172	0.0160964
$829$	42.6047	1.47972	0.739861	−	0.672760i	$-0.234891\pi$
$829$	42.6047	1.47972	0.739861	+	0.672760i	$0.234891\pi$
$830$	−24.4911	−0.850099
$831$	12.9705	0.449942
$832$	2.34864	0.0814246
$833$	8.58376	0.297410
$834$	18.8194	0.651661
$835$	17.9729	0.621978
$836$	−3.15893	−0.109254
$837$	−9.17247	−0.317047
$838$	39.3709	1.36004
$839$	−25.4863	−0.879885	−0.439943	−	0.898026i	$-0.645001\pi$
$839$	−25.4863	−0.879885	−0.439943	+	0.898026i	$0.645001\pi$
$840$	20.0950	0.693342
$841$	49.4030	1.70355
$842$	11.6024	0.399846
$843$	−33.6045	−1.15740
$844$	1.95632	0.0673392
$845$	19.9302	0.685618
$846$	−19.8856	−0.683680
$847$	14.1100	0.484827
$848$	9.45041	0.324528
$849$	−47.1918	−1.61962
$850$	−10.8594	−0.372476
$851$	19.7511	0.677061
$852$	−1.16062	−0.0397623
$853$	10.2920	0.352392	0.176196	−	0.984355i	$-0.443621\pi$
$853$	10.2920	0.352392	0.176196	+	0.984355i	$0.443621\pi$
$854$	3.90404	0.133594
$855$	30.4939	1.04287
$856$	−21.5043	−0.735001
$857$	−10.4662	−0.357520	−0.178760	−	0.983893i	$-0.557209\pi$
$857$	−10.4662	−0.357520	−0.178760	+	0.983893i	$0.557209\pi$
$858$	4.49526	0.153466
$859$	12.6756	0.432484	0.216242	−	0.976340i	$-0.430620\pi$
$859$	12.6756	0.432484	0.216242	+	0.976340i	$0.430620\pi$
$860$	1.35273	0.0461279
$861$	59.9583	2.04337
$862$	−47.3555	−1.61293
$863$	4.43155	0.150852	0.0754258	−	0.997151i	$-0.475968\pi$
$863$	4.43155	0.150852	0.0754258	+	0.997151i	$0.475968\pi$
$864$	0.565324	0.0192327
$865$	−11.2240	−0.381627
$866$	48.2533	1.63971
$867$	20.6815	0.702380
$868$	−1.70660	−0.0579257
$869$	−38.9516	−1.32134
$870$	−46.6909	−1.58297
$871$	−1.94352	−0.0658535
$872$	−10.6053	−0.359140
$873$	44.8838	1.51909
$874$	20.9540	0.708779
$875$	23.5440	0.795932
$876$	0.822957	0.0278052
$877$	49.0742	1.65712	0.828559	−	0.559902i	$-0.189161\pi$
$877$	49.0742	1.65712	0.828559	+	0.559902i	$0.189161\pi$
$878$	−9.84745	−0.332335
$879$	−27.4755	−0.926725
$880$	27.4504	0.925353
$881$	48.2180	1.62451	0.812253	−	0.583305i	$-0.198241\pi$
$881$	48.2180	1.62451	0.812253	+	0.583305i	$0.198241\pi$
$882$	11.0914	0.373467
$883$	19.2102	0.646475	0.323238	−	0.946318i	$-0.395229\pi$
$883$	19.2102	0.646475	0.323238	+	0.946318i	$0.395229\pi$
$884$	−0.0857200	−0.00288308
$885$	−19.4798	−0.654807
$886$	−7.60670	−0.255552
$887$	1.97695	0.0663794	0.0331897	−	0.999449i	$-0.489433\pi$
$887$	1.97695	0.0663794	0.0331897	+	0.999449i	$0.489433\pi$
$888$	−68.4057	−2.29554
$889$	−15.2682	−0.512078
$890$	−26.7267	−0.895880
$891$	−43.0198	−1.44122
$892$	1.02197	0.0342182
$893$	−41.3743	−1.38454
$894$	−23.8020	−0.796057
$895$	20.9688	0.700908
$896$	−24.1476	−0.806714
$897$	−1.37135	−0.0457881
$898$	−8.88316	−0.296435
$899$	−78.2894	−2.61110
$900$	−0.645333	−0.0215111
$901$	6.48014	0.215885
$902$	78.1297	2.60144
$903$	42.8813	1.42700
$904$	18.4478	0.613566
$905$	−12.6086	−0.419125
$906$	−62.0448	−2.06130
$907$	−12.9435	−0.429781	−0.214891	−	0.976638i	$-0.568939\pi$
$907$	−12.9435	−0.429781	−0.214891	+	0.976638i	$0.568939\pi$
$908$	1.41006	0.0467945
$909$	11.5500	0.383090
$910$	1.38751	0.0459955
$911$	49.0291	1.62441	0.812203	−	0.583375i	$-0.198268\pi$
$911$	49.0291	1.62441	0.812203	+	0.583375i	$0.198268\pi$
$912$	−76.0782	−2.51920
$913$	−46.5263	−1.53980
$914$	46.2948	1.53130
$915$	−4.90517	−0.162160
$916$	−1.19363	−0.0394387
$917$	11.6982	0.386309
$918$	4.30889	0.142215
$919$	23.4751	0.774372	0.387186	−	0.922002i	$-0.373447\pi$
$919$	23.4751	0.774372	0.387186	+	0.922002i	$0.373447\pi$
$920$	−7.98820	−0.263363
$921$	15.2137	0.501308
$922$	42.3581	1.39499
$923$	1.58222	0.0520793
$924$	−1.93353	−0.0636085
$925$	−27.5191	−0.904822
$926$	−0.698974	−0.0229697
$927$	−30.2750	−0.994362
$928$	4.82519	0.158395
$929$	49.3896	1.62042	0.810210	−	0.586139i	$-0.199353\pi$
$929$	49.3896	1.62042	0.810210	+	0.586139i	$0.199353\pi$
$930$	46.6232	1.52884
$931$	23.0769	0.756316
$932$	−1.16727	−0.0382353
$933$	41.1378	1.34679
$934$	3.47705	0.113772
$935$	18.8227	0.615569
$936$	2.18684	0.0714792
$937$	−29.7278	−0.971165	−0.485582	−	0.874191i	$-0.661392\pi$
$937$	−29.7278	−0.971165	−0.485582	+	0.874191i	$0.661392\pi$
$938$	18.1768	0.593492
$939$	41.6640	1.35965
$940$	−0.798888	−0.0260568
$941$	−28.0300	−0.913751	−0.456875	−	0.889531i	$-0.651031\pi$
$941$	−28.0300	−0.913751	−0.456875	+	0.889531i	$0.651031\pi$
$942$	51.9482	1.69256
$943$	−23.8348	−0.776167
$944$	22.3770	0.728308
$945$	−3.20765	−0.104345
$946$	55.8772	1.81673
$947$	−28.9321	−0.940167	−0.470084	−	0.882622i	$-0.655776\pi$
$947$	−28.9321	−0.940167	−0.470084	+	0.882622i	$0.655776\pi$
$948$	2.08445	0.0676999
$949$	−1.12190	−0.0364183
$950$	−29.1950	−0.947210
$951$	54.9022	1.78033
$952$	−15.8284	−0.513002
$953$	43.3685	1.40484	0.702422	−	0.711760i	$-0.252102\pi$
$953$	43.3685	1.40484	0.702422	+	0.711760i	$0.252102\pi$
$954$	8.37323	0.271093
$955$	8.41731	0.272378
$956$	0.327755	0.0106004
$957$	−88.6998	−2.86726
$958$	24.9606	0.806440
$959$	12.4411	0.401744
$960$	27.5985	0.890736
$961$	47.1759	1.52180
$962$	−4.72325	−0.152284
$963$	−19.9738	−0.643647
$964$	1.24008	0.0399402
$965$	5.00063	0.160976
$966$	12.8256	0.412656
$967$	−25.8885	−0.832518	−0.416259	−	0.909246i	$-0.636659\pi$
$967$	−25.8885	−0.832518	−0.416259	+	0.909246i	$0.636659\pi$
$968$	19.4263	0.624385
$969$	−52.1668	−1.67584
$970$	39.2075	1.25888
$971$	29.7503	0.954733	0.477367	−	0.878704i	$-0.341591\pi$
$971$	29.7503	0.954733	0.477367	+	0.878704i	$0.341591\pi$
$972$	2.00210	0.0642172
$973$	11.0351	0.353769
$974$	−7.62260	−0.244244
$975$	1.91069	0.0611911
$976$	5.63469	0.180362
$977$	39.0320	1.24874	0.624372	−	0.781127i	$-0.285355\pi$
$977$	39.0320	1.24874	0.624372	+	0.781127i	$0.285355\pi$
$978$	−51.1685	−1.63619
$979$	−50.7733	−1.62272
$980$	0.445588	0.0142338
$981$	−9.85051	−0.314503
$982$	53.1654	1.69658
$983$	−35.6179	−1.13603	−0.568017	−	0.823017i	$-0.692289\pi$
$983$	−35.6179	−1.13603	−0.568017	+	0.823017i	$0.692289\pi$
$984$	82.5488	2.63156
$985$	−21.5884	−0.687863
$986$	36.7775	1.17123
$987$	−25.3245	−0.806087
$988$	−0.230453	−0.00733170
$989$	−17.0463	−0.542040
$990$	24.3215	0.772989
$991$	−45.8253	−1.45569	−0.727845	−	0.685742i	$-0.759478\pi$
$991$	−45.8253	−1.45569	−0.727845	+	0.685742i	$0.759478\pi$
$992$	−4.81819	−0.152978
$993$	−22.1711	−0.703579
$994$	−14.7977	−0.469355
$995$	33.3652	1.05775
$996$	2.48981	0.0788926
$997$	−38.5734	−1.22163	−0.610816	−	0.791772i	$-0.709159\pi$
$997$	−38.5734	−1.22163	−0.610816	+	0.791772i	$0.709159\pi$
$998$	35.1833	1.11371
$999$	10.9192	0.345469

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	401.2.a.a.1.5	✓	12
3.2	odd	2		3609.2.a.b.1.8		12
4.3	odd	2		6416.2.a.k.1.10		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
401.2.a.a.1.5	✓	12	1.1	even	1	trivial
3609.2.a.b.1.8		12	3.2	odd	2
6416.2.a.k.1.10		12	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:	\( N \)	\(=\)	\( 401 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	401.a (trivial)

\(f(q)\)	\(=\)	\(q-1.44790 q^{2} -2.35797 q^{3} +0.0964152 q^{4} -1.54450 q^{5} +3.41411 q^{6} +2.00193 q^{7} +2.75620 q^{8} +2.56004 q^{9} +O(q^{10})\)	\(q-1.44790 q^{2} -2.35797 q^{3} +0.0964152 q^{4} -1.54450 q^{5} +3.41411 q^{6} +2.00193 q^{7} +2.75620 q^{8} +2.56004 q^{9} +2.23628 q^{10} +4.24832 q^{11} -0.227344 q^{12} +0.309927 q^{13} -2.89860 q^{14} +3.64189 q^{15} -4.18353 q^{16} -2.86865 q^{17} -3.70669 q^{18} -7.71219 q^{19} -0.148913 q^{20} -4.72050 q^{21} -6.15114 q^{22} +1.87651 q^{23} -6.49905 q^{24} -2.61452 q^{25} -0.448744 q^{26} +1.03741 q^{27} +0.193017 q^{28} +8.85455 q^{29} -5.27310 q^{30} -8.84171 q^{31} +0.544939 q^{32} -10.0174 q^{33} +4.15351 q^{34} -3.09199 q^{35} +0.246827 q^{36} +10.5255 q^{37} +11.1665 q^{38} -0.730800 q^{39} -4.25695 q^{40} -12.7017 q^{41} +6.83482 q^{42} -9.08404 q^{43} +0.409602 q^{44} -3.95399 q^{45} -2.71699 q^{46} +5.36479 q^{47} +9.86467 q^{48} -2.99227 q^{49} +3.78556 q^{50} +6.76420 q^{51} +0.0298817 q^{52} -2.25895 q^{53} -1.50206 q^{54} -6.56153 q^{55} +5.51773 q^{56} +18.1852 q^{57} -12.8205 q^{58} -5.34882 q^{59} +0.351134 q^{60} -1.34687 q^{61} +12.8019 q^{62} +5.12503 q^{63} +7.57805 q^{64} -0.478683 q^{65} +14.5042 q^{66} -6.27088 q^{67} -0.276581 q^{68} -4.42475 q^{69} +4.47689 q^{70} +5.10513 q^{71} +7.05599 q^{72} -3.61987 q^{73} -15.2399 q^{74} +6.16497 q^{75} -0.743572 q^{76} +8.50485 q^{77} +1.05813 q^{78} -9.16870 q^{79} +6.46147 q^{80} -10.1263 q^{81} +18.3907 q^{82} -10.9517 q^{83} -0.455128 q^{84} +4.43063 q^{85} +13.1528 q^{86} -20.8788 q^{87} +11.7092 q^{88} -11.9514 q^{89} +5.72498 q^{90} +0.620453 q^{91} +0.180924 q^{92} +20.8485 q^{93} -7.76767 q^{94} +11.9115 q^{95} -1.28495 q^{96} +17.5324 q^{97} +4.33250 q^{98} +10.8759 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 3 q^{2} - 5 q^{3} + 5 q^{4} - 7 q^{5} - 9 q^{6} - 20 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) \) 12 * q - 3 * q^2 - 5 * q^3 + 5 * q^4 - 7 * q^5 - 9 * q^6 - 20 * q^7 - 3 * q^8 + 3 * q^9	\( 12 q - 3 q^{2} - 5 q^{3} + 5 q^{4} - 7 q^{5} - 9 q^{6} - 20 q^{7} - 3 q^{8} + 3 q^{9} - 11 q^{10} - 11 q^{11} - 9 q^{12} - 11 q^{13} - 3 q^{14} - 11 q^{15} - 9 q^{16} + q^{17} - q^{18} - 34 q^{19} - 5 q^{20} - 3 q^{21} + 3 q^{22} - 7 q^{23} - 9 q^{24} + 7 q^{25} + 6 q^{26} - 2 q^{27} - 23 q^{28} - 6 q^{29} + 23 q^{30} - 52 q^{31} + 11 q^{32} + 4 q^{33} - 5 q^{34} + 12 q^{35} + 16 q^{36} + 3 q^{37} + 25 q^{38} - 24 q^{39} - 25 q^{40} - 16 q^{41} + 47 q^{42} - 2 q^{43} - 2 q^{44} - 23 q^{45} - 16 q^{46} - 3 q^{47} + 24 q^{48} + 6 q^{49} + 27 q^{50} - 16 q^{51} - 5 q^{52} + 19 q^{53} + 5 q^{54} - 43 q^{55} + 7 q^{56} + 11 q^{57} + 11 q^{58} - q^{59} + 30 q^{60} - 24 q^{61} + 39 q^{62} - 11 q^{63} - q^{64} + 13 q^{65} + 14 q^{66} + 6 q^{67} + 32 q^{68} + 29 q^{69} + 47 q^{70} - 15 q^{71} + 32 q^{72} - 20 q^{73} + 25 q^{74} + 31 q^{75} - 42 q^{76} + 38 q^{77} + 52 q^{78} - 53 q^{79} + 23 q^{80} - 8 q^{81} + 4 q^{82} + 17 q^{83} + 35 q^{84} + 7 q^{85} + 28 q^{86} - 5 q^{87} + 38 q^{88} - q^{89} + 58 q^{90} - 6 q^{91} + 46 q^{92} + 44 q^{93} - 4 q^{94} + 34 q^{95} + 28 q^{96} + 12 q^{97} + 27 q^{98}+O(q^{100}) \) 12 * q - 3 * q^2 - 5 * q^3 + 5 * q^4 - 7 * q^5 - 9 * q^6 - 20 * q^7 - 3 * q^8 + 3 * q^9 - 11 * q^10 - 11 * q^11 - 9 * q^12 - 11 * q^13 - 3 * q^14 - 11 * q^15 - 9 * q^16 + q^17 - q^18 - 34 * q^19 - 5 * q^20 - 3 * q^21 + 3 * q^22 - 7 * q^23 - 9 * q^24 + 7 * q^25 + 6 * q^26 - 2 * q^27 - 23 * q^28 - 6 * q^29 + 23 * q^30 - 52 * q^31 + 11 * q^32 + 4 * q^33 - 5 * q^34 + 12 * q^35 + 16 * q^36 + 3 * q^37 + 25 * q^38 - 24 * q^39 - 25 * q^40 - 16 * q^41 + 47 * q^42 - 2 * q^43 - 2 * q^44 - 23 * q^45 - 16 * q^46 - 3 * q^47 + 24 * q^48 + 6 * q^49 + 27 * q^50 - 16 * q^51 - 5 * q^52 + 19 * q^53 + 5 * q^54 - 43 * q^55 + 7 * q^56 + 11 * q^57 + 11 * q^58 - q^59 + 30 * q^60 - 24 * q^61 + 39 * q^62 - 11 * q^63 - q^64 + 13 * q^65 + 14 * q^66 + 6 * q^67 + 32 * q^68 + 29 * q^69 + 47 * q^70 - 15 * q^71 + 32 * q^72 - 20 * q^73 + 25 * q^74 + 31 * q^75 - 42 * q^76 + 38 * q^77 + 52 * q^78 - 53 * q^79 + 23 * q^80 - 8 * q^81 + 4 * q^82 + 17 * q^83 + 35 * q^84 + 7 * q^85 + 28 * q^86 - 5 * q^87 + 38 * q^88 - q^89 + 58 * q^90 - 6 * q^91 + 46 * q^92 + 44 * q^93 - 4 * q^94 + 34 * q^95 + 28 * q^96 + 12 * q^97 + 27 * q^98

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