LMFDB - Embedded newform 115.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [115,2,Mod(1,115)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(115, 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("115.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 115 = 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	115.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:	$0.918279623245$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 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			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	115.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.61803 q^{2} -1.00000 q^{3} +4.85410 q^{4} -1.00000 q^{5} +2.61803 q^{6} +1.23607 q^{7} -7.47214 q^{8} -2.00000 q^{9} +O(q^{10})$	\(q-2.61803 q^{2} -1.00000 q^{3} +4.85410 q^{4} -1.00000 q^{5} +2.61803 q^{6} +1.23607 q^{7} -7.47214 q^{8} -2.00000 q^{9} +2.61803 q^{10} -3.23607 q^{11} -4.85410 q^{12} -6.23607 q^{13} -3.23607 q^{14} +1.00000 q^{15} +9.85410 q^{16} +2.47214 q^{17} +5.23607 q^{18} -5.70820 q^{19} -4.85410 q^{20} -1.23607 q^{21} +8.47214 q^{22} -1.00000 q^{23} +7.47214 q^{24} +1.00000 q^{25} +16.3262 q^{26} +5.00000 q^{27} +6.00000 q^{28} -0.527864 q^{29} -2.61803 q^{30} +4.23607 q^{31} -10.8541 q^{32} +3.23607 q^{33} -6.47214 q^{34} -1.23607 q^{35} -9.70820 q^{36} -9.70820 q^{37} +14.9443 q^{38} +6.23607 q^{39} +7.47214 q^{40} -7.47214 q^{41} +3.23607 q^{42} +3.70820 q^{43} -15.7082 q^{44} +2.00000 q^{45} +2.61803 q^{46} +9.47214 q^{47} -9.85410 q^{48} -5.47214 q^{49} -2.61803 q^{50} -2.47214 q^{51} -30.2705 q^{52} -6.00000 q^{53} -13.0902 q^{54} +3.23607 q^{55} -9.23607 q^{56} +5.70820 q^{57} +1.38197 q^{58} +8.94427 q^{59} +4.85410 q^{60} +12.1803 q^{61} -11.0902 q^{62} -2.47214 q^{63} +8.70820 q^{64} +6.23607 q^{65} -8.47214 q^{66} +9.70820 q^{67} +12.0000 q^{68} +1.00000 q^{69} +3.23607 q^{70} -6.23607 q^{71} +14.9443 q^{72} -6.70820 q^{73} +25.4164 q^{74} -1.00000 q^{75} -27.7082 q^{76} -4.00000 q^{77} -16.3262 q^{78} +8.76393 q^{79} -9.85410 q^{80} +1.00000 q^{81} +19.5623 q^{82} -6.47214 q^{83} -6.00000 q^{84} -2.47214 q^{85} -9.70820 q^{86} +0.527864 q^{87} +24.1803 q^{88} +2.76393 q^{89} -5.23607 q^{90} -7.70820 q^{91} -4.85410 q^{92} -4.23607 q^{93} -24.7984 q^{94} +5.70820 q^{95} +10.8541 q^{96} -6.18034 q^{97} +14.3262 q^{98} +6.47214 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{2} - 2 q^{3} + 3 q^{4} - 2 q^{5} + 3 q^{6} - 2 q^{7} - 6 q^{8} - 4 q^{9}+O(q^{10}) $ 2 * q - 3 * q^2 - 2 * q^3 + 3 * q^4 - 2 * q^5 + 3 * q^6 - 2 * q^7 - 6 * q^8 - 4 * q^9	\( 2 q - 3 q^{2} - 2 q^{3} + 3 q^{4} - 2 q^{5} + 3 q^{6} - 2 q^{7} - 6 q^{8} - 4 q^{9} + 3 q^{10} - 2 q^{11} - 3 q^{12} - 8 q^{13} - 2 q^{14} + 2 q^{15} + 13 q^{16} - 4 q^{17} + 6 q^{18} + 2 q^{19} - 3 q^{20} + 2 q^{21} + 8 q^{22} - 2 q^{23} + 6 q^{24} + 2 q^{25} + 17 q^{26} + 10 q^{27} + 12 q^{28} - 10 q^{29} - 3 q^{30} + 4 q^{31} - 15 q^{32} + 2 q^{33} - 4 q^{34} + 2 q^{35} - 6 q^{36} - 6 q^{37} + 12 q^{38} + 8 q^{39} + 6 q^{40} - 6 q^{41} + 2 q^{42} - 6 q^{43} - 18 q^{44} + 4 q^{45} + 3 q^{46} + 10 q^{47} - 13 q^{48} - 2 q^{49} - 3 q^{50} + 4 q^{51} - 27 q^{52} - 12 q^{53} - 15 q^{54} + 2 q^{55} - 14 q^{56} - 2 q^{57} + 5 q^{58} + 3 q^{60} + 2 q^{61} - 11 q^{62} + 4 q^{63} + 4 q^{64} + 8 q^{65} - 8 q^{66} + 6 q^{67} + 24 q^{68} + 2 q^{69} + 2 q^{70} - 8 q^{71} + 12 q^{72} + 24 q^{74} - 2 q^{75} - 42 q^{76} - 8 q^{77} - 17 q^{78} + 22 q^{79} - 13 q^{80} + 2 q^{81} + 19 q^{82} - 4 q^{83} - 12 q^{84} + 4 q^{85} - 6 q^{86} + 10 q^{87} + 26 q^{88} + 10 q^{89} - 6 q^{90} - 2 q^{91} - 3 q^{92} - 4 q^{93} - 25 q^{94} - 2 q^{95} + 15 q^{96} + 10 q^{97} + 13 q^{98} + 4 q^{99}+O(q^{100}) \) 2 * q - 3 * q^2 - 2 * q^3 + 3 * q^4 - 2 * q^5 + 3 * q^6 - 2 * q^7 - 6 * q^8 - 4 * q^9 + 3 * q^10 - 2 * q^11 - 3 * q^12 - 8 * q^13 - 2 * q^14 + 2 * q^15 + 13 * q^16 - 4 * q^17 + 6 * q^18 + 2 * q^19 - 3 * q^20 + 2 * q^21 + 8 * q^22 - 2 * q^23 + 6 * q^24 + 2 * q^25 + 17 * q^26 + 10 * q^27 + 12 * q^28 - 10 * q^29 - 3 * q^30 + 4 * q^31 - 15 * q^32 + 2 * q^33 - 4 * q^34 + 2 * q^35 - 6 * q^36 - 6 * q^37 + 12 * q^38 + 8 * q^39 + 6 * q^40 - 6 * q^41 + 2 * q^42 - 6 * q^43 - 18 * q^44 + 4 * q^45 + 3 * q^46 + 10 * q^47 - 13 * q^48 - 2 * q^49 - 3 * q^50 + 4 * q^51 - 27 * q^52 - 12 * q^53 - 15 * q^54 + 2 * q^55 - 14 * q^56 - 2 * q^57 + 5 * q^58 + 3 * q^60 + 2 * q^61 - 11 * q^62 + 4 * q^63 + 4 * q^64 + 8 * q^65 - 8 * q^66 + 6 * q^67 + 24 * q^68 + 2 * q^69 + 2 * q^70 - 8 * q^71 + 12 * q^72 + 24 * q^74 - 2 * q^75 - 42 * q^76 - 8 * q^77 - 17 * q^78 + 22 * q^79 - 13 * q^80 + 2 * q^81 + 19 * q^82 - 4 * q^83 - 12 * q^84 + 4 * q^85 - 6 * q^86 + 10 * q^87 + 26 * q^88 + 10 * q^89 - 6 * q^90 - 2 * q^91 - 3 * q^92 - 4 * q^93 - 25 * q^94 - 2 * q^95 + 15 * q^96 + 10 * q^97 + 13 * q^98 + 4 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.61803	−1.85123	−0.925615	−	0.378467i	$-0.876451\pi$
$2$	−2.61803	−1.85123	−0.925615	+	0.378467i	$0.876451\pi$
$3$	−1.00000	−0.577350	−0.288675	−	0.957427i	$-0.593215\pi$
$3$	−1.00000	−0.577350	−0.288675	+	0.957427i	$0.593215\pi$
$4$	4.85410	2.42705
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	2.61803	1.06881
$7$	1.23607	0.467190	0.233595	−	0.972334i	$-0.424951\pi$
$7$	1.23607	0.467190	0.233595	+	0.972334i	$0.424951\pi$
$8$	−7.47214	−2.64180
$9$	−2.00000	−0.666667
$10$	2.61803	0.827895
$11$	−3.23607	−0.975711	−0.487856	−	0.872924i	$-0.662221\pi$
$11$	−3.23607	−0.975711	−0.487856	+	0.872924i	$0.662221\pi$
$12$	−4.85410	−1.40126
$13$	−6.23607	−1.72957	−0.864787	−	0.502139i	$-0.832547\pi$
$13$	−6.23607	−1.72957	−0.864787	+	0.502139i	$0.832547\pi$
$14$	−3.23607	−0.864876
$15$	1.00000	0.258199
$16$	9.85410	2.46353
$17$	2.47214	0.599581	0.299791	−	0.954005i	$-0.403083\pi$
$17$	2.47214	0.599581	0.299791	+	0.954005i	$0.403083\pi$
$18$	5.23607	1.23415
$19$	−5.70820	−1.30955	−0.654776	−	0.755823i	$-0.727237\pi$
$19$	−5.70820	−1.30955	−0.654776	+	0.755823i	$0.727237\pi$
$20$	−4.85410	−1.08541
$21$	−1.23607	−0.269732
$22$	8.47214	1.80627
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	7.47214	1.52524
$25$	1.00000	0.200000
$26$	16.3262	3.20184
$27$	5.00000	0.962250
$28$	6.00000	1.13389
$29$	−0.527864	−0.0980219	−0.0490109	−	0.998798i	$-0.515607\pi$
$29$	−0.527864	−0.0980219	−0.0490109	+	0.998798i	$0.515607\pi$
$30$	−2.61803	−0.477985
$31$	4.23607	0.760820	0.380410	−	0.924818i	$-0.375783\pi$
$31$	4.23607	0.760820	0.380410	+	0.924818i	$0.375783\pi$
$32$	−10.8541	−1.91875
$33$	3.23607	0.563327
$34$	−6.47214	−1.10996
$35$	−1.23607	−0.208934
$36$	−9.70820	−1.61803
$37$	−9.70820	−1.59602	−0.798009	−	0.602645i	$-0.794114\pi$
$37$	−9.70820	−1.59602	−0.798009	+	0.602645i	$0.794114\pi$
$38$	14.9443	2.42428
$39$	6.23607	0.998570
$40$	7.47214	1.18145
$41$	−7.47214	−1.16695	−0.583476	−	0.812131i	$-0.698308\pi$
$41$	−7.47214	−1.16695	−0.583476	+	0.812131i	$0.698308\pi$
$42$	3.23607	0.499336
$43$	3.70820	0.565496	0.282748	−	0.959194i	$-0.408754\pi$
$43$	3.70820	0.565496	0.282748	+	0.959194i	$0.408754\pi$
$44$	−15.7082	−2.36810
$45$	2.00000	0.298142
$46$	2.61803	0.386008
$47$	9.47214	1.38165	0.690827	−	0.723021i	$-0.257247\pi$
$47$	9.47214	1.38165	0.690827	+	0.723021i	$0.257247\pi$
$48$	−9.85410	−1.42232
$49$	−5.47214	−0.781734
$50$	−2.61803	−0.370246
$51$	−2.47214	−0.346168
$52$	−30.2705	−4.19776
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	−13.0902	−1.78135
$55$	3.23607	0.436351
$56$	−9.23607	−1.23422
$57$	5.70820	0.756070
$58$	1.38197	0.181461
$59$	8.94427	1.16445	0.582223	−	0.813029i	$-0.302183\pi$
$59$	8.94427	1.16445	0.582223	+	0.813029i	$0.302183\pi$
$60$	4.85410	0.626662
$61$	12.1803	1.55953	0.779766	−	0.626071i	$-0.215338\pi$
$61$	12.1803	1.55953	0.779766	+	0.626071i	$0.215338\pi$
$62$	−11.0902	−1.40845
$63$	−2.47214	−0.311460
$64$	8.70820	1.08853
$65$	6.23607	0.773489
$66$	−8.47214	−1.04285
$67$	9.70820	1.18605	0.593023	−	0.805186i	$-0.297934\pi$
$67$	9.70820	1.18605	0.593023	+	0.805186i	$0.297934\pi$
$68$	12.0000	1.45521
$69$	1.00000	0.120386
$70$	3.23607	0.386784
$71$	−6.23607	−0.740085	−0.370043	−	0.929015i	$-0.620657\pi$
$71$	−6.23607	−0.740085	−0.370043	+	0.929015i	$0.620657\pi$
$72$	14.9443	1.76120
$73$	−6.70820	−0.785136	−0.392568	−	0.919723i	$-0.628413\pi$
$73$	−6.70820	−0.785136	−0.392568	+	0.919723i	$0.628413\pi$
$74$	25.4164	2.95460
$75$	−1.00000	−0.115470
$76$	−27.7082	−3.17835
$77$	−4.00000	−0.455842
$78$	−16.3262	−1.84858
$79$	8.76393	0.986019	0.493010	−	0.870024i	$-0.335897\pi$
$79$	8.76393	0.986019	0.493010	+	0.870024i	$0.335897\pi$
$80$	−9.85410	−1.10172
$81$	1.00000	0.111111
$82$	19.5623	2.16030
$83$	−6.47214	−0.710409	−0.355205	−	0.934789i	$-0.615589\pi$
$83$	−6.47214	−0.710409	−0.355205	+	0.934789i	$0.615589\pi$
$84$	−6.00000	−0.654654
$85$	−2.47214	−0.268141
$86$	−9.70820	−1.04686
$87$	0.527864	0.0565930
$88$	24.1803	2.57763
$89$	2.76393	0.292976	0.146488	−	0.989212i	$-0.453203\pi$
$89$	2.76393	0.292976	0.146488	+	0.989212i	$0.453203\pi$
$90$	−5.23607	−0.551930
$91$	−7.70820	−0.808039
$92$	−4.85410	−0.506075
$93$	−4.23607	−0.439260
$94$	−24.7984	−2.55776
$95$	5.70820	0.585649
$96$	10.8541	1.10779
$97$	−6.18034	−0.627518	−0.313759	−	0.949503i	$-0.601588\pi$
$97$	−6.18034	−0.627518	−0.313759	+	0.949503i	$0.601588\pi$
$98$	14.3262	1.44717
$99$	6.47214	0.650474
$100$	4.85410	0.485410
$101$	2.94427	0.292966	0.146483	−	0.989213i	$-0.453205\pi$
$101$	2.94427	0.292966	0.146483	+	0.989213i	$0.453205\pi$
$102$	6.47214	0.640837
$103$	−6.47214	−0.637719	−0.318859	−	0.947802i	$-0.603300\pi$
$103$	−6.47214	−0.637719	−0.318859	+	0.947802i	$0.603300\pi$
$104$	46.5967	4.56919
$105$	1.23607	0.120628
$106$	15.7082	1.52572
$107$	−8.18034	−0.790823	−0.395412	−	0.918504i	$-0.629398\pi$
$107$	−8.18034	−0.790823	−0.395412	+	0.918504i	$0.629398\pi$
$108$	24.2705	2.33543
$109$	−11.7082	−1.12144	−0.560721	−	0.828005i	$-0.689476\pi$
$109$	−11.7082	−1.12144	−0.560721	+	0.828005i	$0.689476\pi$
$110$	−8.47214	−0.807786
$111$	9.70820	0.921462
$112$	12.1803	1.15093
$113$	5.23607	0.492568	0.246284	−	0.969198i	$-0.420790\pi$
$113$	5.23607	0.492568	0.246284	+	0.969198i	$0.420790\pi$
$114$	−14.9443	−1.39966
$115$	1.00000	0.0932505
$116$	−2.56231	−0.237904
$117$	12.4721	1.15305
$118$	−23.4164	−2.15566
$119$	3.05573	0.280118
$120$	−7.47214	−0.682110
$121$	−0.527864	−0.0479876
$122$	−31.8885	−2.88705
$123$	7.47214	0.673740
$124$	20.5623	1.84655
$125$	−1.00000	−0.0894427
$126$	6.47214	0.576584
$127$	−18.4164	−1.63419	−0.817096	−	0.576502i	$-0.804417\pi$
$127$	−18.4164	−1.63419	−0.817096	+	0.576502i	$0.804417\pi$
$128$	−1.09017	−0.0963583
$129$	−3.70820	−0.326489
$130$	−16.3262	−1.43191
$131$	0.236068	0.0206254	0.0103127	−	0.999947i	$-0.496717\pi$
$131$	0.236068	0.0206254	0.0103127	+	0.999947i	$0.496717\pi$
$132$	15.7082	1.36722
$133$	−7.05573	−0.611809
$134$	−25.4164	−2.19564
$135$	−5.00000	−0.430331
$136$	−18.4721	−1.58397
$137$	8.94427	0.764161	0.382080	−	0.924129i	$-0.375208\pi$
$137$	8.94427	0.764161	0.382080	+	0.924129i	$0.375208\pi$
$138$	−2.61803	−0.222862
$139$	−2.70820	−0.229707	−0.114853	−	0.993382i	$-0.536640\pi$
$139$	−2.70820	−0.229707	−0.114853	+	0.993382i	$0.536640\pi$
$140$	−6.00000	−0.507093
$141$	−9.47214	−0.797698
$142$	16.3262	1.37007
$143$	20.1803	1.68756
$144$	−19.7082	−1.64235
$145$	0.527864	0.0438367
$146$	17.5623	1.45347
$147$	5.47214	0.451334
$148$	−47.1246	−3.87362
$149$	−20.4721	−1.67714	−0.838571	−	0.544792i	$-0.816609\pi$
$149$	−20.4721	−1.67714	−0.838571	+	0.544792i	$0.816609\pi$
$150$	2.61803	0.213762
$151$	−18.2361	−1.48403	−0.742015	−	0.670383i	$-0.766130\pi$
$151$	−18.2361	−1.48403	−0.742015	+	0.670383i	$0.766130\pi$
$152$	42.6525	3.45957
$153$	−4.94427	−0.399721
$154$	10.4721	0.843869
$155$	−4.23607	−0.340249
$156$	30.2705	2.42358
$157$	7.70820	0.615182	0.307591	−	0.951519i	$-0.400477\pi$
$157$	7.70820	0.615182	0.307591	+	0.951519i	$0.400477\pi$
$158$	−22.9443	−1.82535
$159$	6.00000	0.475831
$160$	10.8541	0.858092
$161$	−1.23607	−0.0974158
$162$	−2.61803	−0.205692
$163$	8.41641	0.659224	0.329612	−	0.944116i	$-0.393082\pi$
$163$	8.41641	0.659224	0.329612	+	0.944116i	$0.393082\pi$
$164$	−36.2705	−2.83225
$165$	−3.23607	−0.251928
$166$	16.9443	1.31513
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	9.23607	0.712578
$169$	25.8885	1.99143
$170$	6.47214	0.496390
$171$	11.4164	0.873035
$172$	18.0000	1.37249
$173$	−6.94427	−0.527963	−0.263982	−	0.964528i	$-0.585036\pi$
$173$	−6.94427	−0.527963	−0.263982	+	0.964528i	$0.585036\pi$
$174$	−1.38197	−0.104767
$175$	1.23607	0.0934380
$176$	−31.8885	−2.40369
$177$	−8.94427	−0.672293
$178$	−7.23607	−0.542366
$179$	6.23607	0.466106	0.233053	−	0.972464i	$-0.425129\pi$
$179$	6.23607	0.466106	0.233053	+	0.972464i	$0.425129\pi$
$180$	9.70820	0.723607
$181$	20.7639	1.54337	0.771685	−	0.636004i	$-0.219414\pi$
$181$	20.7639	1.54337	0.771685	+	0.636004i	$0.219414\pi$
$182$	20.1803	1.49587
$183$	−12.1803	−0.900397
$184$	7.47214	0.550853
$185$	9.70820	0.713761
$186$	11.0902	0.813171
$187$	−8.00000	−0.585018
$188$	45.9787	3.35334
$189$	6.18034	0.449554
$190$	−14.9443	−1.08417
$191$	−6.00000	−0.434145	−0.217072	−	0.976156i	$-0.569651\pi$
$191$	−6.00000	−0.434145	−0.217072	+	0.976156i	$0.569651\pi$
$192$	−8.70820	−0.628460
$193$	0.708204	0.0509776	0.0254888	−	0.999675i	$-0.491886\pi$
$193$	0.708204	0.0509776	0.0254888	+	0.999675i	$0.491886\pi$
$194$	16.1803	1.16168
$195$	−6.23607	−0.446574
$196$	−26.5623	−1.89731
$197$	−13.7639	−0.980640	−0.490320	−	0.871543i	$-0.663120\pi$
$197$	−13.7639	−0.980640	−0.490320	+	0.871543i	$0.663120\pi$
$198$	−16.9443	−1.20418
$199$	5.70820	0.404644	0.202322	−	0.979319i	$-0.435151\pi$
$199$	5.70820	0.404644	0.202322	+	0.979319i	$0.435151\pi$
$200$	−7.47214	−0.528360
$201$	−9.70820	−0.684764
$202$	−7.70820	−0.542347
$203$	−0.652476	−0.0457948
$204$	−12.0000	−0.840168
$205$	7.47214	0.521877
$206$	16.9443	1.18056
$207$	2.00000	0.139010
$208$	−61.4508	−4.26085
$209$	18.4721	1.27774
$210$	−3.23607	−0.223310
$211$	−9.52786	−0.655925	−0.327963	−	0.944691i	$-0.606362\pi$
$211$	−9.52786	−0.655925	−0.327963	+	0.944691i	$0.606362\pi$
$212$	−29.1246	−2.00029
$213$	6.23607	0.427288
$214$	21.4164	1.46400
$215$	−3.70820	−0.252897
$216$	−37.3607	−2.54207
$217$	5.23607	0.355447
$218$	30.6525	2.07605
$219$	6.70820	0.453298
$220$	15.7082	1.05905
$221$	−15.4164	−1.03702
$222$	−25.4164	−1.70584
$223$	4.00000	0.267860	0.133930	−	0.990991i	$-0.457240\pi$
$223$	4.00000	0.267860	0.133930	+	0.990991i	$0.457240\pi$
$224$	−13.4164	−0.896421
$225$	−2.00000	−0.133333
$226$	−13.7082	−0.911856
$227$	−12.6525	−0.839774	−0.419887	−	0.907576i	$-0.637930\pi$
$227$	−12.6525	−0.839774	−0.419887	+	0.907576i	$0.637930\pi$
$228$	27.7082	1.83502
$229$	−10.1803	−0.672736	−0.336368	−	0.941731i	$-0.609199\pi$
$229$	−10.1803	−0.672736	−0.336368	+	0.941731i	$0.609199\pi$
$230$	−2.61803	−0.172628
$231$	4.00000	0.263181
$232$	3.94427	0.258954
$233$	9.18034	0.601424	0.300712	−	0.953715i	$-0.402776\pi$
$233$	9.18034	0.601424	0.300712	+	0.953715i	$0.402776\pi$
$234$	−32.6525	−2.13456
$235$	−9.47214	−0.617894
$236$	43.4164	2.82617
$237$	−8.76393	−0.569279
$238$	−8.00000	−0.518563
$239$	−2.23607	−0.144639	−0.0723196	−	0.997382i	$-0.523040\pi$
$239$	−2.23607	−0.144639	−0.0723196	+	0.997382i	$0.523040\pi$
$240$	9.85410	0.636080
$241$	20.4721	1.31873	0.659363	−	0.751825i	$-0.270826\pi$
$241$	20.4721	1.31873	0.659363	+	0.751825i	$0.270826\pi$
$242$	1.38197	0.0888361
$243$	−16.0000	−1.02640
$244$	59.1246	3.78507
$245$	5.47214	0.349602
$246$	−19.5623	−1.24725
$247$	35.5967	2.26497
$248$	−31.6525	−2.00993
$249$	6.47214	0.410155
$250$	2.61803	0.165579
$251$	16.9443	1.06951	0.534756	−	0.845006i	$-0.320403\pi$
$251$	16.9443	1.06951	0.534756	+	0.845006i	$0.320403\pi$
$252$	−12.0000	−0.755929
$253$	3.23607	0.203450
$254$	48.2148	3.02526
$255$	2.47214	0.154811
$256$	−14.5623	−0.910144
$257$	−11.1803	−0.697410	−0.348705	−	0.937232i	$-0.613379\pi$
$257$	−11.1803	−0.697410	−0.348705	+	0.937232i	$0.613379\pi$
$258$	9.70820	0.604406
$259$	−12.0000	−0.745644
$260$	30.2705	1.87730
$261$	1.05573	0.0653479
$262$	−0.618034	−0.0381823
$263$	21.7082	1.33859	0.669293	−	0.742999i	$-0.266597\pi$
$263$	21.7082	1.33859	0.669293	+	0.742999i	$0.266597\pi$
$264$	−24.1803	−1.48820
$265$	6.00000	0.368577
$266$	18.4721	1.13260
$267$	−2.76393	−0.169150
$268$	47.1246	2.87859
$269$	−12.5279	−0.763837	−0.381919	−	0.924196i	$-0.624737\pi$
$269$	−12.5279	−0.763837	−0.381919	+	0.924196i	$0.624737\pi$
$270$	13.0902	0.796642
$271$	−7.41641	−0.450515	−0.225257	−	0.974299i	$-0.572322\pi$
$271$	−7.41641	−0.450515	−0.225257	+	0.974299i	$0.572322\pi$
$272$	24.3607	1.47708
$273$	7.70820	0.466522
$274$	−23.4164	−1.41464
$275$	−3.23607	−0.195142
$276$	4.85410	0.292183
$277$	−21.1803	−1.27260	−0.636302	−	0.771440i	$-0.719537\pi$
$277$	−21.1803	−1.27260	−0.636302	+	0.771440i	$0.719537\pi$
$278$	7.09017	0.425240
$279$	−8.47214	−0.507214
$280$	9.23607	0.551961
$281$	−1.23607	−0.0737376	−0.0368688	−	0.999320i	$-0.511738\pi$
$281$	−1.23607	−0.0737376	−0.0368688	+	0.999320i	$0.511738\pi$
$282$	24.7984	1.47672
$283$	11.5279	0.685260	0.342630	−	0.939470i	$-0.388682\pi$
$283$	11.5279	0.685260	0.342630	+	0.939470i	$0.388682\pi$
$284$	−30.2705	−1.79622
$285$	−5.70820	−0.338125
$286$	−52.8328	−3.12407
$287$	−9.23607	−0.545188
$288$	21.7082	1.27917
$289$	−10.8885	−0.640503
$290$	−1.38197	−0.0811518
$291$	6.18034	0.362298
$292$	−32.5623	−1.90556
$293$	−12.9443	−0.756212	−0.378106	−	0.925762i	$-0.623425\pi$
$293$	−12.9443	−0.756212	−0.378106	+	0.925762i	$0.623425\pi$
$294$	−14.3262	−0.835523
$295$	−8.94427	−0.520756
$296$	72.5410	4.21636
$297$	−16.1803	−0.938879
$298$	53.5967	3.10478
$299$	6.23607	0.360641
$300$	−4.85410	−0.280252
$301$	4.58359	0.264194
$302$	47.7426	2.74728
$303$	−2.94427	−0.169144
$304$	−56.2492	−3.22611
$305$	−12.1803	−0.697444
$306$	12.9443	0.739975
$307$	−4.58359	−0.261599	−0.130800	−	0.991409i	$-0.541754\pi$
$307$	−4.58359	−0.261599	−0.130800	+	0.991409i	$0.541754\pi$
$308$	−19.4164	−1.10635
$309$	6.47214	0.368187
$310$	11.0902	0.629879
$311$	−6.81966	−0.386707	−0.193354	−	0.981129i	$-0.561936\pi$
$311$	−6.81966	−0.386707	−0.193354	+	0.981129i	$0.561936\pi$
$312$	−46.5967	−2.63802
$313$	7.41641	0.419200	0.209600	−	0.977787i	$-0.432784\pi$
$313$	7.41641	0.419200	0.209600	+	0.977787i	$0.432784\pi$
$314$	−20.1803	−1.13884
$315$	2.47214	0.139289
$316$	42.5410	2.39312
$317$	−34.9443	−1.96267	−0.981333	−	0.192317i	$-0.938400\pi$
$317$	−34.9443	−1.96267	−0.981333	+	0.192317i	$0.938400\pi$
$318$	−15.7082	−0.880872
$319$	1.70820	0.0956411
$320$	−8.70820	−0.486803
$321$	8.18034	0.456582
$322$	3.23607	0.180339
$323$	−14.1115	−0.785182
$324$	4.85410	0.269672
$325$	−6.23607	−0.345915
$326$	−22.0344	−1.22037
$327$	11.7082	0.647465
$328$	55.8328	3.08285
$329$	11.7082	0.645494
$330$	8.47214	0.466376
$331$	−14.7082	−0.808436	−0.404218	−	0.914663i	$-0.632456\pi$
$331$	−14.7082	−0.808436	−0.404218	+	0.914663i	$0.632456\pi$
$332$	−31.4164	−1.72420
$333$	19.4164	1.06401
$334$	31.4164	1.71903
$335$	−9.70820	−0.530416
$336$	−12.1803	−0.664492
$337$	19.1246	1.04178	0.520892	−	0.853623i	$-0.325599\pi$
$337$	19.1246	1.04178	0.520892	+	0.853623i	$0.325599\pi$
$338$	−67.7771	−3.68659
$339$	−5.23607	−0.284384
$340$	−12.0000	−0.650791
$341$	−13.7082	−0.742341
$342$	−29.8885	−1.61619
$343$	−15.4164	−0.832408
$344$	−27.7082	−1.49393
$345$	−1.00000	−0.0538382
$346$	18.1803	0.977381
$347$	19.0557	1.02297	0.511483	−	0.859294i	$-0.329096\pi$
$347$	19.0557	1.02297	0.511483	+	0.859294i	$0.329096\pi$
$348$	2.56231	0.137354
$349$	21.3607	1.14341	0.571705	−	0.820459i	$-0.306282\pi$
$349$	21.3607	1.14341	0.571705	+	0.820459i	$0.306282\pi$
$350$	−3.23607	−0.172975
$351$	−31.1803	−1.66428
$352$	35.1246	1.87215
$353$	0.347524	0.0184968	0.00924842	−	0.999957i	$-0.497056\pi$
$353$	0.347524	0.0184968	0.00924842	+	0.999957i	$0.497056\pi$
$354$	23.4164	1.24457
$355$	6.23607	0.330976
$356$	13.4164	0.711068
$357$	−3.05573	−0.161726
$358$	−16.3262	−0.862868
$359$	−24.7639	−1.30699	−0.653495	−	0.756931i	$-0.726698\pi$
$359$	−24.7639	−1.30699	−0.653495	+	0.756931i	$0.726698\pi$
$360$	−14.9443	−0.787632
$361$	13.5836	0.714926
$362$	−54.3607	−2.85713
$363$	0.527864	0.0277057
$364$	−37.4164	−1.96115
$365$	6.70820	0.351123
$366$	31.8885	1.66684
$367$	−12.0000	−0.626395	−0.313197	−	0.949688i	$-0.601400\pi$
$367$	−12.0000	−0.626395	−0.313197	+	0.949688i	$0.601400\pi$
$368$	−9.85410	−0.513681
$369$	14.9443	0.777968
$370$	−25.4164	−1.32134
$371$	−7.41641	−0.385041
$372$	−20.5623	−1.06611
$373$	−6.58359	−0.340885	−0.170443	−	0.985368i	$-0.554520\pi$
$373$	−6.58359	−0.340885	−0.170443	+	0.985368i	$0.554520\pi$
$374$	20.9443	1.08300
$375$	1.00000	0.0516398
$376$	−70.7771	−3.65005
$377$	3.29180	0.169536
$378$	−16.1803	−0.832227
$379$	24.0000	1.23280	0.616399	−	0.787434i	$-0.288591\pi$
$379$	24.0000	1.23280	0.616399	+	0.787434i	$0.288591\pi$
$380$	27.7082	1.42140
$381$	18.4164	0.943501
$382$	15.7082	0.803702
$383$	18.1803	0.928972	0.464486	−	0.885580i	$-0.346239\pi$
$383$	18.1803	0.928972	0.464486	+	0.885580i	$0.346239\pi$
$384$	1.09017	0.0556325
$385$	4.00000	0.203859
$386$	−1.85410	−0.0943713
$387$	−7.41641	−0.376997
$388$	−30.0000	−1.52302
$389$	−24.0000	−1.21685	−0.608424	−	0.793612i	$-0.708198\pi$
$389$	−24.0000	−1.21685	−0.608424	+	0.793612i	$0.708198\pi$
$390$	16.3262	0.826711
$391$	−2.47214	−0.125021
$392$	40.8885	2.06518
$393$	−0.236068	−0.0119081
$394$	36.0344	1.81539
$395$	−8.76393	−0.440961
$396$	31.4164	1.57873
$397$	17.7639	0.891546	0.445773	−	0.895146i	$-0.352929\pi$
$397$	17.7639	0.891546	0.445773	+	0.895146i	$0.352929\pi$
$398$	−14.9443	−0.749089
$399$	7.05573	0.353228
$400$	9.85410	0.492705
$401$	−4.47214	−0.223328	−0.111664	−	0.993746i	$-0.535618\pi$
$401$	−4.47214	−0.223328	−0.111664	+	0.993746i	$0.535618\pi$
$402$	25.4164	1.26766
$403$	−26.4164	−1.31590
$404$	14.2918	0.711043
$405$	−1.00000	−0.0496904
$406$	1.70820	0.0847767
$407$	31.4164	1.55725
$408$	18.4721	0.914507
$409$	5.00000	0.247234	0.123617	−	0.992330i	$-0.460551\pi$
$409$	5.00000	0.247234	0.123617	+	0.992330i	$0.460551\pi$
$410$	−19.5623	−0.966113
$411$	−8.94427	−0.441188
$412$	−31.4164	−1.54778
$413$	11.0557	0.544017
$414$	−5.23607	−0.257339
$415$	6.47214	0.317705
$416$	67.6869	3.31862
$417$	2.70820	0.132621
$418$	−48.3607	−2.36540
$419$	−5.23607	−0.255799	−0.127899	−	0.991787i	$-0.540823\pi$
$419$	−5.23607	−0.255799	−0.127899	+	0.991787i	$0.540823\pi$
$420$	6.00000	0.292770
$421$	35.1246	1.71187	0.855934	−	0.517084i	$-0.172983\pi$
$421$	35.1246	1.71187	0.855934	+	0.517084i	$0.172983\pi$
$422$	24.9443	1.21427
$423$	−18.9443	−0.921102
$424$	44.8328	2.17727
$425$	2.47214	0.119916
$426$	−16.3262	−0.791009
$427$	15.0557	0.728598
$428$	−39.7082	−1.91937
$429$	−20.1803	−0.974316
$430$	9.70820	0.468171
$431$	−20.6525	−0.994795	−0.497397	−	0.867523i	$-0.665711\pi$
$431$	−20.6525	−0.994795	−0.497397	+	0.867523i	$0.665711\pi$
$432$	49.2705	2.37053
$433$	8.47214	0.407145	0.203572	−	0.979060i	$-0.434745\pi$
$433$	8.47214	0.407145	0.203572	+	0.979060i	$0.434745\pi$
$434$	−13.7082	−0.658015
$435$	−0.527864	−0.0253091
$436$	−56.8328	−2.72180
$437$	5.70820	0.273060
$438$	−17.5623	−0.839159
$439$	−22.7082	−1.08380	−0.541902	−	0.840442i	$-0.682296\pi$
$439$	−22.7082	−1.08380	−0.541902	+	0.840442i	$0.682296\pi$
$440$	−24.1803	−1.15275
$441$	10.9443	0.521156
$442$	40.3607	1.91976
$443$	8.52786	0.405171	0.202586	−	0.979265i	$-0.435066\pi$
$443$	8.52786	0.405171	0.202586	+	0.979265i	$0.435066\pi$
$444$	47.1246	2.23644
$445$	−2.76393	−0.131023
$446$	−10.4721	−0.495870
$447$	20.4721	0.968299
$448$	10.7639	0.508548
$449$	−20.8328	−0.983161	−0.491581	−	0.870832i	$-0.663581\pi$
$449$	−20.8328	−0.983161	−0.491581	+	0.870832i	$0.663581\pi$
$450$	5.23607	0.246831
$451$	24.1803	1.13861
$452$	25.4164	1.19549
$453$	18.2361	0.856805
$454$	33.1246	1.55462
$455$	7.70820	0.361366
$456$	−42.6525	−1.99739
$457$	−11.4164	−0.534037	−0.267019	−	0.963691i	$-0.586038\pi$
$457$	−11.4164	−0.534037	−0.267019	+	0.963691i	$0.586038\pi$
$458$	26.6525	1.24539
$459$	12.3607	0.576947
$460$	4.85410	0.226324
$461$	21.0000	0.978068	0.489034	−	0.872265i	$-0.337349\pi$
$461$	21.0000	0.978068	0.489034	+	0.872265i	$0.337349\pi$
$462$	−10.4721	−0.487208
$463$	−6.47214	−0.300786	−0.150393	−	0.988626i	$-0.548054\pi$
$463$	−6.47214	−0.300786	−0.150393	+	0.988626i	$0.548054\pi$
$464$	−5.20163	−0.241479
$465$	4.23607	0.196443
$466$	−24.0344	−1.11337
$467$	15.5279	0.718544	0.359272	−	0.933233i	$-0.383025\pi$
$467$	15.5279	0.718544	0.359272	+	0.933233i	$0.383025\pi$
$468$	60.5410	2.79851
$469$	12.0000	0.554109
$470$	24.7984	1.14386
$471$	−7.70820	−0.355175
$472$	−66.8328	−3.07623
$473$	−12.0000	−0.551761
$474$	22.9443	1.05387
$475$	−5.70820	−0.261910
$476$	14.8328	0.679861
$477$	12.0000	0.549442
$478$	5.85410	0.267760
$479$	26.8328	1.22602	0.613011	−	0.790074i	$-0.289958\pi$
$479$	26.8328	1.22602	0.613011	+	0.790074i	$0.289958\pi$
$480$	−10.8541	−0.495420
$481$	60.5410	2.76043
$482$	−53.5967	−2.44126
$483$	1.23607	0.0562430
$484$	−2.56231	−0.116468
$485$	6.18034	0.280635
$486$	41.8885	1.90010
$487$	6.88854	0.312150	0.156075	−	0.987745i	$-0.450116\pi$
$487$	6.88854	0.312150	0.156075	+	0.987745i	$0.450116\pi$
$488$	−91.0132	−4.11997
$489$	−8.41641	−0.380603
$490$	−14.3262	−0.647193
$491$	1.65248	0.0745752	0.0372876	−	0.999305i	$-0.488128\pi$
$491$	1.65248	0.0745752	0.0372876	+	0.999305i	$0.488128\pi$
$492$	36.2705	1.63520
$493$	−1.30495	−0.0587721
$494$	−93.1935	−4.19297
$495$	−6.47214	−0.290901
$496$	41.7426	1.87430
$497$	−7.70820	−0.345760
$498$	−16.9443	−0.759291
$499$	−36.1246	−1.61716	−0.808580	−	0.588386i	$-0.799763\pi$
$499$	−36.1246	−1.61716	−0.808580	+	0.588386i	$0.799763\pi$
$500$	−4.85410	−0.217082
$501$	12.0000	0.536120
$502$	−44.3607	−1.97991
$503$	6.00000	0.267527	0.133763	−	0.991013i	$-0.457294\pi$
$503$	6.00000	0.267527	0.133763	+	0.991013i	$0.457294\pi$
$504$	18.4721	0.822814
$505$	−2.94427	−0.131018
$506$	−8.47214	−0.376632
$507$	−25.8885	−1.14975
$508$	−89.3951	−3.96627
$509$	−11.9443	−0.529421	−0.264710	−	0.964328i	$-0.585276\pi$
$509$	−11.9443	−0.529421	−0.264710	+	0.964328i	$0.585276\pi$
$510$	−6.47214	−0.286591
$511$	−8.29180	−0.366807
$512$	40.3050	1.78124
$513$	−28.5410	−1.26012
$514$	29.2705	1.29107
$515$	6.47214	0.285196
$516$	−18.0000	−0.792406
$517$	−30.6525	−1.34809
$518$	31.4164	1.38036
$519$	6.94427	0.304820
$520$	−46.5967	−2.04340
$521$	−21.8885	−0.958955	−0.479477	−	0.877554i	$-0.659174\pi$
$521$	−21.8885	−0.958955	−0.479477	+	0.877554i	$0.659174\pi$
$522$	−2.76393	−0.120974
$523$	23.1246	1.01117	0.505584	−	0.862777i	$-0.331277\pi$
$523$	23.1246	1.01117	0.505584	+	0.862777i	$0.331277\pi$
$524$	1.14590	0.0500588
$525$	−1.23607	−0.0539464
$526$	−56.8328	−2.47803
$527$	10.4721	0.456173
$528$	31.8885	1.38777
$529$	1.00000	0.0434783
$530$	−15.7082	−0.682321
$531$	−17.8885	−0.776297
$532$	−34.2492	−1.48489
$533$	46.5967	2.01833
$534$	7.23607	0.313135
$535$	8.18034	0.353667
$536$	−72.5410	−3.13329
$537$	−6.23607	−0.269106
$538$	32.7984	1.41404
$539$	17.7082	0.762746
$540$	−24.2705	−1.04444
$541$	−1.11146	−0.0477852	−0.0238926	−	0.999715i	$-0.507606\pi$
$541$	−1.11146	−0.0477852	−0.0238926	+	0.999715i	$0.507606\pi$
$542$	19.4164	0.834006
$543$	−20.7639	−0.891066
$544$	−26.8328	−1.15045
$545$	11.7082	0.501524
$546$	−20.1803	−0.863639
$547$	17.0000	0.726868	0.363434	−	0.931620i	$-0.381604\pi$
$547$	17.0000	0.726868	0.363434	+	0.931620i	$0.381604\pi$
$548$	43.4164	1.85466
$549$	−24.3607	−1.03969
$550$	8.47214	0.361253
$551$	3.01316	0.128365
$552$	−7.47214	−0.318035
$553$	10.8328	0.460658
$554$	55.4508	2.35588
$555$	−9.70820	−0.412090
$556$	−13.1459	−0.557510
$557$	21.5967	0.915084	0.457542	−	0.889188i	$-0.348730\pi$
$557$	21.5967	0.915084	0.457542	+	0.889188i	$0.348730\pi$
$558$	22.1803	0.938969
$559$	−23.1246	−0.978067
$560$	−12.1803	−0.514713
$561$	8.00000	0.337760
$562$	3.23607	0.136505
$563$	24.3607	1.02668	0.513340	−	0.858185i	$-0.328408\pi$
$563$	24.3607	1.02668	0.513340	+	0.858185i	$0.328408\pi$
$564$	−45.9787	−1.93605
$565$	−5.23607	−0.220283
$566$	−30.1803	−1.26857
$567$	1.23607	0.0519100
$568$	46.5967	1.95516
$569$	2.36068	0.0989648	0.0494824	−	0.998775i	$-0.484243\pi$
$569$	2.36068	0.0989648	0.0494824	+	0.998775i	$0.484243\pi$
$570$	14.9443	0.625947
$571$	30.8328	1.29031	0.645157	−	0.764050i	$-0.276792\pi$
$571$	30.8328	1.29031	0.645157	+	0.764050i	$0.276792\pi$
$572$	97.9574	4.09581
$573$	6.00000	0.250654
$574$	24.1803	1.00927
$575$	−1.00000	−0.0417029
$576$	−17.4164	−0.725684
$577$	28.7082	1.19514	0.597569	−	0.801817i	$-0.296133\pi$
$577$	28.7082	1.19514	0.597569	+	0.801817i	$0.296133\pi$
$578$	28.5066	1.18572
$579$	−0.708204	−0.0294320
$580$	2.56231	0.106394
$581$	−8.00000	−0.331896
$582$	−16.1803	−0.670697
$583$	19.4164	0.804145
$584$	50.1246	2.07417
$585$	−12.4721	−0.515659
$586$	33.8885	1.39992
$587$	40.3050	1.66356	0.831782	−	0.555103i	$-0.187321\pi$
$587$	40.3050	1.66356	0.831782	+	0.555103i	$0.187321\pi$
$588$	26.5623	1.09541
$589$	−24.1803	−0.996334
$590$	23.4164	0.964038
$591$	13.7639	0.566173
$592$	−95.6656	−3.93183
$593$	−1.63932	−0.0673188	−0.0336594	−	0.999433i	$-0.510716\pi$
$593$	−1.63932	−0.0673188	−0.0336594	+	0.999433i	$0.510716\pi$
$594$	42.3607	1.73808
$595$	−3.05573	−0.125273
$596$	−99.3738	−4.07051
$597$	−5.70820	−0.233621
$598$	−16.3262	−0.667630
$599$	−47.7771	−1.95212	−0.976059	−	0.217504i	$-0.930208\pi$
$599$	−47.7771	−1.95212	−0.976059	+	0.217504i	$0.930208\pi$
$600$	7.47214	0.305049
$601$	−3.94427	−0.160890	−0.0804451	−	0.996759i	$-0.525634\pi$
$601$	−3.94427	−0.160890	−0.0804451	+	0.996759i	$0.525634\pi$
$602$	−12.0000	−0.489083
$603$	−19.4164	−0.790697
$604$	−88.5197	−3.60182
$605$	0.527864	0.0214607
$606$	7.70820	0.313124
$607$	−18.4721	−0.749761	−0.374880	−	0.927073i	$-0.622316\pi$
$607$	−18.4721	−0.749761	−0.374880	+	0.927073i	$0.622316\pi$
$608$	61.9574	2.51271
$609$	0.652476	0.0264397
$610$	31.8885	1.29113
$611$	−59.0689	−2.38967
$612$	−24.0000	−0.970143
$613$	−17.3050	−0.698940	−0.349470	−	0.936947i	$-0.613638\pi$
$613$	−17.3050	−0.698940	−0.349470	+	0.936947i	$0.613638\pi$
$614$	12.0000	0.484281
$615$	−7.47214	−0.301306
$616$	29.8885	1.20424
$617$	12.1803	0.490362	0.245181	−	0.969477i	$-0.421153\pi$
$617$	12.1803	0.490362	0.245181	+	0.969477i	$0.421153\pi$
$618$	−16.9443	−0.681599
$619$	−36.3607	−1.46146	−0.730730	−	0.682667i	$-0.760820\pi$
$619$	−36.3607	−1.46146	−0.730730	+	0.682667i	$0.760820\pi$
$620$	−20.5623	−0.825802
$621$	−5.00000	−0.200643
$622$	17.8541	0.715884
$623$	3.41641	0.136875
$624$	61.4508	2.46000
$625$	1.00000	0.0400000
$626$	−19.4164	−0.776036
$627$	−18.4721	−0.737706
$628$	37.4164	1.49308
$629$	−24.0000	−0.956943
$630$	−6.47214	−0.257856
$631$	−3.70820	−0.147621	−0.0738106	−	0.997272i	$-0.523516\pi$
$631$	−3.70820	−0.147621	−0.0738106	+	0.997272i	$0.523516\pi$
$632$	−65.4853	−2.60487
$633$	9.52786	0.378699
$634$	91.4853	3.63335
$635$	18.4164	0.730833
$636$	29.1246	1.15487
$637$	34.1246	1.35207
$638$	−4.47214	−0.177054
$639$	12.4721	0.493390
$640$	1.09017	0.0430928
$641$	50.1803	1.98200	0.991002	−	0.133846i	$-0.0427328\pi$
$641$	50.1803	1.98200	0.991002	+	0.133846i	$0.0427328\pi$
$642$	−21.4164	−0.845238
$643$	−20.8328	−0.821566	−0.410783	−	0.911733i	$-0.634745\pi$
$643$	−20.8328	−0.821566	−0.410783	+	0.911733i	$0.634745\pi$
$644$	−6.00000	−0.236433
$645$	3.70820	0.146010
$646$	36.9443	1.45355
$647$	−23.8328	−0.936965	−0.468482	−	0.883473i	$-0.655199\pi$
$647$	−23.8328	−0.936965	−0.468482	+	0.883473i	$0.655199\pi$
$648$	−7.47214	−0.293533
$649$	−28.9443	−1.13616
$650$	16.3262	0.640368
$651$	−5.23607	−0.205218
$652$	40.8541	1.59997
$653$	−1.65248	−0.0646664	−0.0323332	−	0.999477i	$-0.510294\pi$
$653$	−1.65248	−0.0646664	−0.0323332	+	0.999477i	$0.510294\pi$
$654$	−30.6525	−1.19861
$655$	−0.236068	−0.00922394
$656$	−73.6312	−2.87481
$657$	13.4164	0.523424
$658$	−30.6525	−1.19496
$659$	−24.9443	−0.971691	−0.485845	−	0.874045i	$-0.661488\pi$
$659$	−24.9443	−0.971691	−0.485845	+	0.874045i	$0.661488\pi$
$660$	−15.7082	−0.611441
$661$	−26.0000	−1.01128	−0.505641	−	0.862744i	$-0.668744\pi$
$661$	−26.0000	−1.01128	−0.505641	+	0.862744i	$0.668744\pi$
$662$	38.5066	1.49660
$663$	15.4164	0.598724
$664$	48.3607	1.87676
$665$	7.05573	0.273609
$666$	−50.8328	−1.96973
$667$	0.527864	0.0204390
$668$	−58.2492	−2.25373
$669$	−4.00000	−0.154649
$670$	25.4164	0.981922
$671$	−39.4164	−1.52165
$672$	13.4164	0.517549
$673$	−39.5410	−1.52419	−0.762097	−	0.647463i	$-0.775830\pi$
$673$	−39.5410	−1.52419	−0.762097	+	0.647463i	$0.775830\pi$
$674$	−50.0689	−1.92858
$675$	5.00000	0.192450
$676$	125.666	4.83329
$677$	−6.00000	−0.230599	−0.115299	−	0.993331i	$-0.536783\pi$
$677$	−6.00000	−0.230599	−0.115299	+	0.993331i	$0.536783\pi$
$678$	13.7082	0.526460
$679$	−7.63932	−0.293170
$680$	18.4721	0.708374
$681$	12.6525	0.484844
$682$	35.8885	1.37424
$683$	−22.4164	−0.857740	−0.428870	−	0.903366i	$-0.641088\pi$
$683$	−22.4164	−0.857740	−0.428870	+	0.903366i	$0.641088\pi$
$684$	55.4164	2.11890
$685$	−8.94427	−0.341743
$686$	40.3607	1.54098
$687$	10.1803	0.388404
$688$	36.5410	1.39311
$689$	37.4164	1.42545
$690$	2.61803	0.0996669
$691$	−14.4721	−0.550546	−0.275273	−	0.961366i	$-0.588768\pi$
$691$	−14.4721	−0.550546	−0.275273	+	0.961366i	$0.588768\pi$
$692$	−33.7082	−1.28139
$693$	8.00000	0.303895
$694$	−49.8885	−1.89374
$695$	2.70820	0.102728
$696$	−3.94427	−0.149507
$697$	−18.4721	−0.699682
$698$	−55.9230	−2.11672
$699$	−9.18034	−0.347232
$700$	6.00000	0.226779
$701$	−45.5967	−1.72217	−0.861083	−	0.508465i	$-0.830213\pi$
$701$	−45.5967	−1.72217	−0.861083	+	0.508465i	$0.830213\pi$
$702$	81.6312	3.08097
$703$	55.4164	2.09007
$704$	−28.1803	−1.06209
$705$	9.47214	0.356741
$706$	−0.909830	−0.0342419
$707$	3.63932	0.136871
$708$	−43.4164	−1.63169
$709$	−23.7082	−0.890380	−0.445190	−	0.895436i	$-0.646864\pi$
$709$	−23.7082	−0.890380	−0.445190	+	0.895436i	$0.646864\pi$
$710$	−16.3262	−0.612713
$711$	−17.5279	−0.657346
$712$	−20.6525	−0.773984
$713$	−4.23607	−0.158642
$714$	8.00000	0.299392
$715$	−20.1803	−0.754702
$716$	30.2705	1.13126
$717$	2.23607	0.0835075
$718$	64.8328	2.41954
$719$	−7.41641	−0.276585	−0.138293	−	0.990391i	$-0.544161\pi$
$719$	−7.41641	−0.276585	−0.138293	+	0.990391i	$0.544161\pi$
$720$	19.7082	0.734481
$721$	−8.00000	−0.297936
$722$	−35.5623	−1.32349
$723$	−20.4721	−0.761367
$724$	100.790	3.74584
$725$	−0.527864	−0.0196044
$726$	−1.38197	−0.0512896
$727$	−28.8328	−1.06935	−0.534675	−	0.845058i	$-0.679566\pi$
$727$	−28.8328	−1.06935	−0.534675	+	0.845058i	$0.679566\pi$
$728$	57.5967	2.13468
$729$	13.0000	0.481481
$730$	−17.5623	−0.650010
$731$	9.16718	0.339061
$732$	−59.1246	−2.18531
$733$	7.41641	0.273931	0.136966	−	0.990576i	$-0.456265\pi$
$733$	7.41641	0.273931	0.136966	+	0.990576i	$0.456265\pi$
$734$	31.4164	1.15960
$735$	−5.47214	−0.201843
$736$	10.8541	0.400088
$737$	−31.4164	−1.15724
$738$	−39.1246	−1.44020
$739$	44.4853	1.63642	0.818209	−	0.574921i	$-0.194967\pi$
$739$	44.4853	1.63642	0.818209	+	0.574921i	$0.194967\pi$
$740$	47.1246	1.73234
$741$	−35.5967	−1.30768
$742$	19.4164	0.712799
$743$	17.5279	0.643035	0.321517	−	0.946904i	$-0.395807\pi$
$743$	17.5279	0.643035	0.321517	+	0.946904i	$0.395807\pi$
$744$	31.6525	1.16044
$745$	20.4721	0.750041
$746$	17.2361	0.631057
$747$	12.9443	0.473606
$748$	−38.8328	−1.41987
$749$	−10.1115	−0.369465
$750$	−2.61803	−0.0955971
$751$	47.7082	1.74090	0.870449	−	0.492259i	$-0.163829\pi$
$751$	47.7082	1.74090	0.870449	+	0.492259i	$0.163829\pi$
$752$	93.3394	3.40374
$753$	−16.9443	−0.617484
$754$	−8.61803	−0.313850
$755$	18.2361	0.663678
$756$	30.0000	1.09109
$757$	−53.1246	−1.93085	−0.965423	−	0.260687i	$-0.916051\pi$
$757$	−53.1246	−1.93085	−0.965423	+	0.260687i	$0.916051\pi$
$758$	−62.8328	−2.28219
$759$	−3.23607	−0.117462
$760$	−42.6525	−1.54717
$761$	−32.8885	−1.19221	−0.596104	−	0.802907i	$-0.703286\pi$
$761$	−32.8885	−1.19221	−0.596104	+	0.802907i	$0.703286\pi$
$762$	−48.2148	−1.74664
$763$	−14.4721	−0.523926
$764$	−29.1246	−1.05369
$765$	4.94427	0.178761
$766$	−47.5967	−1.71974
$767$	−55.7771	−2.01399
$768$	14.5623	0.525472
$769$	54.3607	1.96030	0.980148	−	0.198267i	$-0.0635312\pi$
$769$	54.3607	1.96030	0.980148	+	0.198267i	$0.0635312\pi$
$770$	−10.4721	−0.377390
$771$	11.1803	0.402650
$772$	3.43769	0.123725
$773$	39.4853	1.42019	0.710094	−	0.704107i	$-0.248653\pi$
$773$	39.4853	1.42019	0.710094	+	0.704107i	$0.248653\pi$
$774$	19.4164	0.697908
$775$	4.23607	0.152164
$776$	46.1803	1.65778
$777$	12.0000	0.430498
$778$	62.8328	2.25267
$779$	42.6525	1.52818
$780$	−30.2705	−1.08386
$781$	20.1803	0.722109
$782$	6.47214	0.231443
$783$	−2.63932	−0.0943216
$784$	−53.9230	−1.92582
$785$	−7.70820	−0.275118
$786$	0.618034	0.0220445
$787$	45.3050	1.61495	0.807474	−	0.589904i	$-0.200834\pi$
$787$	45.3050	1.61495	0.807474	+	0.589904i	$0.200834\pi$
$788$	−66.8115	−2.38006
$789$	−21.7082	−0.772833
$790$	22.9443	0.816321
$791$	6.47214	0.230123
$792$	−48.3607	−1.71842
$793$	−75.9574	−2.69733
$794$	−46.5066	−1.65046
$795$	−6.00000	−0.212798
$796$	27.7082	0.982091
$797$	−9.70820	−0.343882	−0.171941	−	0.985107i	$-0.555004\pi$
$797$	−9.70820	−0.343882	−0.171941	+	0.985107i	$0.555004\pi$
$798$	−18.4721	−0.653907
$799$	23.4164	0.828413
$800$	−10.8541	−0.383750
$801$	−5.52786	−0.195317
$802$	11.7082	0.413431
$803$	21.7082	0.766066
$804$	−47.1246	−1.66196
$805$	1.23607	0.0435657
$806$	69.1591	2.43602
$807$	12.5279	0.441002
$808$	−22.0000	−0.773957
$809$	12.1115	0.425816	0.212908	−	0.977072i	$-0.431707\pi$
$809$	12.1115	0.425816	0.212908	+	0.977072i	$0.431707\pi$
$810$	2.61803	0.0919883
$811$	43.0689	1.51235	0.756177	−	0.654368i	$-0.227065\pi$
$811$	43.0689	1.51235	0.756177	+	0.654368i	$0.227065\pi$
$812$	−3.16718	−0.111146
$813$	7.41641	0.260105
$814$	−82.2492	−2.88283
$815$	−8.41641	−0.294814
$816$	−24.3607	−0.852794
$817$	−21.1672	−0.740546
$818$	−13.0902	−0.457687
$819$	15.4164	0.538693
$820$	36.2705	1.26662
$821$	−0.111456	−0.00388985	−0.00194492	−	0.999998i	$-0.500619\pi$
$821$	−0.111456	−0.00388985	−0.00194492	+	0.999998i	$0.500619\pi$
$822$	23.4164	0.816741
$823$	5.47214	0.190747	0.0953733	−	0.995442i	$-0.469596\pi$
$823$	5.47214	0.190747	0.0953733	+	0.995442i	$0.469596\pi$
$824$	48.3607	1.68472
$825$	3.23607	0.112665
$826$	−28.9443	−1.00710
$827$	49.3050	1.71450	0.857251	−	0.514899i	$-0.172171\pi$
$827$	49.3050	1.71450	0.857251	+	0.514899i	$0.172171\pi$
$828$	9.70820	0.337383
$829$	28.2492	0.981136	0.490568	−	0.871403i	$-0.336789\pi$
$829$	28.2492	0.981136	0.490568	+	0.871403i	$0.336789\pi$
$830$	−16.9443	−0.588144
$831$	21.1803	0.734738
$832$	−54.3050	−1.88269
$833$	−13.5279	−0.468713
$834$	−7.09017	−0.245513
$835$	12.0000	0.415277
$836$	89.6656	3.10115
$837$	21.1803	0.732100
$838$	13.7082	0.473542
$839$	33.8197	1.16758	0.583792	−	0.811903i	$-0.301568\pi$
$839$	33.8197	1.16758	0.583792	+	0.811903i	$0.301568\pi$
$840$	−9.23607	−0.318675
$841$	−28.7214	−0.990392
$842$	−91.9574	−3.16906
$843$	1.23607	0.0425724
$844$	−46.2492	−1.59196
$845$	−25.8885	−0.890593
$846$	49.5967	1.70517
$847$	−0.652476	−0.0224193
$848$	−59.1246	−2.03035
$849$	−11.5279	−0.395635
$850$	−6.47214	−0.221992
$851$	9.70820	0.332793
$852$	30.2705	1.03705
$853$	20.8328	0.713302	0.356651	−	0.934238i	$-0.383919\pi$
$853$	20.8328	0.713302	0.356651	+	0.934238i	$0.383919\pi$
$854$	−39.4164	−1.34880
$855$	−11.4164	−0.390433
$856$	61.1246	2.08920
$857$	48.5967	1.66003	0.830017	−	0.557739i	$-0.188331\pi$
$857$	48.5967	1.66003	0.830017	+	0.557739i	$0.188331\pi$
$858$	52.8328	1.80368
$859$	−32.1246	−1.09608	−0.548039	−	0.836453i	$-0.684625\pi$
$859$	−32.1246	−1.09608	−0.548039	+	0.836453i	$0.684625\pi$
$860$	−18.0000	−0.613795
$861$	9.23607	0.314764
$862$	54.0689	1.84159
$863$	−18.7771	−0.639179	−0.319590	−	0.947556i	$-0.603545\pi$
$863$	−18.7771	−0.639179	−0.319590	+	0.947556i	$0.603545\pi$
$864$	−54.2705	−1.84632
$865$	6.94427	0.236112
$866$	−22.1803	−0.753719
$867$	10.8885	0.369794
$868$	25.4164	0.862689
$869$	−28.3607	−0.962070
$870$	1.38197	0.0468530
$871$	−60.5410	−2.05135
$872$	87.4853	2.96263
$873$	12.3607	0.418346
$874$	−14.9443	−0.505498
$875$	−1.23607	−0.0417867
$876$	32.5623	1.10018
$877$	−2.58359	−0.0872417	−0.0436209	−	0.999048i	$-0.513889\pi$
$877$	−2.58359	−0.0872417	−0.0436209	+	0.999048i	$0.513889\pi$
$878$	59.4508	2.00637
$879$	12.9443	0.436599
$880$	31.8885	1.07496
$881$	17.8885	0.602680	0.301340	−	0.953517i	$-0.402566\pi$
$881$	17.8885	0.602680	0.301340	+	0.953517i	$0.402566\pi$
$882$	−28.6525	−0.964779
$883$	−33.8885	−1.14044	−0.570220	−	0.821492i	$-0.693142\pi$
$883$	−33.8885	−1.14044	−0.570220	+	0.821492i	$0.693142\pi$
$884$	−74.8328	−2.51690
$885$	8.94427	0.300658
$886$	−22.3262	−0.750065
$887$	31.9443	1.07258	0.536292	−	0.844033i	$-0.319825\pi$
$887$	31.9443	1.07258	0.536292	+	0.844033i	$0.319825\pi$
$888$	−72.5410	−2.43432
$889$	−22.7639	−0.763478
$890$	7.23607	0.242554
$891$	−3.23607	−0.108412
$892$	19.4164	0.650109
$893$	−54.0689	−1.80935
$894$	−53.5967	−1.79254
$895$	−6.23607	−0.208449
$896$	−1.34752	−0.0450176
$897$	−6.23607	−0.208216
$898$	54.5410	1.82006
$899$	−2.23607	−0.0745770
$900$	−9.70820	−0.323607
$901$	−14.8328	−0.494153
$902$	−63.3050	−2.10782
$903$	−4.58359	−0.152532
$904$	−39.1246	−1.30127
$905$	−20.7639	−0.690216
$906$	−47.7426	−1.58614
$907$	−27.2361	−0.904359	−0.452179	−	0.891927i	$-0.649353\pi$
$907$	−27.2361	−0.904359	−0.452179	+	0.891927i	$0.649353\pi$
$908$	−61.4164	−2.03818
$909$	−5.88854	−0.195311
$910$	−20.1803	−0.668972
$911$	59.2361	1.96258	0.981289	−	0.192539i	$-0.0616723\pi$
$911$	59.2361	1.96258	0.981289	+	0.192539i	$0.0616723\pi$
$912$	56.2492	1.86260
$913$	20.9443	0.693154
$914$	29.8885	0.988625
$915$	12.1803	0.402670
$916$	−49.4164	−1.63276
$917$	0.291796	0.00963596
$918$	−32.3607	−1.06806
$919$	−12.2918	−0.405469	−0.202734	−	0.979234i	$-0.564983\pi$
$919$	−12.2918	−0.405469	−0.202734	+	0.979234i	$0.564983\pi$
$920$	−7.47214	−0.246349
$921$	4.58359	0.151034
$922$	−54.9787	−1.81063
$923$	38.8885	1.28003
$924$	19.4164	0.638753
$925$	−9.70820	−0.319204
$926$	16.9443	0.556823
$927$	12.9443	0.425146
$928$	5.72949	0.188080
$929$	37.4721	1.22942	0.614710	−	0.788753i	$-0.289273\pi$
$929$	37.4721	1.22942	0.614710	+	0.788753i	$0.289273\pi$
$930$	−11.0902	−0.363661
$931$	31.2361	1.02372
$932$	44.5623	1.45969
$933$	6.81966	0.223266
$934$	−40.6525	−1.33019
$935$	8.00000	0.261628
$936$	−93.1935	−3.04612
$937$	−39.2361	−1.28179	−0.640893	−	0.767630i	$-0.721436\pi$
$937$	−39.2361	−1.28179	−0.640893	+	0.767630i	$0.721436\pi$
$938$	−31.4164	−1.02578
$939$	−7.41641	−0.242025
$940$	−45.9787	−1.49966
$941$	1.52786	0.0498069	0.0249035	−	0.999690i	$-0.492072\pi$
$941$	1.52786	0.0498069	0.0249035	+	0.999690i	$0.492072\pi$
$942$	20.1803	0.657511
$943$	7.47214	0.243326
$944$	88.1378	2.86864
$945$	−6.18034	−0.201046
$946$	31.4164	1.02144
$947$	20.3050	0.659822	0.329911	−	0.944012i	$-0.392981\pi$
$947$	20.3050	0.659822	0.329911	+	0.944012i	$0.392981\pi$
$948$	−42.5410	−1.38167
$949$	41.8328	1.35795
$950$	14.9443	0.484856
$951$	34.9443	1.13315
$952$	−22.8328	−0.740016
$953$	10.3607	0.335615	0.167808	−	0.985820i	$-0.446331\pi$
$953$	10.3607	0.335615	0.167808	+	0.985820i	$0.446331\pi$
$954$	−31.4164	−1.01714
$955$	6.00000	0.194155
$956$	−10.8541	−0.351047
$957$	−1.70820	−0.0552184
$958$	−70.2492	−2.26965
$959$	11.0557	0.357008
$960$	8.70820	0.281056
$961$	−13.0557	−0.421153
$962$	−158.498	−5.11020
$963$	16.3607	0.527216
$964$	99.3738	3.20062
$965$	−0.708204	−0.0227979
$966$	−3.23607	−0.104119
$967$	4.05573	0.130423	0.0652117	−	0.997871i	$-0.479228\pi$
$967$	4.05573	0.130423	0.0652117	+	0.997871i	$0.479228\pi$
$968$	3.94427	0.126774
$969$	14.1115	0.453325
$970$	−16.1803	−0.519519
$971$	−32.8328	−1.05366	−0.526828	−	0.849972i	$-0.676619\pi$
$971$	−32.8328	−1.05366	−0.526828	+	0.849972i	$0.676619\pi$
$972$	−77.6656	−2.49113
$973$	−3.34752	−0.107317
$974$	−18.0344	−0.577861
$975$	6.23607	0.199714
$976$	120.026	3.84195
$977$	−23.8197	−0.762058	−0.381029	−	0.924563i	$-0.624430\pi$
$977$	−23.8197	−0.762058	−0.381029	+	0.924563i	$0.624430\pi$
$978$	22.0344	0.704584
$979$	−8.94427	−0.285860
$980$	26.5623	0.848502
$981$	23.4164	0.747628
$982$	−4.32624	−0.138056
$983$	23.5279	0.750422	0.375211	−	0.926939i	$-0.377570\pi$
$983$	23.5279	0.750422	0.375211	+	0.926939i	$0.377570\pi$
$984$	−55.8328	−1.77989
$985$	13.7639	0.438555
$986$	3.41641	0.108801
$987$	−11.7082	−0.372676
$988$	172.790	5.49719
$989$	−3.70820	−0.117914
$990$	16.9443	0.538524
$991$	−11.0557	−0.351197	−0.175598	−	0.984462i	$-0.556186\pi$
$991$	−11.0557	−0.351197	−0.175598	+	0.984462i	$0.556186\pi$
$992$	−45.9787	−1.45983
$993$	14.7082	0.466751
$994$	20.1803	0.640082
$995$	−5.70820	−0.180962
$996$	31.4164	0.995467
$997$	−36.4721	−1.15508	−0.577542	−	0.816361i	$-0.695988\pi$
$997$	−36.4721	−1.15508	−0.577542	+	0.816361i	$0.695988\pi$
$998$	94.5755	2.99373
$999$	−48.5410	−1.53577

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	115.2.a.b.1.1	✓	2
3.2	odd	2		1035.2.a.m.1.2		2
4.3	odd	2		1840.2.a.p.1.1		2
5.2	odd	4		575.2.b.c.24.1		4
5.3	odd	4		575.2.b.c.24.4		4
5.4	even	2		575.2.a.g.1.2		2
7.6	odd	2		5635.2.a.n.1.1		2
8.3	odd	2		7360.2.a.bf.1.1		2
8.5	even	2		7360.2.a.bt.1.2		2
15.14	odd	2		5175.2.a.bb.1.1		2
20.19	odd	2		9200.2.a.bm.1.2		2
23.22	odd	2		2645.2.a.d.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.2.a.b.1.1	✓	2	1.1	even	1	trivial
575.2.a.g.1.2		2	5.4	even	2
575.2.b.c.24.1		4	5.2	odd	4
575.2.b.c.24.4		4	5.3	odd	4
1035.2.a.m.1.2		2	3.2	odd	2
1840.2.a.p.1.1		2	4.3	odd	2
2645.2.a.d.1.1		2	23.22	odd	2
5175.2.a.bb.1.1		2	15.14	odd	2
5635.2.a.n.1.1		2	7.6	odd	2
7360.2.a.bf.1.1		2	8.3	odd	2
7360.2.a.bt.1.2		2	8.5	even	2
9200.2.a.bm.1.2		2	20.19	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:	\( N \)	\(=\)	\( 115 = 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	115.a (trivial)

\(f(q)\)	\(=\)	\(q-2.61803 q^{2} -1.00000 q^{3} +4.85410 q^{4} -1.00000 q^{5} +2.61803 q^{6} +1.23607 q^{7} -7.47214 q^{8} -2.00000 q^{9} +O(q^{10})\)	\(q-2.61803 q^{2} -1.00000 q^{3} +4.85410 q^{4} -1.00000 q^{5} +2.61803 q^{6} +1.23607 q^{7} -7.47214 q^{8} -2.00000 q^{9} +2.61803 q^{10} -3.23607 q^{11} -4.85410 q^{12} -6.23607 q^{13} -3.23607 q^{14} +1.00000 q^{15} +9.85410 q^{16} +2.47214 q^{17} +5.23607 q^{18} -5.70820 q^{19} -4.85410 q^{20} -1.23607 q^{21} +8.47214 q^{22} -1.00000 q^{23} +7.47214 q^{24} +1.00000 q^{25} +16.3262 q^{26} +5.00000 q^{27} +6.00000 q^{28} -0.527864 q^{29} -2.61803 q^{30} +4.23607 q^{31} -10.8541 q^{32} +3.23607 q^{33} -6.47214 q^{34} -1.23607 q^{35} -9.70820 q^{36} -9.70820 q^{37} +14.9443 q^{38} +6.23607 q^{39} +7.47214 q^{40} -7.47214 q^{41} +3.23607 q^{42} +3.70820 q^{43} -15.7082 q^{44} +2.00000 q^{45} +2.61803 q^{46} +9.47214 q^{47} -9.85410 q^{48} -5.47214 q^{49} -2.61803 q^{50} -2.47214 q^{51} -30.2705 q^{52} -6.00000 q^{53} -13.0902 q^{54} +3.23607 q^{55} -9.23607 q^{56} +5.70820 q^{57} +1.38197 q^{58} +8.94427 q^{59} +4.85410 q^{60} +12.1803 q^{61} -11.0902 q^{62} -2.47214 q^{63} +8.70820 q^{64} +6.23607 q^{65} -8.47214 q^{66} +9.70820 q^{67} +12.0000 q^{68} +1.00000 q^{69} +3.23607 q^{70} -6.23607 q^{71} +14.9443 q^{72} -6.70820 q^{73} +25.4164 q^{74} -1.00000 q^{75} -27.7082 q^{76} -4.00000 q^{77} -16.3262 q^{78} +8.76393 q^{79} -9.85410 q^{80} +1.00000 q^{81} +19.5623 q^{82} -6.47214 q^{83} -6.00000 q^{84} -2.47214 q^{85} -9.70820 q^{86} +0.527864 q^{87} +24.1803 q^{88} +2.76393 q^{89} -5.23607 q^{90} -7.70820 q^{91} -4.85410 q^{92} -4.23607 q^{93} -24.7984 q^{94} +5.70820 q^{95} +10.8541 q^{96} -6.18034 q^{97} +14.3262 q^{98} +6.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{2} - 2 q^{3} + 3 q^{4} - 2 q^{5} + 3 q^{6} - 2 q^{7} - 6 q^{8} - 4 q^{9}+O(q^{10}) \) 2 * q - 3 * q^2 - 2 * q^3 + 3 * q^4 - 2 * q^5 + 3 * q^6 - 2 * q^7 - 6 * q^8 - 4 * q^9	\( 2 q - 3 q^{2} - 2 q^{3} + 3 q^{4} - 2 q^{5} + 3 q^{6} - 2 q^{7} - 6 q^{8} - 4 q^{9} + 3 q^{10} - 2 q^{11} - 3 q^{12} - 8 q^{13} - 2 q^{14} + 2 q^{15} + 13 q^{16} - 4 q^{17} + 6 q^{18} + 2 q^{19} - 3 q^{20} + 2 q^{21} + 8 q^{22} - 2 q^{23} + 6 q^{24} + 2 q^{25} + 17 q^{26} + 10 q^{27} + 12 q^{28} - 10 q^{29} - 3 q^{30} + 4 q^{31} - 15 q^{32} + 2 q^{33} - 4 q^{34} + 2 q^{35} - 6 q^{36} - 6 q^{37} + 12 q^{38} + 8 q^{39} + 6 q^{40} - 6 q^{41} + 2 q^{42} - 6 q^{43} - 18 q^{44} + 4 q^{45} + 3 q^{46} + 10 q^{47} - 13 q^{48} - 2 q^{49} - 3 q^{50} + 4 q^{51} - 27 q^{52} - 12 q^{53} - 15 q^{54} + 2 q^{55} - 14 q^{56} - 2 q^{57} + 5 q^{58} + 3 q^{60} + 2 q^{61} - 11 q^{62} + 4 q^{63} + 4 q^{64} + 8 q^{65} - 8 q^{66} + 6 q^{67} + 24 q^{68} + 2 q^{69} + 2 q^{70} - 8 q^{71} + 12 q^{72} + 24 q^{74} - 2 q^{75} - 42 q^{76} - 8 q^{77} - 17 q^{78} + 22 q^{79} - 13 q^{80} + 2 q^{81} + 19 q^{82} - 4 q^{83} - 12 q^{84} + 4 q^{85} - 6 q^{86} + 10 q^{87} + 26 q^{88} + 10 q^{89} - 6 q^{90} - 2 q^{91} - 3 q^{92} - 4 q^{93} - 25 q^{94} - 2 q^{95} + 15 q^{96} + 10 q^{97} + 13 q^{98} + 4 q^{99}+O(q^{100}) \) 2 * q - 3 * q^2 - 2 * q^3 + 3 * q^4 - 2 * q^5 + 3 * q^6 - 2 * q^7 - 6 * q^8 - 4 * q^9 + 3 * q^10 - 2 * q^11 - 3 * q^12 - 8 * q^13 - 2 * q^14 + 2 * q^15 + 13 * q^16 - 4 * q^17 + 6 * q^18 + 2 * q^19 - 3 * q^20 + 2 * q^21 + 8 * q^22 - 2 * q^23 + 6 * q^24 + 2 * q^25 + 17 * q^26 + 10 * q^27 + 12 * q^28 - 10 * q^29 - 3 * q^30 + 4 * q^31 - 15 * q^32 + 2 * q^33 - 4 * q^34 + 2 * q^35 - 6 * q^36 - 6 * q^37 + 12 * q^38 + 8 * q^39 + 6 * q^40 - 6 * q^41 + 2 * q^42 - 6 * q^43 - 18 * q^44 + 4 * q^45 + 3 * q^46 + 10 * q^47 - 13 * q^48 - 2 * q^49 - 3 * q^50 + 4 * q^51 - 27 * q^52 - 12 * q^53 - 15 * q^54 + 2 * q^55 - 14 * q^56 - 2 * q^57 + 5 * q^58 + 3 * q^60 + 2 * q^61 - 11 * q^62 + 4 * q^63 + 4 * q^64 + 8 * q^65 - 8 * q^66 + 6 * q^67 + 24 * q^68 + 2 * q^69 + 2 * q^70 - 8 * q^71 + 12 * q^72 + 24 * q^74 - 2 * q^75 - 42 * q^76 - 8 * q^77 - 17 * q^78 + 22 * q^79 - 13 * q^80 + 2 * q^81 + 19 * q^82 - 4 * q^83 - 12 * q^84 + 4 * q^85 - 6 * q^86 + 10 * q^87 + 26 * q^88 + 10 * q^89 - 6 * q^90 - 2 * q^91 - 3 * q^92 - 4 * q^93 - 25 * q^94 - 2 * q^95 + 15 * q^96 + 10 * q^97 + 13 * q^98 + 4 * q^99

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