LMFDB - Embedded newform 4030.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4030,2,Mod(1,4030)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4030, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4030.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4030 = 2 \cdot 5 \cdot 13 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4030.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:	$32.1797120146$
Analytic rank:	$0$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	4030.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.23607 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.23607 q^{6} +2.85410 q^{7} -1.00000 q^{8} -1.47214 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.23607 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.23607 q^{6} +2.85410 q^{7} -1.00000 q^{8} -1.47214 q^{9} +1.00000 q^{10} -0.763932 q^{11} -1.23607 q^{12} +1.00000 q^{13} -2.85410 q^{14} +1.23607 q^{15} +1.00000 q^{16} +4.61803 q^{17} +1.47214 q^{18} -7.09017 q^{19} -1.00000 q^{20} -3.52786 q^{21} +0.763932 q^{22} +0.854102 q^{23} +1.23607 q^{24} +1.00000 q^{25} -1.00000 q^{26} +5.52786 q^{27} +2.85410 q^{28} +7.38197 q^{29} -1.23607 q^{30} -1.00000 q^{31} -1.00000 q^{32} +0.944272 q^{33} -4.61803 q^{34} -2.85410 q^{35} -1.47214 q^{36} -10.0902 q^{37} +7.09017 q^{38} -1.23607 q^{39} +1.00000 q^{40} -1.70820 q^{41} +3.52786 q^{42} +5.23607 q^{43} -0.763932 q^{44} +1.47214 q^{45} -0.854102 q^{46} -11.0902 q^{47} -1.23607 q^{48} +1.14590 q^{49} -1.00000 q^{50} -5.70820 q^{51} +1.00000 q^{52} +10.0000 q^{53} -5.52786 q^{54} +0.763932 q^{55} -2.85410 q^{56} +8.76393 q^{57} -7.38197 q^{58} +6.85410 q^{59} +1.23607 q^{60} +6.56231 q^{61} +1.00000 q^{62} -4.20163 q^{63} +1.00000 q^{64} -1.00000 q^{65} -0.944272 q^{66} -0.472136 q^{67} +4.61803 q^{68} -1.05573 q^{69} +2.85410 q^{70} -6.94427 q^{71} +1.47214 q^{72} -3.23607 q^{73} +10.0902 q^{74} -1.23607 q^{75} -7.09017 q^{76} -2.18034 q^{77} +1.23607 q^{78} +16.4721 q^{79} -1.00000 q^{80} -2.41641 q^{81} +1.70820 q^{82} -1.85410 q^{83} -3.52786 q^{84} -4.61803 q^{85} -5.23607 q^{86} -9.12461 q^{87} +0.763932 q^{88} +10.8541 q^{89} -1.47214 q^{90} +2.85410 q^{91} +0.854102 q^{92} +1.23607 q^{93} +11.0902 q^{94} +7.09017 q^{95} +1.23607 q^{96} -9.38197 q^{97} -1.14590 q^{98} +1.12461 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} - q^{7} - 2 q^{8} + 6 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 - q^7 - 2 * q^8 + 6 * q^9	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} - q^{7} - 2 q^{8} + 6 q^{9} + 2 q^{10} - 6 q^{11} + 2 q^{12} + 2 q^{13} + q^{14} - 2 q^{15} + 2 q^{16} + 7 q^{17} - 6 q^{18} - 3 q^{19} - 2 q^{20} - 16 q^{21} + 6 q^{22} - 5 q^{23} - 2 q^{24} + 2 q^{25} - 2 q^{26} + 20 q^{27} - q^{28} + 17 q^{29} + 2 q^{30} - 2 q^{31} - 2 q^{32} - 16 q^{33} - 7 q^{34} + q^{35} + 6 q^{36} - 9 q^{37} + 3 q^{38} + 2 q^{39} + 2 q^{40} + 10 q^{41} + 16 q^{42} + 6 q^{43} - 6 q^{44} - 6 q^{45} + 5 q^{46} - 11 q^{47} + 2 q^{48} + 9 q^{49} - 2 q^{50} + 2 q^{51} + 2 q^{52} + 20 q^{53} - 20 q^{54} + 6 q^{55} + q^{56} + 22 q^{57} - 17 q^{58} + 7 q^{59} - 2 q^{60} - 7 q^{61} + 2 q^{62} - 33 q^{63} + 2 q^{64} - 2 q^{65} + 16 q^{66} + 8 q^{67} + 7 q^{68} - 20 q^{69} - q^{70} + 4 q^{71} - 6 q^{72} - 2 q^{73} + 9 q^{74} + 2 q^{75} - 3 q^{76} + 18 q^{77} - 2 q^{78} + 24 q^{79} - 2 q^{80} + 22 q^{81} - 10 q^{82} + 3 q^{83} - 16 q^{84} - 7 q^{85} - 6 q^{86} + 22 q^{87} + 6 q^{88} + 15 q^{89} + 6 q^{90} - q^{91} - 5 q^{92} - 2 q^{93} + 11 q^{94} + 3 q^{95} - 2 q^{96} - 21 q^{97} - 9 q^{98} - 38 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 - q^7 - 2 * q^8 + 6 * q^9 + 2 * q^10 - 6 * q^11 + 2 * q^12 + 2 * q^13 + q^14 - 2 * q^15 + 2 * q^16 + 7 * q^17 - 6 * q^18 - 3 * q^19 - 2 * q^20 - 16 * q^21 + 6 * q^22 - 5 * q^23 - 2 * q^24 + 2 * q^25 - 2 * q^26 + 20 * q^27 - q^28 + 17 * q^29 + 2 * q^30 - 2 * q^31 - 2 * q^32 - 16 * q^33 - 7 * q^34 + q^35 + 6 * q^36 - 9 * q^37 + 3 * q^38 + 2 * q^39 + 2 * q^40 + 10 * q^41 + 16 * q^42 + 6 * q^43 - 6 * q^44 - 6 * q^45 + 5 * q^46 - 11 * q^47 + 2 * q^48 + 9 * q^49 - 2 * q^50 + 2 * q^51 + 2 * q^52 + 20 * q^53 - 20 * q^54 + 6 * q^55 + q^56 + 22 * q^57 - 17 * q^58 + 7 * q^59 - 2 * q^60 - 7 * q^61 + 2 * q^62 - 33 * q^63 + 2 * q^64 - 2 * q^65 + 16 * q^66 + 8 * q^67 + 7 * q^68 - 20 * q^69 - q^70 + 4 * q^71 - 6 * q^72 - 2 * q^73 + 9 * q^74 + 2 * q^75 - 3 * q^76 + 18 * q^77 - 2 * q^78 + 24 * q^79 - 2 * q^80 + 22 * q^81 - 10 * q^82 + 3 * q^83 - 16 * q^84 - 7 * q^85 - 6 * q^86 + 22 * q^87 + 6 * q^88 + 15 * q^89 + 6 * q^90 - q^91 - 5 * q^92 - 2 * q^93 + 11 * q^94 + 3 * q^95 - 2 * q^96 - 21 * q^97 - 9 * q^98 - 38 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.23607	−0.713644	−0.356822	−	0.934172i	$-0.616140\pi$
$3$	−1.23607	−0.713644	−0.356822	+	0.934172i	$0.616140\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	1.23607	0.504623
$7$	2.85410	1.07875	0.539375	−	0.842066i	$-0.318661\pi$
$7$	2.85410	1.07875	0.539375	+	0.842066i	$0.318661\pi$
$8$	−1.00000	−0.353553
$9$	−1.47214	−0.490712
$10$	1.00000	0.316228
$11$	−0.763932	−0.230334	−0.115167	−	0.993346i	$-0.536740\pi$
$11$	−0.763932	−0.230334	−0.115167	+	0.993346i	$0.536740\pi$
$12$	−1.23607	−0.356822
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−2.85410	−0.762791
$15$	1.23607	0.319151
$16$	1.00000	0.250000
$17$	4.61803	1.12004	0.560019	−	0.828480i	$-0.310794\pi$
$17$	4.61803	1.12004	0.560019	+	0.828480i	$0.310794\pi$
$18$	1.47214	0.346986
$19$	−7.09017	−1.62660	−0.813298	−	0.581847i	$-0.802330\pi$
$19$	−7.09017	−1.62660	−0.813298	+	0.581847i	$0.802330\pi$
$20$	−1.00000	−0.223607
$21$	−3.52786	−0.769843
$22$	0.763932	0.162871
$23$	0.854102	0.178093	0.0890463	−	0.996027i	$-0.471618\pi$
$23$	0.854102	0.178093	0.0890463	+	0.996027i	$0.471618\pi$
$24$	1.23607	0.252311
$25$	1.00000	0.200000
$26$	−1.00000	−0.196116
$27$	5.52786	1.06384
$28$	2.85410	0.539375
$29$	7.38197	1.37080	0.685398	−	0.728168i	$-0.259628\pi$
$29$	7.38197	1.37080	0.685398	+	0.728168i	$0.259628\pi$
$30$	−1.23607	−0.225674
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	−1.00000	−0.176777
$33$	0.944272	0.164377
$34$	−4.61803	−0.791986
$35$	−2.85410	−0.482431
$36$	−1.47214	−0.245356
$37$	−10.0902	−1.65881	−0.829407	−	0.558645i	$-0.811321\pi$
$37$	−10.0902	−1.65881	−0.829407	+	0.558645i	$0.811321\pi$
$38$	7.09017	1.15018
$39$	−1.23607	−0.197929
$40$	1.00000	0.158114
$41$	−1.70820	−0.266777	−0.133388	−	0.991064i	$-0.542586\pi$
$41$	−1.70820	−0.266777	−0.133388	+	0.991064i	$0.542586\pi$
$42$	3.52786	0.544361
$43$	5.23607	0.798493	0.399246	−	0.916844i	$-0.369272\pi$
$43$	5.23607	0.798493	0.399246	+	0.916844i	$0.369272\pi$
$44$	−0.763932	−0.115167
$45$	1.47214	0.219453
$46$	−0.854102	−0.125930
$47$	−11.0902	−1.61767	−0.808834	−	0.588037i	$-0.799901\pi$
$47$	−11.0902	−1.61767	−0.808834	+	0.588037i	$0.799901\pi$
$48$	−1.23607	−0.178411
$49$	1.14590	0.163700
$50$	−1.00000	−0.141421
$51$	−5.70820	−0.799308
$52$	1.00000	0.138675
$53$	10.0000	1.37361	0.686803	−	0.726844i	$-0.259014\pi$
$53$	10.0000	1.37361	0.686803	+	0.726844i	$0.259014\pi$
$54$	−5.52786	−0.752247
$55$	0.763932	0.103009
$56$	−2.85410	−0.381395
$57$	8.76393	1.16081
$58$	−7.38197	−0.969300
$59$	6.85410	0.892328	0.446164	−	0.894951i	$-0.352790\pi$
$59$	6.85410	0.892328	0.446164	+	0.894951i	$0.352790\pi$
$60$	1.23607	0.159576
$61$	6.56231	0.840217	0.420109	−	0.907474i	$-0.361992\pi$
$61$	6.56231	0.840217	0.420109	+	0.907474i	$0.361992\pi$
$62$	1.00000	0.127000
$63$	−4.20163	−0.529355
$64$	1.00000	0.125000
$65$	−1.00000	−0.124035
$66$	−0.944272	−0.116232
$67$	−0.472136	−0.0576806	−0.0288403	−	0.999584i	$-0.509181\pi$
$67$	−0.472136	−0.0576806	−0.0288403	+	0.999584i	$0.509181\pi$
$68$	4.61803	0.560019
$69$	−1.05573	−0.127095
$70$	2.85410	0.341130
$71$	−6.94427	−0.824133	−0.412067	−	0.911154i	$-0.635193\pi$
$71$	−6.94427	−0.824133	−0.412067	+	0.911154i	$0.635193\pi$
$72$	1.47214	0.173493
$73$	−3.23607	−0.378753	−0.189377	−	0.981905i	$-0.560647\pi$
$73$	−3.23607	−0.378753	−0.189377	+	0.981905i	$0.560647\pi$
$74$	10.0902	1.17296
$75$	−1.23607	−0.142729
$76$	−7.09017	−0.813298
$77$	−2.18034	−0.248473
$78$	1.23607	0.139957
$79$	16.4721	1.85326	0.926630	−	0.375974i	$-0.122692\pi$
$79$	16.4721	1.85326	0.926630	+	0.375974i	$0.122692\pi$
$80$	−1.00000	−0.111803
$81$	−2.41641	−0.268490
$82$	1.70820	0.188640
$83$	−1.85410	−0.203514	−0.101757	−	0.994809i	$-0.532446\pi$
$83$	−1.85410	−0.203514	−0.101757	+	0.994809i	$0.532446\pi$
$84$	−3.52786	−0.384922
$85$	−4.61803	−0.500896
$86$	−5.23607	−0.564620
$87$	−9.12461	−0.978261
$88$	0.763932	0.0814354
$89$	10.8541	1.15053	0.575266	−	0.817966i	$-0.304898\pi$
$89$	10.8541	1.15053	0.575266	+	0.817966i	$0.304898\pi$
$90$	−1.47214	−0.155177
$91$	2.85410	0.299191
$92$	0.854102	0.0890463
$93$	1.23607	0.128174
$94$	11.0902	1.14386
$95$	7.09017	0.727436
$96$	1.23607	0.126156
$97$	−9.38197	−0.952594	−0.476297	−	0.879284i	$-0.658021\pi$
$97$	−9.38197	−0.952594	−0.476297	+	0.879284i	$0.658021\pi$
$98$	−1.14590	−0.115753
$99$	1.12461	0.113028
$100$	1.00000	0.100000
$101$	−2.47214	−0.245987	−0.122993	−	0.992407i	$-0.539249\pi$
$101$	−2.47214	−0.245987	−0.122993	+	0.992407i	$0.539249\pi$
$102$	5.70820	0.565196
$103$	0.944272	0.0930419	0.0465209	−	0.998917i	$-0.485187\pi$
$103$	0.944272	0.0930419	0.0465209	+	0.998917i	$0.485187\pi$
$104$	−1.00000	−0.0980581
$105$	3.52786	0.344284
$106$	−10.0000	−0.971286
$107$	−9.38197	−0.906989	−0.453494	−	0.891259i	$-0.649823\pi$
$107$	−9.38197	−0.906989	−0.453494	+	0.891259i	$0.649823\pi$
$108$	5.52786	0.531919
$109$	−4.38197	−0.419716	−0.209858	−	0.977732i	$-0.567300\pi$
$109$	−4.38197	−0.419716	−0.209858	+	0.977732i	$0.567300\pi$
$110$	−0.763932	−0.0728381
$111$	12.4721	1.18380
$112$	2.85410	0.269687
$113$	0.291796	0.0274499	0.0137249	−	0.999906i	$-0.495631\pi$
$113$	0.291796	0.0274499	0.0137249	+	0.999906i	$0.495631\pi$
$114$	−8.76393	−0.820817
$115$	−0.854102	−0.0796454
$116$	7.38197	0.685398
$117$	−1.47214	−0.136099
$118$	−6.85410	−0.630971
$119$	13.1803	1.20824
$120$	−1.23607	−0.112837
$121$	−10.4164	−0.946946
$122$	−6.56231	−0.594123
$123$	2.11146	0.190384
$124$	−1.00000	−0.0898027
$125$	−1.00000	−0.0894427
$126$	4.20163	0.374311
$127$	6.47214	0.574309	0.287155	−	0.957884i	$-0.407291\pi$
$127$	6.47214	0.574309	0.287155	+	0.957884i	$0.407291\pi$
$128$	−1.00000	−0.0883883
$129$	−6.47214	−0.569840
$130$	1.00000	0.0877058
$131$	−10.7639	−0.940449	−0.470225	−	0.882547i	$-0.655827\pi$
$131$	−10.7639	−0.940449	−0.470225	+	0.882547i	$0.655827\pi$
$132$	0.944272	0.0821883
$133$	−20.2361	−1.75469
$134$	0.472136	0.0407863
$135$	−5.52786	−0.475763
$136$	−4.61803	−0.395993
$137$	16.0000	1.36697	0.683486	−	0.729964i	$-0.260463\pi$
$137$	16.0000	1.36697	0.683486	+	0.729964i	$0.260463\pi$
$138$	1.05573	0.0898695
$139$	−4.79837	−0.406993	−0.203496	−	0.979076i	$-0.565231\pi$
$139$	−4.79837	−0.406993	−0.203496	+	0.979076i	$0.565231\pi$
$140$	−2.85410	−0.241216
$141$	13.7082	1.15444
$142$	6.94427	0.582750
$143$	−0.763932	−0.0638832
$144$	−1.47214	−0.122678
$145$	−7.38197	−0.613039
$146$	3.23607	0.267819
$147$	−1.41641	−0.116823
$148$	−10.0902	−0.829407
$149$	17.1459	1.40465	0.702323	−	0.711858i	$-0.252146\pi$
$149$	17.1459	1.40465	0.702323	+	0.711858i	$0.252146\pi$
$150$	1.23607	0.100925
$151$	12.3820	1.00763	0.503815	−	0.863812i	$-0.331929\pi$
$151$	12.3820	1.00763	0.503815	+	0.863812i	$0.331929\pi$
$152$	7.09017	0.575089
$153$	−6.79837	−0.549616
$154$	2.18034	0.175697
$155$	1.00000	0.0803219
$156$	−1.23607	−0.0989646
$157$	−21.5623	−1.72086	−0.860430	−	0.509569i	$-0.829805\pi$
$157$	−21.5623	−1.72086	−0.860430	+	0.509569i	$0.829805\pi$
$158$	−16.4721	−1.31045
$159$	−12.3607	−0.980266
$160$	1.00000	0.0790569
$161$	2.43769	0.192117
$162$	2.41641	0.189851
$163$	1.05573	0.0826910	0.0413455	−	0.999145i	$-0.486836\pi$
$163$	1.05573	0.0826910	0.0413455	+	0.999145i	$0.486836\pi$
$164$	−1.70820	−0.133388
$165$	−0.944272	−0.0735115
$166$	1.85410	0.143906
$167$	−18.4721	−1.42942	−0.714708	−	0.699423i	$-0.753441\pi$
$167$	−18.4721	−1.42942	−0.714708	+	0.699423i	$0.753441\pi$
$168$	3.52786	0.272181
$169$	1.00000	0.0769231
$170$	4.61803	0.354187
$171$	10.4377	0.798190
$172$	5.23607	0.399246
$173$	12.8541	0.977279	0.488640	−	0.872486i	$-0.337493\pi$
$173$	12.8541	0.977279	0.488640	+	0.872486i	$0.337493\pi$
$174$	9.12461	0.691735
$175$	2.85410	0.215750
$176$	−0.763932	−0.0575835
$177$	−8.47214	−0.636805
$178$	−10.8541	−0.813549
$179$	19.3262	1.44451	0.722255	−	0.691626i	$-0.243106\pi$
$179$	19.3262	1.44451	0.722255	+	0.691626i	$0.243106\pi$
$180$	1.47214	0.109727
$181$	3.52786	0.262224	0.131112	−	0.991368i	$-0.458145\pi$
$181$	3.52786	0.262224	0.131112	+	0.991368i	$0.458145\pi$
$182$	−2.85410	−0.211560
$183$	−8.11146	−0.599616
$184$	−0.854102	−0.0629652
$185$	10.0902	0.741844
$186$	−1.23607	−0.0906329
$187$	−3.52786	−0.257983
$188$	−11.0902	−0.808834
$189$	15.7771	1.14761
$190$	−7.09017	−0.514375
$191$	4.79837	0.347198	0.173599	−	0.984816i	$-0.444460\pi$
$191$	4.79837	0.347198	0.173599	+	0.984816i	$0.444460\pi$
$192$	−1.23607	−0.0892055
$193$	14.6180	1.05223	0.526115	−	0.850414i	$-0.323648\pi$
$193$	14.6180	1.05223	0.526115	+	0.850414i	$0.323648\pi$
$194$	9.38197	0.673586
$195$	1.23607	0.0885167
$196$	1.14590	0.0818499
$197$	−18.0344	−1.28490	−0.642450	−	0.766327i	$-0.722082\pi$
$197$	−18.0344	−1.28490	−0.642450	+	0.766327i	$0.722082\pi$
$198$	−1.12461	−0.0799227
$199$	17.4164	1.23462	0.617308	−	0.786721i	$-0.288223\pi$
$199$	17.4164	1.23462	0.617308	+	0.786721i	$0.288223\pi$
$200$	−1.00000	−0.0707107
$201$	0.583592	0.0411634
$202$	2.47214	0.173939
$203$	21.0689	1.47875
$204$	−5.70820	−0.399654
$205$	1.70820	0.119306
$206$	−0.944272	−0.0657905
$207$	−1.25735	−0.0873922
$208$	1.00000	0.0693375
$209$	5.41641	0.374661
$210$	−3.52786	−0.243446
$211$	−4.76393	−0.327963	−0.163981	−	0.986463i	$-0.552434\pi$
$211$	−4.76393	−0.327963	−0.163981	+	0.986463i	$0.552434\pi$
$212$	10.0000	0.686803
$213$	8.58359	0.588138
$214$	9.38197	0.641338
$215$	−5.23607	−0.357097
$216$	−5.52786	−0.376124
$217$	−2.85410	−0.193749
$218$	4.38197	0.296784
$219$	4.00000	0.270295
$220$	0.763932	0.0515043
$221$	4.61803	0.310643
$222$	−12.4721	−0.837075
$223$	11.1246	0.744959	0.372480	−	0.928040i	$-0.378508\pi$
$223$	11.1246	0.744959	0.372480	+	0.928040i	$0.378508\pi$
$224$	−2.85410	−0.190698
$225$	−1.47214	−0.0981424
$226$	−0.291796	−0.0194100
$227$	23.1246	1.53483	0.767417	−	0.641148i	$-0.221542\pi$
$227$	23.1246	1.53483	0.767417	+	0.641148i	$0.221542\pi$
$228$	8.76393	0.580406
$229$	−14.1803	−0.937063	−0.468532	−	0.883447i	$-0.655217\pi$
$229$	−14.1803	−0.937063	−0.468532	+	0.883447i	$0.655217\pi$
$230$	0.854102	0.0563178
$231$	2.69505	0.177321
$232$	−7.38197	−0.484650
$233$	10.0000	0.655122	0.327561	−	0.944830i	$-0.393773\pi$
$233$	10.0000	0.655122	0.327561	+	0.944830i	$0.393773\pi$
$234$	1.47214	0.0962365
$235$	11.0902	0.723443
$236$	6.85410	0.446164
$237$	−20.3607	−1.32257
$238$	−13.1803	−0.854355
$239$	20.3262	1.31480	0.657398	−	0.753544i	$-0.271657\pi$
$239$	20.3262	1.31480	0.657398	+	0.753544i	$0.271657\pi$
$240$	1.23607	0.0797878
$241$	8.56231	0.551547	0.275773	−	0.961223i	$-0.411066\pi$
$241$	8.56231	0.551547	0.275773	+	0.961223i	$0.411066\pi$
$242$	10.4164	0.669592
$243$	−13.5967	−0.872232
$244$	6.56231	0.420109
$245$	−1.14590	−0.0732087
$246$	−2.11146	−0.134622
$247$	−7.09017	−0.451137
$248$	1.00000	0.0635001
$249$	2.29180	0.145237
$250$	1.00000	0.0632456
$251$	−2.32624	−0.146831	−0.0734154	−	0.997301i	$-0.523390\pi$
$251$	−2.32624	−0.146831	−0.0734154	+	0.997301i	$0.523390\pi$
$252$	−4.20163	−0.264678
$253$	−0.652476	−0.0410208
$254$	−6.47214	−0.406098
$255$	5.70820	0.357462
$256$	1.00000	0.0625000
$257$	27.4164	1.71019	0.855094	−	0.518473i	$-0.173499\pi$
$257$	27.4164	1.71019	0.855094	+	0.518473i	$0.173499\pi$
$258$	6.47214	0.402938
$259$	−28.7984	−1.78944
$260$	−1.00000	−0.0620174
$261$	−10.8673	−0.672666
$262$	10.7639	0.664998
$263$	−7.27051	−0.448319	−0.224159	−	0.974552i	$-0.571964\pi$
$263$	−7.27051	−0.448319	−0.224159	+	0.974552i	$0.571964\pi$
$264$	−0.944272	−0.0581159
$265$	−10.0000	−0.614295
$266$	20.2361	1.24075
$267$	−13.4164	−0.821071
$268$	−0.472136	−0.0288403
$269$	25.9787	1.58395	0.791975	−	0.610553i	$-0.209053\pi$
$269$	25.9787	1.58395	0.791975	+	0.610553i	$0.209053\pi$
$270$	5.52786	0.336415
$271$	7.09017	0.430697	0.215349	−	0.976537i	$-0.430911\pi$
$271$	7.09017	0.430697	0.215349	+	0.976537i	$0.430911\pi$
$272$	4.61803	0.280009
$273$	−3.52786	−0.213516
$274$	−16.0000	−0.966595
$275$	−0.763932	−0.0460668
$276$	−1.05573	−0.0635474
$277$	0.944272	0.0567358	0.0283679	−	0.999598i	$-0.490969\pi$
$277$	0.944272	0.0567358	0.0283679	+	0.999598i	$0.490969\pi$
$278$	4.79837	0.287787
$279$	1.47214	0.0881345
$280$	2.85410	0.170565
$281$	28.3607	1.69186	0.845928	−	0.533297i	$-0.179047\pi$
$281$	28.3607	1.69186	0.845928	+	0.533297i	$0.179047\pi$
$282$	−13.7082	−0.816312
$283$	−17.3820	−1.03325	−0.516625	−	0.856212i	$-0.672812\pi$
$283$	−17.3820	−1.03325	−0.516625	+	0.856212i	$0.672812\pi$
$284$	−6.94427	−0.412067
$285$	−8.76393	−0.519131
$286$	0.763932	0.0451722
$287$	−4.87539	−0.287785
$288$	1.47214	0.0867464
$289$	4.32624	0.254485
$290$	7.38197	0.433484
$291$	11.5967	0.679813
$292$	−3.23607	−0.189377
$293$	33.8885	1.97979	0.989895	−	0.141803i	$-0.0452899\pi$
$293$	33.8885	1.97979	0.989895	+	0.141803i	$0.0452899\pi$
$294$	1.41641	0.0826066
$295$	−6.85410	−0.399061
$296$	10.0902	0.586479
$297$	−4.22291	−0.245038
$298$	−17.1459	−0.993235
$299$	0.854102	0.0493940
$300$	−1.23607	−0.0713644
$301$	14.9443	0.861374
$302$	−12.3820	−0.712502
$303$	3.05573	0.175547
$304$	−7.09017	−0.406649
$305$	−6.56231	−0.375757
$306$	6.79837	0.388637
$307$	−8.18034	−0.466877	−0.233438	−	0.972372i	$-0.574998\pi$
$307$	−8.18034	−0.466877	−0.233438	+	0.972372i	$0.574998\pi$
$308$	−2.18034	−0.124236
$309$	−1.16718	−0.0663988
$310$	−1.00000	−0.0567962
$311$	33.9787	1.92676	0.963378	−	0.268147i	$-0.0864112\pi$
$311$	33.9787	1.92676	0.963378	+	0.268147i	$0.0864112\pi$
$312$	1.23607	0.0699786
$313$	4.47214	0.252780	0.126390	−	0.991981i	$-0.459661\pi$
$313$	4.47214	0.252780	0.126390	+	0.991981i	$0.459661\pi$
$314$	21.5623	1.21683
$315$	4.20163	0.236735
$316$	16.4721	0.926630
$317$	−2.47214	−0.138849	−0.0694245	−	0.997587i	$-0.522116\pi$
$317$	−2.47214	−0.138849	−0.0694245	+	0.997587i	$0.522116\pi$
$318$	12.3607	0.693153
$319$	−5.63932	−0.315741
$320$	−1.00000	−0.0559017
$321$	11.5967	0.647267
$322$	−2.43769	−0.135847
$323$	−32.7426	−1.82185
$324$	−2.41641	−0.134245
$325$	1.00000	0.0554700
$326$	−1.05573	−0.0584714
$327$	5.41641	0.299528
$328$	1.70820	0.0943198
$329$	−31.6525	−1.74506
$330$	0.944272	0.0519805
$331$	−10.4721	−0.575601	−0.287800	−	0.957690i	$-0.592924\pi$
$331$	−10.4721	−0.575601	−0.287800	+	0.957690i	$0.592924\pi$
$332$	−1.85410	−0.101757
$333$	14.8541	0.814000
$334$	18.4721	1.01075
$335$	0.472136	0.0257955
$336$	−3.52786	−0.192461
$337$	−6.74265	−0.367295	−0.183648	−	0.982992i	$-0.558791\pi$
$337$	−6.74265	−0.367295	−0.183648	+	0.982992i	$0.558791\pi$
$338$	−1.00000	−0.0543928
$339$	−0.360680	−0.0195894
$340$	−4.61803	−0.250448
$341$	0.763932	0.0413692
$342$	−10.4377	−0.564406
$343$	−16.7082	−0.902158
$344$	−5.23607	−0.282310
$345$	1.05573	0.0568385
$346$	−12.8541	−0.691041
$347$	22.4721	1.20637	0.603184	−	0.797602i	$-0.293899\pi$
$347$	22.4721	1.20637	0.603184	+	0.797602i	$0.293899\pi$
$348$	−9.12461	−0.489131
$349$	10.6180	0.568370	0.284185	−	0.958769i	$-0.408277\pi$
$349$	10.6180	0.568370	0.284185	+	0.958769i	$0.408277\pi$
$350$	−2.85410	−0.152558
$351$	5.52786	0.295056
$352$	0.763932	0.0407177
$353$	15.5967	0.830131	0.415066	−	0.909792i	$-0.363759\pi$
$353$	15.5967	0.830131	0.415066	+	0.909792i	$0.363759\pi$
$354$	8.47214	0.450289
$355$	6.94427	0.368564
$356$	10.8541	0.575266
$357$	−16.2918	−0.862253
$358$	−19.3262	−1.02142
$359$	3.52786	0.186194	0.0930968	−	0.995657i	$-0.470323\pi$
$359$	3.52786	0.186194	0.0930968	+	0.995657i	$0.470323\pi$
$360$	−1.47214	−0.0775884
$361$	31.2705	1.64582
$362$	−3.52786	−0.185420
$363$	12.8754	0.675783
$364$	2.85410	0.149596
$365$	3.23607	0.169384
$366$	8.11146	0.423993
$367$	10.6180	0.554257	0.277128	−	0.960833i	$-0.410617\pi$
$367$	10.6180	0.554257	0.277128	+	0.960833i	$0.410617\pi$
$368$	0.854102	0.0445231
$369$	2.51471	0.130910
$370$	−10.0902	−0.524563
$371$	28.5410	1.48178
$372$	1.23607	0.0640871
$373$	−28.8541	−1.49401	−0.747004	−	0.664819i	$-0.768509\pi$
$373$	−28.8541	−1.49401	−0.747004	+	0.664819i	$0.768509\pi$
$374$	3.52786	0.182422
$375$	1.23607	0.0638303
$376$	11.0902	0.571932
$377$	7.38197	0.380191
$378$	−15.7771	−0.811486
$379$	16.5623	0.850749	0.425374	−	0.905018i	$-0.360142\pi$
$379$	16.5623	0.850749	0.425374	+	0.905018i	$0.360142\pi$
$380$	7.09017	0.363718
$381$	−8.00000	−0.409852
$382$	−4.79837	−0.245506
$383$	−16.2918	−0.832472	−0.416236	−	0.909257i	$-0.636651\pi$
$383$	−16.2918	−0.832472	−0.416236	+	0.909257i	$0.636651\pi$
$384$	1.23607	0.0630778
$385$	2.18034	0.111120
$386$	−14.6180	−0.744038
$387$	−7.70820	−0.391830
$388$	−9.38197	−0.476297
$389$	21.4164	1.08585	0.542927	−	0.839780i	$-0.317316\pi$
$389$	21.4164	1.08585	0.542927	+	0.839780i	$0.317316\pi$
$390$	−1.23607	−0.0625907
$391$	3.94427	0.199470
$392$	−1.14590	−0.0578766
$393$	13.3050	0.671146
$394$	18.0344	0.908562
$395$	−16.4721	−0.828803
$396$	1.12461	0.0565139
$397$	31.3050	1.57115	0.785575	−	0.618766i	$-0.212367\pi$
$397$	31.3050	1.57115	0.785575	+	0.618766i	$0.212367\pi$
$398$	−17.4164	−0.873006
$399$	25.0132	1.25222
$400$	1.00000	0.0500000
$401$	6.90983	0.345060	0.172530	−	0.985004i	$-0.444806\pi$
$401$	6.90983	0.345060	0.172530	+	0.985004i	$0.444806\pi$
$402$	−0.583592	−0.0291069
$403$	−1.00000	−0.0498135
$404$	−2.47214	−0.122993
$405$	2.41641	0.120072
$406$	−21.0689	−1.04563
$407$	7.70820	0.382081
$408$	5.70820	0.282598
$409$	12.4721	0.616707	0.308354	−	0.951272i	$-0.400222\pi$
$409$	12.4721	0.616707	0.308354	+	0.951272i	$0.400222\pi$
$410$	−1.70820	−0.0843622
$411$	−19.7771	−0.975532
$412$	0.944272	0.0465209
$413$	19.5623	0.962598
$414$	1.25735	0.0617956
$415$	1.85410	0.0910143
$416$	−1.00000	−0.0490290
$417$	5.93112	0.290448
$418$	−5.41641	−0.264925
$419$	12.1803	0.595049	0.297524	−	0.954714i	$-0.403839\pi$
$419$	12.1803	0.595049	0.297524	+	0.954714i	$0.403839\pi$
$420$	3.52786	0.172142
$421$	−24.7984	−1.20860	−0.604299	−	0.796757i	$-0.706547\pi$
$421$	−24.7984	−1.20860	−0.604299	+	0.796757i	$0.706547\pi$
$422$	4.76393	0.231905
$423$	16.3262	0.793809
$424$	−10.0000	−0.485643
$425$	4.61803	0.224008
$426$	−8.58359	−0.415876
$427$	18.7295	0.906384
$428$	−9.38197	−0.453494
$429$	0.944272	0.0455899
$430$	5.23607	0.252506
$431$	19.7082	0.949311	0.474655	−	0.880172i	$-0.342573\pi$
$431$	19.7082	0.949311	0.474655	+	0.880172i	$0.342573\pi$
$432$	5.52786	0.265959
$433$	27.5066	1.32188	0.660941	−	0.750438i	$-0.270157\pi$
$433$	27.5066	1.32188	0.660941	+	0.750438i	$0.270157\pi$
$434$	2.85410	0.137001
$435$	9.12461	0.437492
$436$	−4.38197	−0.209858
$437$	−6.05573	−0.289685
$438$	−4.00000	−0.191127
$439$	17.7984	0.849470	0.424735	−	0.905318i	$-0.360367\pi$
$439$	17.7984	0.849470	0.424735	+	0.905318i	$0.360367\pi$
$440$	−0.763932	−0.0364190
$441$	−1.68692	−0.0803294
$442$	−4.61803	−0.219657
$443$	26.4721	1.25773	0.628865	−	0.777515i	$-0.283520\pi$
$443$	26.4721	1.25773	0.628865	+	0.777515i	$0.283520\pi$
$444$	12.4721	0.591901
$445$	−10.8541	−0.514534
$446$	−11.1246	−0.526766
$447$	−21.1935	−1.00242
$448$	2.85410	0.134844
$449$	−27.5623	−1.30075	−0.650373	−	0.759615i	$-0.725387\pi$
$449$	−27.5623	−1.30075	−0.650373	+	0.759615i	$0.725387\pi$
$450$	1.47214	0.0693972
$451$	1.30495	0.0614478
$452$	0.291796	0.0137249
$453$	−15.3050	−0.719089
$454$	−23.1246	−1.08529
$455$	−2.85410	−0.133802
$456$	−8.76393	−0.410409
$457$	37.8885	1.77235	0.886176	−	0.463349i	$-0.153353\pi$
$457$	37.8885	1.77235	0.886176	+	0.463349i	$0.153353\pi$
$458$	14.1803	0.662604
$459$	25.5279	1.19154
$460$	−0.854102	−0.0398227
$461$	−10.1803	−0.474146	−0.237073	−	0.971492i	$-0.576188\pi$
$461$	−10.1803	−0.474146	−0.237073	+	0.971492i	$0.576188\pi$
$462$	−2.69505	−0.125385
$463$	−11.0557	−0.513803	−0.256902	−	0.966438i	$-0.582702\pi$
$463$	−11.0557	−0.513803	−0.256902	+	0.966438i	$0.582702\pi$
$464$	7.38197	0.342699
$465$	−1.23607	−0.0573213
$466$	−10.0000	−0.463241
$467$	−10.2016	−0.472075	−0.236037	−	0.971744i	$-0.575849\pi$
$467$	−10.2016	−0.472075	−0.236037	+	0.971744i	$0.575849\pi$
$468$	−1.47214	−0.0680495
$469$	−1.34752	−0.0622229
$470$	−11.0902	−0.511551
$471$	26.6525	1.22808
$472$	−6.85410	−0.315486
$473$	−4.00000	−0.183920
$474$	20.3607	0.935197
$475$	−7.09017	−0.325319
$476$	13.1803	0.604120
$477$	−14.7214	−0.674045
$478$	−20.3262	−0.929700
$479$	26.7639	1.22288	0.611438	−	0.791293i	$-0.290592\pi$
$479$	26.7639	1.22288	0.611438	+	0.791293i	$0.290592\pi$
$480$	−1.23607	−0.0564185
$481$	−10.0902	−0.460072
$482$	−8.56231	−0.390002
$483$	−3.01316	−0.137103
$484$	−10.4164	−0.473473
$485$	9.38197	0.426013
$486$	13.5967	0.616761
$487$	38.0689	1.72507	0.862533	−	0.506001i	$-0.168877\pi$
$487$	38.0689	1.72507	0.862533	+	0.506001i	$0.168877\pi$
$488$	−6.56231	−0.297062
$489$	−1.30495	−0.0590120
$490$	1.14590	0.0517664
$491$	−5.79837	−0.261677	−0.130838	−	0.991404i	$-0.541767\pi$
$491$	−5.79837	−0.261677	−0.130838	+	0.991404i	$0.541767\pi$
$492$	2.11146	0.0951918
$493$	34.0902	1.53534
$494$	7.09017	0.319002
$495$	−1.12461	−0.0505475
$496$	−1.00000	−0.0449013
$497$	−19.8197	−0.889033
$498$	−2.29180	−0.102698
$499$	−21.4164	−0.958730	−0.479365	−	0.877616i	$-0.659133\pi$
$499$	−21.4164	−0.958730	−0.479365	+	0.877616i	$0.659133\pi$
$500$	−1.00000	−0.0447214
$501$	22.8328	1.02009
$502$	2.32624	0.103825
$503$	35.7082	1.59215	0.796075	−	0.605198i	$-0.206906\pi$
$503$	35.7082	1.59215	0.796075	+	0.605198i	$0.206906\pi$
$504$	4.20163	0.187155
$505$	2.47214	0.110009
$506$	0.652476	0.0290061
$507$	−1.23607	−0.0548957
$508$	6.47214	0.287155
$509$	−3.81966	−0.169303	−0.0846517	−	0.996411i	$-0.526978\pi$
$509$	−3.81966	−0.169303	−0.0846517	+	0.996411i	$0.526978\pi$
$510$	−5.70820	−0.252764
$511$	−9.23607	−0.408580
$512$	−1.00000	−0.0441942
$513$	−39.1935	−1.73044
$514$	−27.4164	−1.20929
$515$	−0.944272	−0.0416096
$516$	−6.47214	−0.284920
$517$	8.47214	0.372604
$518$	28.7984	1.26533
$519$	−15.8885	−0.697430
$520$	1.00000	0.0438529
$521$	−10.4377	−0.457284	−0.228642	−	0.973511i	$-0.573428\pi$
$521$	−10.4377	−0.457284	−0.228642	+	0.973511i	$0.573428\pi$
$522$	10.8673	0.475647
$523$	−17.7082	−0.774326	−0.387163	−	0.922011i	$-0.626545\pi$
$523$	−17.7082	−0.774326	−0.387163	+	0.922011i	$0.626545\pi$
$524$	−10.7639	−0.470225
$525$	−3.52786	−0.153969
$526$	7.27051	0.317009
$527$	−4.61803	−0.201165
$528$	0.944272	0.0410942
$529$	−22.2705	−0.968283
$530$	10.0000	0.434372
$531$	−10.0902	−0.437876
$532$	−20.2361	−0.877345
$533$	−1.70820	−0.0739905
$534$	13.4164	0.580585
$535$	9.38197	0.405618
$536$	0.472136	0.0203932
$537$	−23.8885	−1.03087
$538$	−25.9787	−1.12002
$539$	−0.875388	−0.0377056
$540$	−5.52786	−0.237881
$541$	−20.5623	−0.884043	−0.442021	−	0.897005i	$-0.645738\pi$
$541$	−20.5623	−0.884043	−0.442021	+	0.897005i	$0.645738\pi$
$542$	−7.09017	−0.304549
$543$	−4.36068	−0.187135
$544$	−4.61803	−0.197997
$545$	4.38197	0.187703
$546$	3.52786	0.150979
$547$	−35.3262	−1.51044	−0.755220	−	0.655471i	$-0.772470\pi$
$547$	−35.3262	−1.51044	−0.755220	+	0.655471i	$0.772470\pi$
$548$	16.0000	0.683486
$549$	−9.66061	−0.412305
$550$	0.763932	0.0325742
$551$	−52.3394	−2.22973
$552$	1.05573	0.0449348
$553$	47.0132	1.99920
$554$	−0.944272	−0.0401183
$555$	−12.4721	−0.529413
$556$	−4.79837	−0.203496
$557$	−11.0344	−0.467544	−0.233772	−	0.972291i	$-0.575107\pi$
$557$	−11.0344	−0.467544	−0.233772	+	0.972291i	$0.575107\pi$
$558$	−1.47214	−0.0623205
$559$	5.23607	0.221462
$560$	−2.85410	−0.120608
$561$	4.36068	0.184108
$562$	−28.3607	−1.19632
$563$	−42.2492	−1.78059	−0.890296	−	0.455382i	$-0.849503\pi$
$563$	−42.2492	−1.78059	−0.890296	+	0.455382i	$0.849503\pi$
$564$	13.7082	0.577220
$565$	−0.291796	−0.0122760
$566$	17.3820	0.730619
$567$	−6.89667	−0.289633
$568$	6.94427	0.291375
$569$	29.1246	1.22097	0.610484	−	0.792029i	$-0.290975\pi$
$569$	29.1246	1.22097	0.610484	+	0.792029i	$0.290975\pi$
$570$	8.76393	0.367081
$571$	−34.2492	−1.43329	−0.716643	−	0.697440i	$-0.754322\pi$
$571$	−34.2492	−1.43329	−0.716643	+	0.697440i	$0.754322\pi$
$572$	−0.763932	−0.0319416
$573$	−5.93112	−0.247776
$574$	4.87539	0.203495
$575$	0.854102	0.0356185
$576$	−1.47214	−0.0613390
$577$	−32.9230	−1.37060	−0.685301	−	0.728260i	$-0.740329\pi$
$577$	−32.9230	−1.37060	−0.685301	+	0.728260i	$0.740329\pi$
$578$	−4.32624	−0.179948
$579$	−18.0689	−0.750917
$580$	−7.38197	−0.306519
$581$	−5.29180	−0.219541
$582$	−11.5967	−0.480701
$583$	−7.63932	−0.316388
$584$	3.23607	0.133909
$585$	1.47214	0.0608653
$586$	−33.8885	−1.39992
$587$	−22.4508	−0.926646	−0.463323	−	0.886190i	$-0.653343\pi$
$587$	−22.4508	−0.926646	−0.463323	+	0.886190i	$0.653343\pi$
$588$	−1.41641	−0.0584117
$589$	7.09017	0.292145
$590$	6.85410	0.282179
$591$	22.2918	0.916962
$592$	−10.0902	−0.414703
$593$	−6.85410	−0.281464	−0.140732	−	0.990048i	$-0.544946\pi$
$593$	−6.85410	−0.281464	−0.140732	+	0.990048i	$0.544946\pi$
$594$	4.22291	0.173268
$595$	−13.1803	−0.540341
$596$	17.1459	0.702323
$597$	−21.5279	−0.881077
$598$	−0.854102	−0.0349268
$599$	−15.9098	−0.650058	−0.325029	−	0.945704i	$-0.605374\pi$
$599$	−15.9098	−0.650058	−0.325029	+	0.945704i	$0.605374\pi$
$600$	1.23607	0.0504623
$601$	16.2918	0.664556	0.332278	−	0.943181i	$-0.392183\pi$
$601$	16.2918	0.664556	0.332278	+	0.943181i	$0.392183\pi$
$602$	−14.9443	−0.609083
$603$	0.695048	0.0283046
$604$	12.3820	0.503815
$605$	10.4164	0.423487
$606$	−3.05573	−0.124130
$607$	1.34752	0.0546943	0.0273472	−	0.999626i	$-0.491294\pi$
$607$	1.34752	0.0546943	0.0273472	+	0.999626i	$0.491294\pi$
$608$	7.09017	0.287544
$609$	−26.0426	−1.05530
$610$	6.56231	0.265700
$611$	−11.0902	−0.448660
$612$	−6.79837	−0.274808
$613$	−0.562306	−0.0227113	−0.0113557	−	0.999936i	$-0.503615\pi$
$613$	−0.562306	−0.0227113	−0.0113557	+	0.999936i	$0.503615\pi$
$614$	8.18034	0.330132
$615$	−2.11146	−0.0851421
$616$	2.18034	0.0878484
$617$	44.9230	1.80853	0.904266	−	0.426970i	$-0.140419\pi$
$617$	44.9230	1.80853	0.904266	+	0.426970i	$0.140419\pi$
$618$	1.16718	0.0469510
$619$	−32.8328	−1.31966	−0.659831	−	0.751414i	$-0.729372\pi$
$619$	−32.8328	−1.31966	−0.659831	+	0.751414i	$0.729372\pi$
$620$	1.00000	0.0401610
$621$	4.72136	0.189462
$622$	−33.9787	−1.36242
$623$	30.9787	1.24114
$624$	−1.23607	−0.0494823
$625$	1.00000	0.0400000
$626$	−4.47214	−0.178743
$627$	−6.69505	−0.267374
$628$	−21.5623	−0.860430
$629$	−46.5967	−1.85793
$630$	−4.20163	−0.167397
$631$	20.3262	0.809175	0.404587	−	0.914499i	$-0.367415\pi$
$631$	20.3262	0.809175	0.404587	+	0.914499i	$0.367415\pi$
$632$	−16.4721	−0.655226
$633$	5.88854	0.234049
$634$	2.47214	0.0981811
$635$	−6.47214	−0.256839
$636$	−12.3607	−0.490133
$637$	1.14590	0.0454021
$638$	5.63932	0.223263
$639$	10.2229	0.404412
$640$	1.00000	0.0395285
$641$	−17.8885	−0.706555	−0.353278	−	0.935519i	$-0.614933\pi$
$641$	−17.8885	−0.706555	−0.353278	+	0.935519i	$0.614933\pi$
$642$	−11.5967	−0.457687
$643$	2.79837	0.110357	0.0551785	−	0.998477i	$-0.482427\pi$
$643$	2.79837	0.110357	0.0551785	+	0.998477i	$0.482427\pi$
$644$	2.43769	0.0960586
$645$	6.47214	0.254840
$646$	32.7426	1.28824
$647$	−28.9443	−1.13792	−0.568958	−	0.822366i	$-0.692653\pi$
$647$	−28.9443	−1.13792	−0.568958	+	0.822366i	$0.692653\pi$
$648$	2.41641	0.0949255
$649$	−5.23607	−0.205534
$650$	−1.00000	−0.0392232
$651$	3.52786	0.138268
$652$	1.05573	0.0413455
$653$	8.11146	0.317426	0.158713	−	0.987325i	$-0.449266\pi$
$653$	8.11146	0.317426	0.158713	+	0.987325i	$0.449266\pi$
$654$	−5.41641	−0.211798
$655$	10.7639	0.420582
$656$	−1.70820	−0.0666942
$657$	4.76393	0.185859
$658$	31.6525	1.23394
$659$	−40.0689	−1.56086	−0.780431	−	0.625242i	$-0.785000\pi$
$659$	−40.0689	−1.56086	−0.780431	+	0.625242i	$0.785000\pi$
$660$	−0.944272	−0.0367557
$661$	23.6180	0.918635	0.459318	−	0.888272i	$-0.348094\pi$
$661$	23.6180	0.918635	0.459318	+	0.888272i	$0.348094\pi$
$662$	10.4721	0.407011
$663$	−5.70820	−0.221688
$664$	1.85410	0.0719531
$665$	20.2361	0.784721
$666$	−14.8541	−0.575585
$667$	6.30495	0.244129
$668$	−18.4721	−0.714708
$669$	−13.7508	−0.531636
$670$	−0.472136	−0.0182402
$671$	−5.01316	−0.193531
$672$	3.52786	0.136090
$673$	18.9230	0.729427	0.364714	−	0.931120i	$-0.381167\pi$
$673$	18.9230	0.729427	0.364714	+	0.931120i	$0.381167\pi$
$674$	6.74265	0.259717
$675$	5.52786	0.212768
$676$	1.00000	0.0384615
$677$	−13.3475	−0.512987	−0.256494	−	0.966546i	$-0.582567\pi$
$677$	−13.3475	−0.512987	−0.256494	+	0.966546i	$0.582567\pi$
$678$	0.360680	0.0138518
$679$	−26.7771	−1.02761
$680$	4.61803	0.177094
$681$	−28.5836	−1.09533
$682$	−0.763932	−0.0292525
$683$	−17.5967	−0.673321	−0.336660	−	0.941626i	$-0.609297\pi$
$683$	−17.5967	−0.673321	−0.336660	+	0.941626i	$0.609297\pi$
$684$	10.4377	0.399095
$685$	−16.0000	−0.611329
$686$	16.7082	0.637922
$687$	17.5279	0.668730
$688$	5.23607	0.199623
$689$	10.0000	0.380970
$690$	−1.05573	−0.0401909
$691$	−7.79837	−0.296664	−0.148332	−	0.988938i	$-0.547390\pi$
$691$	−7.79837	−0.296664	−0.148332	+	0.988938i	$0.547390\pi$
$692$	12.8541	0.488640
$693$	3.20976	0.121929
$694$	−22.4721	−0.853031
$695$	4.79837	0.182013
$696$	9.12461	0.345868
$697$	−7.88854	−0.298800
$698$	−10.6180	−0.401899
$699$	−12.3607	−0.467524
$700$	2.85410	0.107875
$701$	21.7082	0.819908	0.409954	−	0.912106i	$-0.365545\pi$
$701$	21.7082	0.819908	0.409954	+	0.912106i	$0.365545\pi$
$702$	−5.52786	−0.208636
$703$	71.5410	2.69822
$704$	−0.763932	−0.0287918
$705$	−13.7082	−0.516281
$706$	−15.5967	−0.586991
$707$	−7.05573	−0.265358
$708$	−8.47214	−0.318402
$709$	29.0132	1.08961	0.544806	−	0.838562i	$-0.316603\pi$
$709$	29.0132	1.08961	0.544806	+	0.838562i	$0.316603\pi$
$710$	−6.94427	−0.260614
$711$	−24.2492	−0.909417
$712$	−10.8541	−0.406775
$713$	−0.854102	−0.0319864
$714$	16.2918	0.609705
$715$	0.763932	0.0285694
$716$	19.3262	0.722255
$717$	−25.1246	−0.938296
$718$	−3.52786	−0.131659
$719$	43.4164	1.61916	0.809579	−	0.587010i	$-0.199695\pi$
$719$	43.4164	1.61916	0.809579	+	0.587010i	$0.199695\pi$
$720$	1.47214	0.0548633
$721$	2.69505	0.100369
$722$	−31.2705	−1.16377
$723$	−10.5836	−0.393608
$724$	3.52786	0.131112
$725$	7.38197	0.274159
$726$	−12.8754	−0.477850
$727$	19.0557	0.706738	0.353369	−	0.935484i	$-0.385036\pi$
$727$	19.0557	0.706738	0.353369	+	0.935484i	$0.385036\pi$
$728$	−2.85410	−0.105780
$729$	24.0557	0.890953
$730$	−3.23607	−0.119772
$731$	24.1803	0.894342
$732$	−8.11146	−0.299808
$733$	43.5967	1.61028	0.805141	−	0.593083i	$-0.202089\pi$
$733$	43.5967	1.61028	0.805141	+	0.593083i	$0.202089\pi$
$734$	−10.6180	−0.391919
$735$	1.41641	0.0522450
$736$	−0.854102	−0.0314826
$737$	0.360680	0.0132858
$738$	−2.51471	−0.0925677
$739$	8.83282	0.324920	0.162460	−	0.986715i	$-0.448057\pi$
$739$	8.83282	0.324920	0.162460	+	0.986715i	$0.448057\pi$
$740$	10.0902	0.370922
$741$	8.76393	0.321951
$742$	−28.5410	−1.04777
$743$	10.4721	0.384185	0.192093	−	0.981377i	$-0.438473\pi$
$743$	10.4721	0.384185	0.192093	+	0.981377i	$0.438473\pi$
$744$	−1.23607	−0.0453165
$745$	−17.1459	−0.628177
$746$	28.8541	1.05642
$747$	2.72949	0.0998668
$748$	−3.52786	−0.128991
$749$	−26.7771	−0.978413
$750$	−1.23607	−0.0451348
$751$	43.2705	1.57896	0.789482	−	0.613774i	$-0.210349\pi$
$751$	43.2705	1.57896	0.789482	+	0.613774i	$0.210349\pi$
$752$	−11.0902	−0.404417
$753$	2.87539	0.104785
$754$	−7.38197	−0.268835
$755$	−12.3820	−0.450626
$756$	15.7771	0.573807
$757$	50.3607	1.83039	0.915195	−	0.403011i	$-0.132036\pi$
$757$	50.3607	1.83039	0.915195	+	0.403011i	$0.132036\pi$
$758$	−16.5623	−0.601570
$759$	0.806504	0.0292743
$760$	−7.09017	−0.257187
$761$	51.9787	1.88423	0.942113	−	0.335294i	$-0.108836\pi$
$761$	51.9787	1.88423	0.942113	+	0.335294i	$0.108836\pi$
$762$	8.00000	0.289809
$763$	−12.5066	−0.452769
$764$	4.79837	0.173599
$765$	6.79837	0.245796
$766$	16.2918	0.588647
$767$	6.85410	0.247487
$768$	−1.23607	−0.0446028
$769$	21.8885	0.789321	0.394661	−	0.918827i	$-0.370862\pi$
$769$	21.8885	0.789321	0.394661	+	0.918827i	$0.370862\pi$
$770$	−2.18034	−0.0785740
$771$	−33.8885	−1.22047
$772$	14.6180	0.526115
$773$	22.9787	0.826487	0.413243	−	0.910621i	$-0.364396\pi$
$773$	22.9787	0.826487	0.413243	+	0.910621i	$0.364396\pi$
$774$	7.70820	0.277066
$775$	−1.00000	−0.0359211
$776$	9.38197	0.336793
$777$	35.5967	1.27703
$778$	−21.4164	−0.767815
$779$	12.1115	0.433938
$780$	1.23607	0.0442583
$781$	5.30495	0.189826
$782$	−3.94427	−0.141047
$783$	40.8065	1.45831
$784$	1.14590	0.0409249
$785$	21.5623	0.769592
$786$	−13.3050	−0.474572
$787$	−23.6738	−0.843878	−0.421939	−	0.906624i	$-0.638650\pi$
$787$	−23.6738	−0.843878	−0.421939	+	0.906624i	$0.638650\pi$
$788$	−18.0344	−0.642450
$789$	8.98684	0.319940
$790$	16.4721	0.586052
$791$	0.832816	0.0296115
$792$	−1.12461	−0.0399613
$793$	6.56231	0.233034
$794$	−31.3050	−1.11097
$795$	12.3607	0.438388
$796$	17.4164	0.617308
$797$	−36.4721	−1.29191	−0.645955	−	0.763376i	$-0.723541\pi$
$797$	−36.4721	−1.29191	−0.645955	+	0.763376i	$0.723541\pi$
$798$	−25.0132	−0.885456
$799$	−51.2148	−1.81185
$800$	−1.00000	−0.0353553
$801$	−15.9787	−0.564580
$802$	−6.90983	−0.243995
$803$	2.47214	0.0872398
$804$	0.583592	0.0205817
$805$	−2.43769	−0.0859174
$806$	1.00000	0.0352235
$807$	−32.1115	−1.13038
$808$	2.47214	0.0869694
$809$	−14.0689	−0.494636	−0.247318	−	0.968934i	$-0.579549\pi$
$809$	−14.0689	−0.494636	−0.247318	+	0.968934i	$0.579549\pi$
$810$	−2.41641	−0.0849039
$811$	−50.8328	−1.78498	−0.892491	−	0.451066i	$-0.851044\pi$
$811$	−50.8328	−1.78498	−0.892491	+	0.451066i	$0.851044\pi$
$812$	21.0689	0.739373
$813$	−8.76393	−0.307365
$814$	−7.70820	−0.270172
$815$	−1.05573	−0.0369805
$816$	−5.70820	−0.199827
$817$	−37.1246	−1.29883
$818$	−12.4721	−0.436078
$819$	−4.20163	−0.146817
$820$	1.70820	0.0596531
$821$	−30.3607	−1.05960	−0.529798	−	0.848124i	$-0.677732\pi$
$821$	−30.3607	−1.05960	−0.529798	+	0.848124i	$0.677732\pi$
$822$	19.7771	0.689805
$823$	−42.5623	−1.48363	−0.741814	−	0.670605i	$-0.766034\pi$
$823$	−42.5623	−1.48363	−0.741814	+	0.670605i	$0.766034\pi$
$824$	−0.944272	−0.0328953
$825$	0.944272	0.0328753
$826$	−19.5623	−0.680660
$827$	−41.1459	−1.43078	−0.715392	−	0.698724i	$-0.753752\pi$
$827$	−41.1459	−1.43078	−0.715392	+	0.698724i	$0.753752\pi$
$828$	−1.25735	−0.0436961
$829$	10.4377	0.362516	0.181258	−	0.983436i	$-0.441983\pi$
$829$	10.4377	0.362516	0.181258	+	0.983436i	$0.441983\pi$
$830$	−1.85410	−0.0643568
$831$	−1.16718	−0.0404892
$832$	1.00000	0.0346688
$833$	5.29180	0.183350
$834$	−5.93112	−0.205378
$835$	18.4721	0.639255
$836$	5.41641	0.187330
$837$	−5.52786	−0.191071
$838$	−12.1803	−0.420763
$839$	−31.7771	−1.09707	−0.548533	−	0.836129i	$-0.684814\pi$
$839$	−31.7771	−1.09707	−0.548533	+	0.836129i	$0.684814\pi$
$840$	−3.52786	−0.121723
$841$	25.4934	0.879084
$842$	24.7984	0.854608
$843$	−35.0557	−1.20738
$844$	−4.76393	−0.163981
$845$	−1.00000	−0.0344010
$846$	−16.3262	−0.561308
$847$	−29.7295	−1.02152
$848$	10.0000	0.343401
$849$	21.4853	0.737373
$850$	−4.61803	−0.158397
$851$	−8.61803	−0.295422
$852$	8.58359	0.294069
$853$	32.8328	1.12417	0.562087	−	0.827078i	$-0.309999\pi$
$853$	32.8328	1.12417	0.562087	+	0.827078i	$0.309999\pi$
$854$	−18.7295	−0.640910
$855$	−10.4377	−0.356962
$856$	9.38197	0.320669
$857$	9.70820	0.331626	0.165813	−	0.986157i	$-0.446975\pi$
$857$	9.70820	0.331626	0.165813	+	0.986157i	$0.446975\pi$
$858$	−0.944272	−0.0322369
$859$	−20.2705	−0.691621	−0.345810	−	0.938304i	$-0.612396\pi$
$859$	−20.2705	−0.691621	−0.345810	+	0.938304i	$0.612396\pi$
$860$	−5.23607	−0.178548
$861$	6.02631	0.205376
$862$	−19.7082	−0.671264
$863$	−25.7771	−0.877462	−0.438731	−	0.898618i	$-0.644572\pi$
$863$	−25.7771	−0.877462	−0.438731	+	0.898618i	$0.644572\pi$
$864$	−5.52786	−0.188062
$865$	−12.8541	−0.437053
$866$	−27.5066	−0.934712
$867$	−5.34752	−0.181611
$868$	−2.85410	−0.0968745
$869$	−12.5836	−0.426869
$870$	−9.12461	−0.309353
$871$	−0.472136	−0.0159977
$872$	4.38197	0.148392
$873$	13.8115	0.467449
$874$	6.05573	0.204838
$875$	−2.85410	−0.0964863
$876$	4.00000	0.135147
$877$	−50.3607	−1.70056	−0.850280	−	0.526331i	$-0.823567\pi$
$877$	−50.3607	−1.70056	−0.850280	+	0.526331i	$0.823567\pi$
$878$	−17.7984	−0.600666
$879$	−41.8885	−1.41287
$880$	0.763932	0.0257521
$881$	−33.4164	−1.12583	−0.562914	−	0.826516i	$-0.690320\pi$
$881$	−33.4164	−1.12583	−0.562914	+	0.826516i	$0.690320\pi$
$882$	1.68692	0.0568015
$883$	45.9574	1.54659	0.773295	−	0.634046i	$-0.218607\pi$
$883$	45.9574	1.54659	0.773295	+	0.634046i	$0.218607\pi$
$884$	4.61803	0.155321
$885$	8.47214	0.284788
$886$	−26.4721	−0.889349
$887$	−58.2492	−1.95582	−0.977909	−	0.209032i	$-0.932969\pi$
$887$	−58.2492	−1.95582	−0.977909	+	0.209032i	$0.932969\pi$
$888$	−12.4721	−0.418537
$889$	18.4721	0.619536
$890$	10.8541	0.363830
$891$	1.84597	0.0618424
$892$	11.1246	0.372480
$893$	78.6312	2.63129
$894$	21.1935	0.708817
$895$	−19.3262	−0.646005
$896$	−2.85410	−0.0953489
$897$	−1.05573	−0.0352497
$898$	27.5623	0.919766
$899$	−7.38197	−0.246202
$900$	−1.47214	−0.0490712
$901$	46.1803	1.53849
$902$	−1.30495	−0.0434501
$903$	−18.4721	−0.614714
$904$	−0.291796	−0.00970499
$905$	−3.52786	−0.117270
$906$	15.3050	0.508473
$907$	−38.3820	−1.27445	−0.637226	−	0.770677i	$-0.719918\pi$
$907$	−38.3820	−1.27445	−0.637226	+	0.770677i	$0.719918\pi$
$908$	23.1246	0.767417
$909$	3.63932	0.120709
$910$	2.85410	0.0946126
$911$	22.0000	0.728893	0.364446	−	0.931224i	$-0.381258\pi$
$911$	22.0000	0.728893	0.364446	+	0.931224i	$0.381258\pi$
$912$	8.76393	0.290203
$913$	1.41641	0.0468763
$914$	−37.8885	−1.25324
$915$	8.11146	0.268156
$916$	−14.1803	−0.468532
$917$	−30.7214	−1.01451
$918$	−25.5279	−0.842545
$919$	4.49342	0.148224	0.0741122	−	0.997250i	$-0.476388\pi$
$919$	4.49342	0.148224	0.0741122	+	0.997250i	$0.476388\pi$
$920$	0.854102	0.0281589
$921$	10.1115	0.333184
$922$	10.1803	0.335272
$923$	−6.94427	−0.228573
$924$	2.69505	0.0886606
$925$	−10.0902	−0.331763
$926$	11.0557	0.363314
$927$	−1.39010	−0.0456568
$928$	−7.38197	−0.242325
$929$	55.0476	1.80605	0.903027	−	0.429585i	$-0.141340\pi$
$929$	55.0476	1.80605	0.903027	+	0.429585i	$0.141340\pi$
$930$	1.23607	0.0405323
$931$	−8.12461	−0.266273
$932$	10.0000	0.327561
$933$	−42.0000	−1.37502
$934$	10.2016	0.333807
$935$	3.52786	0.115373
$936$	1.47214	0.0481183
$937$	−26.9443	−0.880231	−0.440115	−	0.897941i	$-0.645063\pi$
$937$	−26.9443	−0.880231	−0.440115	+	0.897941i	$0.645063\pi$
$938$	1.34752	0.0439982
$939$	−5.52786	−0.180395
$940$	11.0902	0.361721
$941$	−1.70820	−0.0556859	−0.0278429	−	0.999612i	$-0.508864\pi$
$941$	−1.70820	−0.0556859	−0.0278429	+	0.999612i	$0.508864\pi$
$942$	−26.6525	−0.868385
$943$	−1.45898	−0.0475109
$944$	6.85410	0.223082
$945$	−15.7771	−0.513229
$946$	4.00000	0.130051
$947$	−12.0000	−0.389948	−0.194974	−	0.980808i	$-0.562462\pi$
$947$	−12.0000	−0.389948	−0.194974	+	0.980808i	$0.562462\pi$
$948$	−20.3607	−0.661284
$949$	−3.23607	−0.105047
$950$	7.09017	0.230035
$951$	3.05573	0.0990888
$952$	−13.1803	−0.427177
$953$	−21.8541	−0.707924	−0.353962	−	0.935260i	$-0.615166\pi$
$953$	−21.8541	−0.707924	−0.353962	+	0.935260i	$0.615166\pi$
$954$	14.7214	0.476622
$955$	−4.79837	−0.155272
$956$	20.3262	0.657398
$957$	6.97058	0.225327
$958$	−26.7639	−0.864703
$959$	45.6656	1.47462
$960$	1.23607	0.0398939
$961$	1.00000	0.0322581
$962$	10.0902	0.325320
$963$	13.8115	0.445070
$964$	8.56231	0.275773
$965$	−14.6180	−0.470571
$966$	3.01316	0.0969467
$967$	4.06888	0.130846	0.0654232	−	0.997858i	$-0.479160\pi$
$967$	4.06888	0.130846	0.0654232	+	0.997858i	$0.479160\pi$
$968$	10.4164	0.334796
$969$	40.4721	1.30015
$970$	−9.38197	−0.301237
$971$	−16.4721	−0.528616	−0.264308	−	0.964438i	$-0.585144\pi$
$971$	−16.4721	−0.528616	−0.264308	+	0.964438i	$0.585144\pi$
$972$	−13.5967	−0.436116
$973$	−13.6950	−0.439043
$974$	−38.0689	−1.21981
$975$	−1.23607	−0.0395859
$976$	6.56231	0.210054
$977$	32.2016	1.03022	0.515111	−	0.857124i	$-0.327751\pi$
$977$	32.2016	1.03022	0.515111	+	0.857124i	$0.327751\pi$
$978$	1.30495	0.0417278
$979$	−8.29180	−0.265007
$980$	−1.14590	−0.0366044
$981$	6.45085	0.205960
$982$	5.79837	0.185034
$983$	−7.34752	−0.234350	−0.117175	−	0.993111i	$-0.537384\pi$
$983$	−7.34752	−0.234350	−0.117175	+	0.993111i	$0.537384\pi$
$984$	−2.11146	−0.0673108
$985$	18.0344	0.574625
$986$	−34.0902	−1.08565
$987$	39.1246	1.24535
$988$	−7.09017	−0.225568
$989$	4.47214	0.142206
$990$	1.12461	0.0357425
$991$	11.4164	0.362654	0.181327	−	0.983423i	$-0.441961\pi$
$991$	11.4164	0.362654	0.181327	+	0.983423i	$0.441961\pi$
$992$	1.00000	0.0317500
$993$	12.9443	0.410774
$994$	19.8197	0.628641
$995$	−17.4164	−0.552137
$996$	2.29180	0.0726183
$997$	−47.3820	−1.50060	−0.750301	−	0.661096i	$-0.770091\pi$
$997$	−47.3820	−1.50060	−0.750301	+	0.661096i	$0.770091\pi$
$998$	21.4164	0.677925
$999$	−55.7771	−1.76471

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4030.2.a.b.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4030.2.a.b.1.1	✓	2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4030 = 2 \cdot 5 \cdot 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4030.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.23607 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.23607 q^{6} +2.85410 q^{7} -1.00000 q^{8} -1.47214 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.23607 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.23607 q^{6} +2.85410 q^{7} -1.00000 q^{8} -1.47214 q^{9} +1.00000 q^{10} -0.763932 q^{11} -1.23607 q^{12} +1.00000 q^{13} -2.85410 q^{14} +1.23607 q^{15} +1.00000 q^{16} +4.61803 q^{17} +1.47214 q^{18} -7.09017 q^{19} -1.00000 q^{20} -3.52786 q^{21} +0.763932 q^{22} +0.854102 q^{23} +1.23607 q^{24} +1.00000 q^{25} -1.00000 q^{26} +5.52786 q^{27} +2.85410 q^{28} +7.38197 q^{29} -1.23607 q^{30} -1.00000 q^{31} -1.00000 q^{32} +0.944272 q^{33} -4.61803 q^{34} -2.85410 q^{35} -1.47214 q^{36} -10.0902 q^{37} +7.09017 q^{38} -1.23607 q^{39} +1.00000 q^{40} -1.70820 q^{41} +3.52786 q^{42} +5.23607 q^{43} -0.763932 q^{44} +1.47214 q^{45} -0.854102 q^{46} -11.0902 q^{47} -1.23607 q^{48} +1.14590 q^{49} -1.00000 q^{50} -5.70820 q^{51} +1.00000 q^{52} +10.0000 q^{53} -5.52786 q^{54} +0.763932 q^{55} -2.85410 q^{56} +8.76393 q^{57} -7.38197 q^{58} +6.85410 q^{59} +1.23607 q^{60} +6.56231 q^{61} +1.00000 q^{62} -4.20163 q^{63} +1.00000 q^{64} -1.00000 q^{65} -0.944272 q^{66} -0.472136 q^{67} +4.61803 q^{68} -1.05573 q^{69} +2.85410 q^{70} -6.94427 q^{71} +1.47214 q^{72} -3.23607 q^{73} +10.0902 q^{74} -1.23607 q^{75} -7.09017 q^{76} -2.18034 q^{77} +1.23607 q^{78} +16.4721 q^{79} -1.00000 q^{80} -2.41641 q^{81} +1.70820 q^{82} -1.85410 q^{83} -3.52786 q^{84} -4.61803 q^{85} -5.23607 q^{86} -9.12461 q^{87} +0.763932 q^{88} +10.8541 q^{89} -1.47214 q^{90} +2.85410 q^{91} +0.854102 q^{92} +1.23607 q^{93} +11.0902 q^{94} +7.09017 q^{95} +1.23607 q^{96} -9.38197 q^{97} -1.14590 q^{98} +1.12461 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} - q^{7} - 2 q^{8} + 6 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 - q^7 - 2 * q^8 + 6 * q^9	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} - q^{7} - 2 q^{8} + 6 q^{9} + 2 q^{10} - 6 q^{11} + 2 q^{12} + 2 q^{13} + q^{14} - 2 q^{15} + 2 q^{16} + 7 q^{17} - 6 q^{18} - 3 q^{19} - 2 q^{20} - 16 q^{21} + 6 q^{22} - 5 q^{23} - 2 q^{24} + 2 q^{25} - 2 q^{26} + 20 q^{27} - q^{28} + 17 q^{29} + 2 q^{30} - 2 q^{31} - 2 q^{32} - 16 q^{33} - 7 q^{34} + q^{35} + 6 q^{36} - 9 q^{37} + 3 q^{38} + 2 q^{39} + 2 q^{40} + 10 q^{41} + 16 q^{42} + 6 q^{43} - 6 q^{44} - 6 q^{45} + 5 q^{46} - 11 q^{47} + 2 q^{48} + 9 q^{49} - 2 q^{50} + 2 q^{51} + 2 q^{52} + 20 q^{53} - 20 q^{54} + 6 q^{55} + q^{56} + 22 q^{57} - 17 q^{58} + 7 q^{59} - 2 q^{60} - 7 q^{61} + 2 q^{62} - 33 q^{63} + 2 q^{64} - 2 q^{65} + 16 q^{66} + 8 q^{67} + 7 q^{68} - 20 q^{69} - q^{70} + 4 q^{71} - 6 q^{72} - 2 q^{73} + 9 q^{74} + 2 q^{75} - 3 q^{76} + 18 q^{77} - 2 q^{78} + 24 q^{79} - 2 q^{80} + 22 q^{81} - 10 q^{82} + 3 q^{83} - 16 q^{84} - 7 q^{85} - 6 q^{86} + 22 q^{87} + 6 q^{88} + 15 q^{89} + 6 q^{90} - q^{91} - 5 q^{92} - 2 q^{93} + 11 q^{94} + 3 q^{95} - 2 q^{96} - 21 q^{97} - 9 q^{98} - 38 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 - q^7 - 2 * q^8 + 6 * q^9 + 2 * q^10 - 6 * q^11 + 2 * q^12 + 2 * q^13 + q^14 - 2 * q^15 + 2 * q^16 + 7 * q^17 - 6 * q^18 - 3 * q^19 - 2 * q^20 - 16 * q^21 + 6 * q^22 - 5 * q^23 - 2 * q^24 + 2 * q^25 - 2 * q^26 + 20 * q^27 - q^28 + 17 * q^29 + 2 * q^30 - 2 * q^31 - 2 * q^32 - 16 * q^33 - 7 * q^34 + q^35 + 6 * q^36 - 9 * q^37 + 3 * q^38 + 2 * q^39 + 2 * q^40 + 10 * q^41 + 16 * q^42 + 6 * q^43 - 6 * q^44 - 6 * q^45 + 5 * q^46 - 11 * q^47 + 2 * q^48 + 9 * q^49 - 2 * q^50 + 2 * q^51 + 2 * q^52 + 20 * q^53 - 20 * q^54 + 6 * q^55 + q^56 + 22 * q^57 - 17 * q^58 + 7 * q^59 - 2 * q^60 - 7 * q^61 + 2 * q^62 - 33 * q^63 + 2 * q^64 - 2 * q^65 + 16 * q^66 + 8 * q^67 + 7 * q^68 - 20 * q^69 - q^70 + 4 * q^71 - 6 * q^72 - 2 * q^73 + 9 * q^74 + 2 * q^75 - 3 * q^76 + 18 * q^77 - 2 * q^78 + 24 * q^79 - 2 * q^80 + 22 * q^81 - 10 * q^82 + 3 * q^83 - 16 * q^84 - 7 * q^85 - 6 * q^86 + 22 * q^87 + 6 * q^88 + 15 * q^89 + 6 * q^90 - q^91 - 5 * q^92 - 2 * q^93 + 11 * q^94 + 3 * q^95 - 2 * q^96 - 21 * q^97 - 9 * q^98 - 38 * q^99

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