LMFDB - Embedded newform 8045.2.a.b.1.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8045,2,Mod(1,8045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8045, 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("8045.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8045 = 5 \cdot 1609 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8045.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:	$64.2396484261$
Analytic rank:	$1$
Dimension:	$126$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
Character	$\chi$	$=$	8045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.03743 q^{2} +2.30287 q^{3} +2.15111 q^{4} -1.00000 q^{5} -4.69193 q^{6} -3.80447 q^{7} -0.307877 q^{8} +2.30320 q^{9} +O(q^{10})$	\(q-2.03743 q^{2} +2.30287 q^{3} +2.15111 q^{4} -1.00000 q^{5} -4.69193 q^{6} -3.80447 q^{7} -0.307877 q^{8} +2.30320 q^{9} +2.03743 q^{10} -3.67798 q^{11} +4.95373 q^{12} +5.18841 q^{13} +7.75134 q^{14} -2.30287 q^{15} -3.67494 q^{16} +1.49136 q^{17} -4.69261 q^{18} +0.771807 q^{19} -2.15111 q^{20} -8.76121 q^{21} +7.49363 q^{22} -4.19421 q^{23} -0.709000 q^{24} +1.00000 q^{25} -10.5710 q^{26} -1.60463 q^{27} -8.18385 q^{28} +9.03381 q^{29} +4.69193 q^{30} -4.46079 q^{31} +8.10319 q^{32} -8.46992 q^{33} -3.03854 q^{34} +3.80447 q^{35} +4.95445 q^{36} -1.76051 q^{37} -1.57250 q^{38} +11.9482 q^{39} +0.307877 q^{40} +4.51269 q^{41} +17.8503 q^{42} +2.32501 q^{43} -7.91175 q^{44} -2.30320 q^{45} +8.54540 q^{46} +5.84554 q^{47} -8.46291 q^{48} +7.47403 q^{49} -2.03743 q^{50} +3.43441 q^{51} +11.1608 q^{52} -1.84116 q^{53} +3.26931 q^{54} +3.67798 q^{55} +1.17131 q^{56} +1.77737 q^{57} -18.4057 q^{58} -3.61920 q^{59} -4.95373 q^{60} +8.39987 q^{61} +9.08854 q^{62} -8.76249 q^{63} -9.15977 q^{64} -5.18841 q^{65} +17.2568 q^{66} -3.79997 q^{67} +3.20808 q^{68} -9.65872 q^{69} -7.75134 q^{70} -8.96489 q^{71} -0.709104 q^{72} +4.06595 q^{73} +3.58692 q^{74} +2.30287 q^{75} +1.66024 q^{76} +13.9928 q^{77} -24.3436 q^{78} +4.41518 q^{79} +3.67494 q^{80} -10.6049 q^{81} -9.19429 q^{82} -11.0688 q^{83} -18.8463 q^{84} -1.49136 q^{85} -4.73704 q^{86} +20.8037 q^{87} +1.13237 q^{88} +0.424685 q^{89} +4.69261 q^{90} -19.7392 q^{91} -9.02221 q^{92} -10.2726 q^{93} -11.9099 q^{94} -0.771807 q^{95} +18.6606 q^{96} -5.61715 q^{97} -15.2278 q^{98} -8.47115 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 126 q + 5 q^{2} - 9 q^{3} + 109 q^{4} - 126 q^{5} - 21 q^{6} - 23 q^{7} + 12 q^{8} + 109 q^{9}+O(q^{10}) $ 126 * q + 5 * q^2 - 9 * q^3 + 109 * q^4 - 126 * q^5 - 21 * q^6 - 23 * q^7 + 12 * q^8 + 109 * q^9	\( 126 q + 5 q^{2} - 9 q^{3} + 109 q^{4} - 126 q^{5} - 21 q^{6} - 23 q^{7} + 12 q^{8} + 109 q^{9} - 5 q^{10} - 44 q^{11} - 11 q^{12} - 35 q^{13} - 14 q^{14} + 9 q^{15} + 75 q^{16} + 11 q^{17} - 15 q^{18} - 130 q^{19} - 109 q^{20} - 44 q^{21} - 14 q^{22} + 75 q^{23} - 63 q^{24} + 126 q^{25} - 43 q^{26} - 42 q^{27} - 77 q^{28} - 24 q^{29} + 21 q^{30} - 78 q^{31} + 24 q^{32} - 29 q^{33} - 57 q^{34} + 23 q^{35} + 50 q^{36} - 31 q^{37} - 3 q^{38} - 57 q^{39} - 12 q^{40} - 38 q^{41} - 10 q^{42} - 100 q^{43} - 90 q^{44} - 109 q^{45} - 96 q^{46} + 12 q^{47} - 22 q^{48} + 65 q^{49} + 5 q^{50} - 74 q^{51} - 112 q^{52} + 20 q^{53} - 90 q^{54} + 44 q^{55} - 57 q^{56} + 6 q^{57} - 35 q^{58} - 97 q^{59} + 11 q^{60} - 102 q^{61} - 16 q^{62} - 15 q^{63} + 4 q^{64} + 35 q^{65} - 83 q^{66} - 121 q^{67} + 41 q^{68} - 71 q^{69} + 14 q^{70} - 32 q^{71} - 32 q^{72} - 85 q^{73} - 42 q^{74} - 9 q^{75} - 275 q^{76} + 13 q^{77} + 10 q^{78} - 97 q^{79} - 75 q^{80} + 86 q^{81} - 55 q^{82} - 73 q^{83} - 111 q^{84} - 11 q^{85} - 56 q^{86} - q^{87} - 37 q^{88} - 67 q^{89} + 15 q^{90} - 180 q^{91} + 98 q^{92} - 44 q^{93} - 86 q^{94} + 130 q^{95} - 179 q^{96} - 50 q^{97} + 18 q^{98} - 217 q^{99}+O(q^{100}) \) 126 * q + 5 * q^2 - 9 * q^3 + 109 * q^4 - 126 * q^5 - 21 * q^6 - 23 * q^7 + 12 * q^8 + 109 * q^9 - 5 * q^10 - 44 * q^11 - 11 * q^12 - 35 * q^13 - 14 * q^14 + 9 * q^15 + 75 * q^16 + 11 * q^17 - 15 * q^18 - 130 * q^19 - 109 * q^20 - 44 * q^21 - 14 * q^22 + 75 * q^23 - 63 * q^24 + 126 * q^25 - 43 * q^26 - 42 * q^27 - 77 * q^28 - 24 * q^29 + 21 * q^30 - 78 * q^31 + 24 * q^32 - 29 * q^33 - 57 * q^34 + 23 * q^35 + 50 * q^36 - 31 * q^37 - 3 * q^38 - 57 * q^39 - 12 * q^40 - 38 * q^41 - 10 * q^42 - 100 * q^43 - 90 * q^44 - 109 * q^45 - 96 * q^46 + 12 * q^47 - 22 * q^48 + 65 * q^49 + 5 * q^50 - 74 * q^51 - 112 * q^52 + 20 * q^53 - 90 * q^54 + 44 * q^55 - 57 * q^56 + 6 * q^57 - 35 * q^58 - 97 * q^59 + 11 * q^60 - 102 * q^61 - 16 * q^62 - 15 * q^63 + 4 * q^64 + 35 * q^65 - 83 * q^66 - 121 * q^67 + 41 * q^68 - 71 * q^69 + 14 * q^70 - 32 * q^71 - 32 * q^72 - 85 * q^73 - 42 * q^74 - 9 * q^75 - 275 * q^76 + 13 * q^77 + 10 * q^78 - 97 * q^79 - 75 * q^80 + 86 * q^81 - 55 * q^82 - 73 * q^83 - 111 * q^84 - 11 * q^85 - 56 * q^86 - q^87 - 37 * q^88 - 67 * q^89 + 15 * q^90 - 180 * q^91 + 98 * q^92 - 44 * q^93 - 86 * q^94 + 130 * q^95 - 179 * q^96 - 50 * q^97 + 18 * q^98 - 217 * 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 $

$2$	−2.03743	−1.44068	−0.720339	−	0.693622i	$-0.756014\pi$
$2$	−2.03743	−1.44068	−0.720339	+	0.693622i	$0.756014\pi$
$3$	2.30287	1.32956	0.664781	−	0.747038i	$-0.268525\pi$
$3$	2.30287	1.32956	0.664781	+	0.747038i	$0.268525\pi$
$4$	2.15111	1.07556
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−4.69193	−1.91547
$7$	−3.80447	−1.43796	−0.718978	−	0.695033i	$-0.755390\pi$
$7$	−3.80447	−1.43796	−0.718978	+	0.695033i	$0.755390\pi$
$8$	−0.307877	−0.108851
$9$	2.30320	0.767735
$10$	2.03743	0.644291
$11$	−3.67798	−1.10895	−0.554477	−	0.832199i	$-0.687082\pi$
$11$	−3.67798	−1.10895	−0.554477	+	0.832199i	$0.687082\pi$
$12$	4.95373	1.43002
$13$	5.18841	1.43900	0.719502	−	0.694490i	$-0.244370\pi$
$13$	5.18841	1.43900	0.719502	+	0.694490i	$0.244370\pi$
$14$	7.75134	2.07163
$15$	−2.30287	−0.594598
$16$	−3.67494	−0.918736
$17$	1.49136	0.361708	0.180854	−	0.983510i	$-0.442114\pi$
$17$	1.49136	0.361708	0.180854	+	0.983510i	$0.442114\pi$
$18$	−4.69261	−1.10606
$19$	0.771807	0.177065	0.0885324	−	0.996073i	$-0.471782\pi$
$19$	0.771807	0.177065	0.0885324	+	0.996073i	$0.471782\pi$
$20$	−2.15111	−0.481003
$21$	−8.76121	−1.91185
$22$	7.49363	1.59765
$23$	−4.19421	−0.874553	−0.437277	−	0.899327i	$-0.644057\pi$
$23$	−4.19421	−0.874553	−0.437277	+	0.899327i	$0.644057\pi$
$24$	−0.709000	−0.144724
$25$	1.00000	0.200000
$26$	−10.5710	−2.07314
$27$	−1.60463	−0.308811
$28$	−8.18385	−1.54660
$29$	9.03381	1.67754	0.838768	−	0.544489i	$-0.183276\pi$
$29$	9.03381	1.67754	0.838768	+	0.544489i	$0.183276\pi$
$30$	4.69193	0.856625
$31$	−4.46079	−0.801182	−0.400591	−	0.916257i	$-0.631195\pi$
$31$	−4.46079	−0.801182	−0.400591	+	0.916257i	$0.631195\pi$
$32$	8.10319	1.43245
$33$	−8.46992	−1.47442
$34$	−3.03854	−0.521105
$35$	3.80447	0.643074
$36$	4.95445	0.825741
$37$	−1.76051	−0.289427	−0.144713	−	0.989474i	$-0.546226\pi$
$37$	−1.76051	−0.289427	−0.144713	+	0.989474i	$0.546226\pi$
$38$	−1.57250	−0.255093
$39$	11.9482	1.91325
$40$	0.307877	0.0486796
$41$	4.51269	0.704764	0.352382	−	0.935856i	$-0.385372\pi$
$41$	4.51269	0.704764	0.352382	+	0.935856i	$0.385372\pi$
$42$	17.8503	2.75436
$43$	2.32501	0.354561	0.177281	−	0.984160i	$-0.443270\pi$
$43$	2.32501	0.354561	0.177281	+	0.984160i	$0.443270\pi$
$44$	−7.91175	−1.19274
$45$	−2.30320	−0.343342
$46$	8.54540	1.25995
$47$	5.84554	0.852659	0.426329	−	0.904568i	$-0.359806\pi$
$47$	5.84554	0.852659	0.426329	+	0.904568i	$0.359806\pi$
$48$	−8.46291	−1.22152
$49$	7.47403	1.06772
$50$	−2.03743	−0.288136
$51$	3.43441	0.480913
$52$	11.1608	1.54773
$53$	−1.84116	−0.252903	−0.126452	−	0.991973i	$-0.540359\pi$
$53$	−1.84116	−0.252903	−0.126452	+	0.991973i	$0.540359\pi$
$54$	3.26931	0.444897
$55$	3.67798	0.495939
$56$	1.17131	0.156523
$57$	1.77737	0.235418
$58$	−18.4057	−2.41679
$59$	−3.61920	−0.471180	−0.235590	−	0.971853i	$-0.575702\pi$
$59$	−3.61920	−0.471180	−0.235590	+	0.971853i	$0.575702\pi$
$60$	−4.95373	−0.639523
$61$	8.39987	1.07549	0.537747	−	0.843106i	$-0.319276\pi$
$61$	8.39987	1.07549	0.537747	+	0.843106i	$0.319276\pi$
$62$	9.08854	1.15425
$63$	−8.76249	−1.10397
$64$	−9.15977	−1.14497
$65$	−5.18841	−0.643543
$66$	17.2568	2.12417
$67$	−3.79997	−0.464240	−0.232120	−	0.972687i	$-0.574566\pi$
$67$	−3.79997	−0.464240	−0.232120	+	0.972687i	$0.574566\pi$
$68$	3.20808	0.389037
$69$	−9.65872	−1.16277
$70$	−7.75134	−0.926463
$71$	−8.96489	−1.06394	−0.531968	−	0.846764i	$-0.678548\pi$
$71$	−8.96489	−1.06394	−0.531968	+	0.846764i	$0.678548\pi$
$72$	−0.709104	−0.0835687
$73$	4.06595	0.475883	0.237942	−	0.971279i	$-0.423527\pi$
$73$	4.06595	0.475883	0.237942	+	0.971279i	$0.423527\pi$
$74$	3.58692	0.416971
$75$	2.30287	0.265912
$76$	1.66024	0.190443
$77$	13.9928	1.59463
$78$	−24.3436	−2.75637
$79$	4.41518	0.496746	0.248373	−	0.968664i	$-0.420104\pi$
$79$	4.41518	0.496746	0.248373	+	0.968664i	$0.420104\pi$
$80$	3.67494	0.410871
$81$	−10.6049	−1.17832
$82$	−9.19429	−1.01534
$83$	−11.0688	−1.21496	−0.607480	−	0.794335i	$-0.707820\pi$
$83$	−11.0688	−1.21496	−0.607480	+	0.794335i	$0.707820\pi$
$84$	−18.8463	−2.05630
$85$	−1.49136	−0.161761
$86$	−4.73704	−0.510809
$87$	20.8037	2.23039
$88$	1.13237	0.120711
$89$	0.424685	0.0450165	0.0225083	−	0.999747i	$-0.492835\pi$
$89$	0.424685	0.0450165	0.0225083	+	0.999747i	$0.492835\pi$
$90$	4.69261	0.494645
$91$	−19.7392	−2.06923
$92$	−9.02221	−0.940631
$93$	−10.2726	−1.06522
$94$	−11.9099	−1.22841
$95$	−0.771807	−0.0791857
$96$	18.6606	1.90454
$97$	−5.61715	−0.570336	−0.285168	−	0.958478i	$-0.592049\pi$
$97$	−5.61715	−0.570336	−0.285168	+	0.958478i	$0.592049\pi$
$98$	−15.2278	−1.53824
$99$	−8.47115	−0.851383
$100$	2.15111	0.215111
$101$	7.87327	0.783420	0.391710	−	0.920089i	$-0.371884\pi$
$101$	7.87327	0.783420	0.391710	+	0.920089i	$0.371884\pi$
$102$	−6.99735	−0.692841
$103$	0.334114	0.0329212	0.0164606	−	0.999865i	$-0.494760\pi$
$103$	0.334114	0.0329212	0.0164606	+	0.999865i	$0.494760\pi$
$104$	−1.59739	−0.156637
$105$	8.76121	0.855006
$106$	3.75124	0.364352
$107$	11.0330	1.06660	0.533298	−	0.845927i	$-0.320952\pi$
$107$	11.0330	1.06660	0.533298	+	0.845927i	$0.320952\pi$
$108$	−3.45173	−0.332143
$109$	1.80358	0.172752	0.0863758	−	0.996263i	$-0.472471\pi$
$109$	1.80358	0.172752	0.0863758	+	0.996263i	$0.472471\pi$
$110$	−7.49363	−0.714489
$111$	−4.05423	−0.384811
$112$	13.9812	1.32110
$113$	−12.5184	−1.17763	−0.588817	−	0.808266i	$-0.700406\pi$
$113$	−12.5184	−1.17763	−0.588817	+	0.808266i	$0.700406\pi$
$114$	−3.62126	−0.339162
$115$	4.19421	0.391112
$116$	19.4327	1.80428
$117$	11.9500	1.10477
$118$	7.37386	0.678819
$119$	−5.67384	−0.520120
$120$	0.709000	0.0647226
$121$	2.52757	0.229779
$122$	−17.1141	−1.54944
$123$	10.3921	0.937028
$124$	−9.59565	−0.861715
$125$	−1.00000	−0.0894427
$126$	17.8529	1.59047
$127$	−13.5256	−1.20020	−0.600099	−	0.799925i	$-0.704872\pi$
$127$	−13.5256	−1.20020	−0.600099	+	0.799925i	$0.704872\pi$
$128$	2.45598	0.217080
$129$	5.35420	0.471411
$130$	10.5710	0.927138
$131$	21.5249	1.88064	0.940318	−	0.340296i	$-0.110527\pi$
$131$	21.5249	1.88064	0.940318	+	0.340296i	$0.110527\pi$
$132$	−18.2197	−1.58582
$133$	−2.93632	−0.254611
$134$	7.74216	0.668821
$135$	1.60463	0.138104
$136$	−0.459155	−0.0393722
$137$	1.34368	0.114798	0.0573990	−	0.998351i	$-0.481719\pi$
$137$	1.34368	0.114798	0.0573990	+	0.998351i	$0.481719\pi$
$138$	19.6789	1.67518
$139$	−6.05929	−0.513942	−0.256971	−	0.966419i	$-0.582725\pi$
$139$	−6.05929	−0.513942	−0.256971	+	0.966419i	$0.582725\pi$
$140$	8.18385	0.691661
$141$	13.4615	1.13366
$142$	18.2653	1.53279
$143$	−19.0829	−1.59579
$144$	−8.46415	−0.705346
$145$	−9.03381	−0.750217
$146$	−8.28407	−0.685595
$147$	17.2117	1.41960
$148$	−3.78706	−0.311294
$149$	−12.3594	−1.01252	−0.506259	−	0.862381i	$-0.668972\pi$
$149$	−12.3594	−1.01252	−0.506259	+	0.862381i	$0.668972\pi$
$150$	−4.69193	−0.383094
$151$	0.523389	0.0425928	0.0212964	−	0.999773i	$-0.493221\pi$
$151$	0.523389	0.0425928	0.0212964	+	0.999773i	$0.493221\pi$
$152$	−0.237622	−0.0192737
$153$	3.43491	0.277696
$154$	−28.5093	−2.29735
$155$	4.46079	0.358299
$156$	25.7019	2.05780
$157$	−12.7726	−1.01937	−0.509684	−	0.860362i	$-0.670238\pi$
$157$	−12.7726	−1.01937	−0.509684	+	0.860362i	$0.670238\pi$
$158$	−8.99560	−0.715652
$159$	−4.23996	−0.336251
$160$	−8.10319	−0.640613
$161$	15.9568	1.25757
$162$	21.6066	1.69758
$163$	15.1867	1.18951	0.594755	−	0.803907i	$-0.297249\pi$
$163$	15.1867	1.18951	0.594755	+	0.803907i	$0.297249\pi$
$164$	9.70730	0.758013
$165$	8.46992	0.659382
$166$	22.5519	1.75037
$167$	−6.06616	−0.469413	−0.234707	−	0.972066i	$-0.575413\pi$
$167$	−6.06616	−0.469413	−0.234707	+	0.972066i	$0.575413\pi$
$168$	2.69737	0.208107
$169$	13.9196	1.07074
$170$	3.03854	0.233045
$171$	1.77763	0.135939
$172$	5.00136	0.381350
$173$	17.3366	1.31807	0.659037	−	0.752110i	$-0.270964\pi$
$173$	17.3366	1.31807	0.659037	+	0.752110i	$0.270964\pi$
$174$	−42.3860	−3.21327
$175$	−3.80447	−0.287591
$176$	13.5164	1.01884
$177$	−8.33455	−0.626463
$178$	−0.865265	−0.0648543
$179$	19.9107	1.48819	0.744097	−	0.668071i	$-0.232880\pi$
$179$	19.9107	1.48819	0.744097	+	0.668071i	$0.232880\pi$
$180$	−4.95445	−0.369283
$181$	−10.6184	−0.789260	−0.394630	−	0.918840i	$-0.629127\pi$
$181$	−10.6184	−0.789260	−0.394630	+	0.918840i	$0.629127\pi$
$182$	40.2171	2.98109
$183$	19.3438	1.42993
$184$	1.29130	0.0951959
$185$	1.76051	0.129436
$186$	20.9297	1.53464
$187$	−5.48520	−0.401117
$188$	12.5744	0.917082
$189$	6.10477	0.444056
$190$	1.57250	0.114081
$191$	1.18147	0.0854881	0.0427441	−	0.999086i	$-0.486390\pi$
$191$	1.18147	0.0854881	0.0427441	+	0.999086i	$0.486390\pi$
$192$	−21.0937	−1.52231
$193$	−4.35980	−0.313825	−0.156913	−	0.987612i	$-0.550154\pi$
$193$	−4.35980	−0.313825	−0.156913	+	0.987612i	$0.550154\pi$
$194$	11.4445	0.821670
$195$	−11.9482	−0.855630
$196$	16.0775	1.14839
$197$	−12.0789	−0.860588	−0.430294	−	0.902689i	$-0.641590\pi$
$197$	−12.0789	−0.860588	−0.430294	+	0.902689i	$0.641590\pi$
$198$	17.2594	1.22657
$199$	−20.3682	−1.44387	−0.721933	−	0.691963i	$-0.756746\pi$
$199$	−20.3682	−1.44387	−0.721933	+	0.691963i	$0.756746\pi$
$200$	−0.307877	−0.0217702
$201$	−8.75083	−0.617236
$202$	−16.0412	−1.12866
$203$	−34.3689	−2.41222
$204$	7.38779	0.517248
$205$	−4.51269	−0.315180
$206$	−0.680732	−0.0474289
$207$	−9.66013	−0.671425
$208$	−19.0671	−1.32207
$209$	−2.83869	−0.196357
$210$	−17.8503	−1.23179
$211$	−27.5667	−1.89777	−0.948884	−	0.315624i	$-0.897786\pi$
$211$	−27.5667	−1.89777	−0.948884	+	0.315624i	$0.897786\pi$
$212$	−3.96055	−0.272011
$213$	−20.6450	−1.41457
$214$	−22.4789	−1.53662
$215$	−2.32501	−0.158565
$216$	0.494028	0.0336143
$217$	16.9710	1.15206
$218$	−3.67466	−0.248879
$219$	9.36335	0.632716
$220$	7.91175	0.533410
$221$	7.73778	0.520499
$222$	8.26021	0.554389
$223$	−1.06046	−0.0710137	−0.0355068	−	0.999369i	$-0.511305\pi$
$223$	−1.06046	−0.0710137	−0.0355068	+	0.999369i	$0.511305\pi$
$224$	−30.8284	−2.05981
$225$	2.30320	0.153547
$226$	25.5054	1.69659
$227$	13.7755	0.914314	0.457157	−	0.889386i	$-0.348868\pi$
$227$	13.7755	0.914314	0.457157	+	0.889386i	$0.348868\pi$
$228$	3.82332	0.253206
$229$	2.48850	0.164444	0.0822222	−	0.996614i	$-0.473798\pi$
$229$	2.48850	0.164444	0.0822222	+	0.996614i	$0.473798\pi$
$230$	−8.54540	−0.563467
$231$	32.2236	2.12016
$232$	−2.78130	−0.182601
$233$	16.0571	1.05194	0.525968	−	0.850504i	$-0.323703\pi$
$233$	16.0571	1.05194	0.525968	+	0.850504i	$0.323703\pi$
$234$	−24.3472	−1.59163
$235$	−5.84554	−0.381321
$236$	−7.78530	−0.506780
$237$	10.1676	0.660455
$238$	11.5600	0.749326
$239$	−22.8199	−1.47610	−0.738048	−	0.674749i	$-0.764252\pi$
$239$	−22.8199	−1.47610	−0.738048	+	0.674749i	$0.764252\pi$
$240$	8.46291	0.546279
$241$	−15.2872	−0.984735	−0.492368	−	0.870387i	$-0.663868\pi$
$241$	−15.2872	−0.984735	−0.492368	+	0.870387i	$0.663868\pi$
$242$	−5.14974	−0.331038
$243$	−19.6077	−1.25784
$244$	18.0691	1.15675
$245$	−7.47403	−0.477498
$246$	−21.1732	−1.34996
$247$	4.00445	0.254797
$248$	1.37337	0.0872094
$249$	−25.4900	−1.61537
$250$	2.03743	0.128858
$251$	−28.8330	−1.81993	−0.909963	−	0.414690i	$-0.863890\pi$
$251$	−28.8330	−1.81993	−0.909963	+	0.414690i	$0.863890\pi$
$252$	−18.8491	−1.18738
$253$	15.4262	0.969839
$254$	27.5573	1.72910
$255$	−3.43441	−0.215071
$256$	13.3156	0.832227
$257$	13.8799	0.865807	0.432903	−	0.901440i	$-0.357489\pi$
$257$	13.8799	0.865807	0.432903	+	0.901440i	$0.357489\pi$
$258$	−10.9088	−0.679152
$259$	6.69783	0.416183
$260$	−11.1608	−0.692166
$261$	20.8067	1.28790
$262$	−43.8554	−2.70939
$263$	−11.3107	−0.697451	−0.348725	−	0.937225i	$-0.613385\pi$
$263$	−11.3107	−0.697451	−0.348725	+	0.937225i	$0.613385\pi$
$264$	2.60769	0.160492
$265$	1.84116	0.113102
$266$	5.98254	0.366813
$267$	0.977993	0.0598522
$268$	−8.17415	−0.499316
$269$	12.8782	0.785197	0.392599	−	0.919710i	$-0.371576\pi$
$269$	12.8782	0.785197	0.392599	+	0.919710i	$0.371576\pi$
$270$	−3.26931	−0.198964
$271$	−4.73851	−0.287844	−0.143922	−	0.989589i	$-0.545971\pi$
$271$	−4.73851	−0.287844	−0.143922	+	0.989589i	$0.545971\pi$
$272$	−5.48066	−0.332314
$273$	−45.4567	−2.75116
$274$	−2.73764	−0.165387
$275$	−3.67798	−0.221791
$276$	−20.7770	−1.25063
$277$	25.1346	1.51019	0.755097	−	0.655614i	$-0.227590\pi$
$277$	25.1346	1.51019	0.755097	+	0.655614i	$0.227590\pi$
$278$	12.3454	0.740426
$279$	−10.2741	−0.615095
$280$	−1.17131	−0.0699992
$281$	−15.0397	−0.897192	−0.448596	−	0.893735i	$-0.648076\pi$
$281$	−15.0397	−0.897192	−0.448596	+	0.893735i	$0.648076\pi$
$282$	−27.4268	−1.63324
$283$	1.63568	0.0972309	0.0486155	−	0.998818i	$-0.484519\pi$
$283$	1.63568	0.0972309	0.0486155	+	0.998818i	$0.484519\pi$
$284$	−19.2845	−1.14432
$285$	−1.77737	−0.105282
$286$	38.8800	2.29902
$287$	−17.1684	−1.01342
$288$	18.6633	1.09975
$289$	−14.7758	−0.869167
$290$	18.4057	1.08082
$291$	−12.9356	−0.758296
$292$	8.74630	0.511839
$293$	−31.9101	−1.86421	−0.932103	−	0.362193i	$-0.882028\pi$
$293$	−31.9101	−1.86421	−0.932103	+	0.362193i	$0.882028\pi$
$294$	−35.0676	−2.04518
$295$	3.61920	0.210718
$296$	0.542022	0.0315044
$297$	5.90180	0.342457
$298$	25.1813	1.45871
$299$	−21.7613	−1.25849
$300$	4.95373	0.286003
$301$	−8.84545	−0.509843
$302$	−1.06637	−0.0613625
$303$	18.1311	1.04161
$304$	−2.83635	−0.162676
$305$	−8.39987	−0.480975
$306$	−6.99837	−0.400070
$307$	17.2565	0.984878	0.492439	−	0.870347i	$-0.336105\pi$
$307$	17.2565	0.984878	0.492439	+	0.870347i	$0.336105\pi$
$308$	30.1001	1.71511
$309$	0.769420	0.0437708
$310$	−9.08854	−0.516194
$311$	10.6053	0.601373	0.300687	−	0.953723i	$-0.402784\pi$
$311$	10.6053	0.601373	0.300687	+	0.953723i	$0.402784\pi$
$312$	−3.67858	−0.208259
$313$	−28.0652	−1.58634	−0.793169	−	0.609002i	$-0.791570\pi$
$313$	−28.0652	−1.58634	−0.793169	+	0.609002i	$0.791570\pi$
$314$	26.0233	1.46858
$315$	8.76249	0.493710
$316$	9.49753	0.534278
$317$	−16.2809	−0.914428	−0.457214	−	0.889357i	$-0.651153\pi$
$317$	−16.2809	−0.914428	−0.457214	+	0.889357i	$0.651153\pi$
$318$	8.63861	0.484429
$319$	−33.2262	−1.86031
$320$	9.15977	0.512046
$321$	25.4075	1.41811
$322$	−32.5108	−1.81175
$323$	1.15104	0.0640457
$324$	−22.8122	−1.26735
$325$	5.18841	0.287801
$326$	−30.9417	−1.71370
$327$	4.15341	0.229684
$328$	−1.38935	−0.0767142
$329$	−22.2392	−1.22609
$330$	−17.2568	−0.949958
$331$	−26.6546	−1.46507	−0.732534	−	0.680730i	$-0.761663\pi$
$331$	−26.6546	−1.46507	−0.732534	+	0.680730i	$0.761663\pi$
$332$	−23.8103	−1.30676
$333$	−4.05483	−0.222203
$334$	12.3594	0.676274
$335$	3.79997	0.207614
$336$	32.1969	1.75649
$337$	−17.3550	−0.945387	−0.472694	−	0.881227i	$-0.656718\pi$
$337$	−17.3550	−0.945387	−0.472694	+	0.881227i	$0.656718\pi$
$338$	−28.3601	−1.54259
$339$	−28.8283	−1.56574
$340$	−3.20808	−0.173983
$341$	16.4067	0.888474
$342$	−3.62179	−0.195844
$343$	−1.80343	−0.0973761
$344$	−0.715818	−0.0385943
$345$	9.65872	0.520008
$346$	−35.3220	−1.89892
$347$	−25.6676	−1.37791	−0.688954	−	0.724805i	$-0.741930\pi$
$347$	−25.6676	−1.37791	−0.688954	+	0.724805i	$0.741930\pi$
$348$	44.7510	2.39891
$349$	−13.5814	−0.726993	−0.363497	−	0.931595i	$-0.618417\pi$
$349$	−13.5814	−0.726993	−0.363497	+	0.931595i	$0.618417\pi$
$350$	7.75134	0.414327
$351$	−8.32546	−0.444380
$352$	−29.8034	−1.58853
$353$	7.99345	0.425448	0.212724	−	0.977112i	$-0.431766\pi$
$353$	7.99345	0.425448	0.212724	+	0.977112i	$0.431766\pi$
$354$	16.9810	0.902532
$355$	8.96489	0.475807
$356$	0.913544	0.0484177
$357$	−13.0661	−0.691532
$358$	−40.5666	−2.14401
$359$	−30.5992	−1.61496	−0.807482	−	0.589892i	$-0.799170\pi$
$359$	−30.5992	−1.61496	−0.807482	+	0.589892i	$0.799170\pi$
$360$	0.709104	0.0373730
$361$	−18.4043	−0.968648
$362$	21.6342	1.13707
$363$	5.82066	0.305505
$364$	−42.4611	−2.22557
$365$	−4.06595	−0.212821
$366$	−39.4116	−2.06008
$367$	−32.4407	−1.69339	−0.846697	−	0.532076i	$-0.821412\pi$
$367$	−32.4407	−1.69339	−0.846697	+	0.532076i	$0.821412\pi$
$368$	15.4135	0.803484
$369$	10.3937	0.541072
$370$	−3.58692	−0.186475
$371$	7.00466	0.363664
$372$	−22.0975	−1.14570
$373$	19.6667	1.01830	0.509151	−	0.860677i	$-0.329959\pi$
$373$	19.6667	1.01830	0.509151	+	0.860677i	$0.329959\pi$
$374$	11.1757	0.577881
$375$	−2.30287	−0.118920
$376$	−1.79971	−0.0928127
$377$	46.8711	2.41398
$378$	−12.4380	−0.639743
$379$	15.7828	0.810710	0.405355	−	0.914159i	$-0.367148\pi$
$379$	15.7828	0.810710	0.405355	+	0.914159i	$0.367148\pi$
$380$	−1.66024	−0.0851686
$381$	−31.1476	−1.59574
$382$	−2.40716	−0.123161
$383$	22.1357	1.13108	0.565542	−	0.824720i	$-0.308667\pi$
$383$	22.1357	1.13108	0.565542	+	0.824720i	$0.308667\pi$
$384$	5.65581	0.288622
$385$	−13.9928	−0.713139
$386$	8.88278	0.452122
$387$	5.35498	0.272209
$388$	−12.0831	−0.613427
$389$	10.5687	0.535855	0.267927	−	0.963439i	$-0.413661\pi$
$389$	10.5687	0.535855	0.267927	+	0.963439i	$0.413661\pi$
$390$	24.3436	1.23269
$391$	−6.25508	−0.316333
$392$	−2.30108	−0.116222
$393$	49.5690	2.50042
$394$	24.6099	1.23983
$395$	−4.41518	−0.222152
$396$	−18.2224	−0.915709
$397$	−6.84387	−0.343484	−0.171742	−	0.985142i	$-0.554940\pi$
$397$	−6.84387	−0.343484	−0.171742	+	0.985142i	$0.554940\pi$
$398$	41.4988	2.08015
$399$	−6.76196	−0.338522
$400$	−3.67494	−0.183747
$401$	−5.66705	−0.282999	−0.141500	−	0.989938i	$-0.545192\pi$
$401$	−5.66705	−0.282999	−0.141500	+	0.989938i	$0.545192\pi$
$402$	17.8292	0.889238
$403$	−23.1444	−1.15290
$404$	16.9363	0.842612
$405$	10.6049	0.526960
$406$	70.0241	3.47524
$407$	6.47514	0.320961
$408$	−1.05737	−0.0523478
$409$	7.67963	0.379733	0.189867	−	0.981810i	$-0.439194\pi$
$409$	7.67963	0.379733	0.189867	+	0.981810i	$0.439194\pi$
$410$	9.19429	0.454073
$411$	3.09431	0.152631
$412$	0.718716	0.0354086
$413$	13.7692	0.677536
$414$	19.6818	0.967308
$415$	11.0688	0.543347
$416$	42.0426	2.06131
$417$	−13.9537	−0.683318
$418$	5.78363	0.282887
$419$	7.69527	0.375939	0.187969	−	0.982175i	$-0.439809\pi$
$419$	7.69527	0.375939	0.187969	+	0.982175i	$0.439809\pi$
$420$	18.8463	0.919606
$421$	−29.4174	−1.43372	−0.716859	−	0.697218i	$-0.754421\pi$
$421$	−29.4174	−1.43372	−0.716859	+	0.697218i	$0.754421\pi$
$422$	56.1651	2.73407
$423$	13.4635	0.654616
$424$	0.566852	0.0275287
$425$	1.49136	0.0723416
$426$	42.0626	2.03794
$427$	−31.9571	−1.54651
$428$	23.7331	1.14718
$429$	−43.9454	−2.12170
$430$	4.73704	0.228441
$431$	10.8034	0.520383	0.260192	−	0.965557i	$-0.416214\pi$
$431$	10.8034	0.520383	0.260192	+	0.965557i	$0.416214\pi$
$432$	5.89692	0.283716
$433$	6.32388	0.303906	0.151953	−	0.988388i	$-0.451444\pi$
$433$	6.32388	0.303906	0.151953	+	0.988388i	$0.451444\pi$
$434$	−34.5771	−1.65975
$435$	−20.8037	−0.997460
$436$	3.87970	0.185804
$437$	−3.23712	−0.154853
$438$	−19.0771	−0.911541
$439$	2.82671	0.134912	0.0674558	−	0.997722i	$-0.478512\pi$
$439$	2.82671	0.134912	0.0674558	+	0.997722i	$0.478512\pi$
$440$	−1.13237	−0.0539834
$441$	17.2142	0.819725
$442$	−15.7652	−0.749872
$443$	15.0943	0.717154	0.358577	−	0.933500i	$-0.383262\pi$
$443$	15.0943	0.717154	0.358577	+	0.933500i	$0.383262\pi$
$444$	−8.72110	−0.413885
$445$	−0.424685	−0.0201320
$446$	2.16061	0.102308
$447$	−28.4620	−1.34621
$448$	34.8481	1.64642
$449$	−18.4546	−0.870926	−0.435463	−	0.900207i	$-0.643415\pi$
$449$	−18.4546	−0.870926	−0.435463	+	0.900207i	$0.643415\pi$
$450$	−4.69261	−0.221212
$451$	−16.5976	−0.781551
$452$	−26.9285	−1.26661
$453$	1.20530	0.0566298
$454$	−28.0667	−1.31723
$455$	19.7392	0.925386
$456$	−0.547211	−0.0256255
$457$	−12.2756	−0.574228	−0.287114	−	0.957896i	$-0.592696\pi$
$457$	−12.2756	−0.574228	−0.287114	+	0.957896i	$0.592696\pi$
$458$	−5.07013	−0.236912
$459$	−2.39308	−0.111699
$460$	9.02221	0.420663
$461$	−4.44895	−0.207208	−0.103604	−	0.994619i	$-0.533037\pi$
$461$	−4.44895	−0.207208	−0.103604	+	0.994619i	$0.533037\pi$
$462$	−65.6532	−3.05446
$463$	−0.00658194	−0.000305889 0	−0.000152944	−	1.00000i	$-0.500049\pi$
$463$	−0.00658194	−0.000305889 0	−0.000152944		1.00000i	$0.500049\pi$
$464$	−33.1987	−1.54121
$465$	10.2726	0.476381
$466$	−32.7152	−1.51550
$467$	2.28455	0.105716	0.0528581	−	0.998602i	$-0.483167\pi$
$467$	2.28455	0.105716	0.0528581	+	0.998602i	$0.483167\pi$
$468$	25.7057	1.18825
$469$	14.4569	0.667557
$470$	11.9099	0.549361
$471$	−29.4137	−1.35531
$472$	1.11427	0.0512884
$473$	−8.55136	−0.393192
$474$	−20.7157	−0.951503
$475$	0.771807	0.0354129
$476$	−12.2051	−0.559418
$477$	−4.24058	−0.194163
$478$	46.4938	2.12658
$479$	−8.87437	−0.405480	−0.202740	−	0.979233i	$-0.564985\pi$
$479$	−8.87437	−0.405480	−0.202740	+	0.979233i	$0.564985\pi$
$480$	−18.6606	−0.851735
$481$	−9.13426	−0.416487
$482$	31.1466	1.41869
$483$	36.7463	1.67202
$484$	5.43708	0.247140
$485$	5.61715	0.255062
$486$	39.9493	1.81214
$487$	20.7422	0.939918	0.469959	−	0.882688i	$-0.344269\pi$
$487$	20.7422	0.939918	0.469959	+	0.882688i	$0.344269\pi$
$488$	−2.58613	−0.117068
$489$	34.9729	1.58153
$490$	15.2278	0.687922
$491$	−16.8366	−0.759824	−0.379912	−	0.925023i	$-0.624046\pi$
$491$	−16.8366	−0.759824	−0.379912	+	0.925023i	$0.624046\pi$
$492$	22.3546	1.00783
$493$	13.4727	0.606778
$494$	−8.15877	−0.367081
$495$	8.47115	0.380750
$496$	16.3932	0.736075
$497$	34.1067	1.52989
$498$	51.9341	2.32722
$499$	−28.7277	−1.28603	−0.643013	−	0.765855i	$-0.722316\pi$
$499$	−28.7277	−1.28603	−0.643013	+	0.765855i	$0.722316\pi$
$500$	−2.15111	−0.0962006
$501$	−13.9696	−0.624114
$502$	58.7452	2.62193
$503$	−12.5655	−0.560269	−0.280135	−	0.959961i	$-0.590379\pi$
$503$	−12.5655	−0.560269	−0.280135	+	0.959961i	$0.590379\pi$
$504$	2.69777	0.120168
$505$	−7.87327	−0.350356
$506$	−31.4298	−1.39723
$507$	32.0549	1.42361
$508$	−29.0950	−1.29088
$509$	26.3227	1.16673	0.583367	−	0.812208i	$-0.301735\pi$
$509$	26.3227	1.16673	0.583367	+	0.812208i	$0.301735\pi$
$510$	6.99735	0.309848
$511$	−15.4688	−0.684299
$512$	−32.0416	−1.41605
$513$	−1.23846	−0.0546795
$514$	−28.2794	−1.24735
$515$	−0.334114	−0.0147228
$516$	11.5175	0.507028
$517$	−21.4998	−0.945560
$518$	−13.6463	−0.599586
$519$	39.9238	1.75246
$520$	1.59739	0.0700502
$521$	−34.2844	−1.50202	−0.751012	−	0.660288i	$-0.770434\pi$
$521$	−34.2844	−1.50202	−0.751012	+	0.660288i	$0.770434\pi$
$522$	−42.3922	−1.85545
$523$	−12.2303	−0.534795	−0.267397	−	0.963586i	$-0.586164\pi$
$523$	−12.2303	−0.534795	−0.267397	+	0.963586i	$0.586164\pi$
$524$	46.3024	2.02273
$525$	−8.76121	−0.382370
$526$	23.0448	1.00480
$527$	−6.65264	−0.289794
$528$	31.1265	1.35461
$529$	−5.40859	−0.235156
$530$	−3.75124	−0.162943
$531$	−8.33576	−0.361741
$532$	−6.31635	−0.273849
$533$	23.4137	1.01416
$534$	−1.99259	−0.0862278
$535$	−11.0330	−0.476996
$536$	1.16992	0.0505329
$537$	45.8517	1.97865
$538$	−26.2384	−1.13122
$539$	−27.4894	−1.18405
$540$	3.45173	0.148539
$541$	4.70554	0.202307	0.101154	−	0.994871i	$-0.467747\pi$
$541$	4.70554	0.202307	0.101154	+	0.994871i	$0.467747\pi$
$542$	9.65437	0.414691
$543$	−24.4528	−1.04937
$544$	12.0848	0.518130
$545$	−1.80358	−0.0772568
$546$	92.6147	3.96354
$547$	−7.35339	−0.314408	−0.157204	−	0.987566i	$-0.550248\pi$
$547$	−7.35339	−0.314408	−0.157204	+	0.987566i	$0.550248\pi$
$548$	2.89040	0.123472
$549$	19.3466	0.825694
$550$	7.49363	0.319529
$551$	6.97236	0.297032
$552$	2.97370	0.126569
$553$	−16.7974	−0.714299
$554$	−51.2100	−2.17570
$555$	4.05423	0.172093
$556$	−13.0342	−0.552773
$557$	19.0065	0.805333	0.402667	−	0.915347i	$-0.368083\pi$
$557$	19.0065	0.805333	0.402667	+	0.915347i	$0.368083\pi$
$558$	20.9328	0.886155
$559$	12.0631	0.510215
$560$	−13.9812	−0.590815
$561$	−12.6317	−0.533310
$562$	30.6423	1.29257
$563$	−10.8001	−0.455171	−0.227586	−	0.973758i	$-0.573083\pi$
$563$	−10.8001	−0.455171	−0.227586	+	0.973758i	$0.573083\pi$
$564$	28.9572	1.21932
$565$	12.5184	0.526654
$566$	−3.33257	−0.140079
$567$	40.3459	1.69437
$568$	2.76008	0.115811
$569$	0.528685	0.0221636	0.0110818	−	0.999939i	$-0.496472\pi$
$569$	0.528685	0.0221636	0.0110818	+	0.999939i	$0.496472\pi$
$570$	3.62126	0.151678
$571$	−3.33973	−0.139763	−0.0698817	−	0.997555i	$-0.522262\pi$
$571$	−3.33973	−0.139763	−0.0698817	+	0.997555i	$0.522262\pi$
$572$	−41.0494	−1.71636
$573$	2.72077	0.113662
$574$	34.9794	1.46001
$575$	−4.19421	−0.174911
$576$	−21.0968	−0.879034
$577$	5.15312	0.214527	0.107264	−	0.994231i	$-0.465791\pi$
$577$	5.15312	0.214527	0.107264	+	0.994231i	$0.465791\pi$
$578$	30.1047	1.25219
$579$	−10.0400	−0.417250
$580$	−19.4327	−0.806900
$581$	42.1111	1.74706
$582$	26.3553	1.09246
$583$	6.77177	0.280458
$584$	−1.25181	−0.0518003
$585$	−11.9500	−0.494070
$586$	65.0145	2.68572
$587$	32.3037	1.33331	0.666657	−	0.745364i	$-0.267724\pi$
$587$	32.3037	1.33331	0.666657	+	0.745364i	$0.267724\pi$
$588$	37.0243	1.52686
$589$	−3.44287	−0.141861
$590$	−7.37386	−0.303577
$591$	−27.8162	−1.14421
$592$	6.46979	0.265907
$593$	−6.77049	−0.278031	−0.139015	−	0.990290i	$-0.544394\pi$
$593$	−6.77049	−0.278031	−0.139015	+	0.990290i	$0.544394\pi$
$594$	−12.0245	−0.493370
$595$	5.67384	0.232605
$596$	−26.5864	−1.08902
$597$	−46.9054	−1.91971
$598$	44.3370	1.81308
$599$	2.34181	0.0956837	0.0478418	−	0.998855i	$-0.484766\pi$
$599$	2.34181	0.0956837	0.0478418	+	0.998855i	$0.484766\pi$
$600$	−0.709000	−0.0289448
$601$	35.2991	1.43988	0.719941	−	0.694036i	$-0.244169\pi$
$601$	35.2991	1.43988	0.719941	+	0.694036i	$0.244169\pi$
$602$	18.0220	0.734520
$603$	−8.75210	−0.356413
$604$	1.12587	0.0458109
$605$	−2.52757	−0.102760
$606$	−36.9408	−1.50062
$607$	−15.5606	−0.631587	−0.315793	−	0.948828i	$-0.602271\pi$
$607$	−15.5606	−0.631587	−0.315793	+	0.948828i	$0.602271\pi$
$608$	6.25410	0.253637
$609$	−79.1470	−3.20720
$610$	17.1141	0.692931
$611$	30.3290	1.22698
$612$	7.38886	0.298677
$613$	4.65427	0.187984	0.0939920	−	0.995573i	$-0.470037\pi$
$613$	4.65427	0.187984	0.0939920	+	0.995573i	$0.470037\pi$
$614$	−35.1588	−1.41889
$615$	−10.3921	−0.419052
$616$	−4.30806	−0.173577
$617$	14.2750	0.574689	0.287344	−	0.957827i	$-0.407228\pi$
$617$	14.2750	0.574689	0.287344	+	0.957827i	$0.407228\pi$
$618$	−1.56764	−0.0630596
$619$	20.8500	0.838033	0.419017	−	0.907979i	$-0.362375\pi$
$619$	20.8500	0.838033	0.419017	+	0.907979i	$0.362375\pi$
$620$	9.59565	0.385371
$621$	6.73015	0.270071
$622$	−21.6076	−0.866386
$623$	−1.61570	−0.0647318
$624$	−43.9090	−1.75777
$625$	1.00000	0.0400000
$626$	57.1808	2.28540
$627$	−6.53714	−0.261068
$628$	−27.4754	−1.09639
$629$	−2.62556	−0.104688
$630$	−17.8529	−0.711278
$631$	−26.2943	−1.04676	−0.523380	−	0.852100i	$-0.675329\pi$
$631$	−26.2943	−1.04676	−0.523380	+	0.852100i	$0.675329\pi$
$632$	−1.35933	−0.0540713
$633$	−63.4825	−2.52320
$634$	33.1712	1.31740
$635$	13.5256	0.536745
$636$	−9.12062	−0.361656
$637$	38.7783	1.53645
$638$	67.6960	2.68011
$639$	−20.6480	−0.816822
$640$	−2.45598	−0.0970813
$641$	20.1066	0.794164	0.397082	−	0.917783i	$-0.370023\pi$
$641$	20.1066	0.794164	0.397082	+	0.917783i	$0.370023\pi$
$642$	−51.7659	−2.04304
$643$	−11.3258	−0.446645	−0.223323	−	0.974745i	$-0.571690\pi$
$643$	−11.3258	−0.446645	−0.223323	+	0.974745i	$0.571690\pi$
$644$	34.3248	1.35259
$645$	−5.35420	−0.210821
$646$	−2.34516	−0.0922693
$647$	10.0761	0.396132	0.198066	−	0.980189i	$-0.436534\pi$
$647$	10.0761	0.396132	0.198066	+	0.980189i	$0.436534\pi$
$648$	3.26499	0.128261
$649$	13.3114	0.522517
$650$	−10.5710	−0.414629
$651$	39.0819	1.53174
$652$	32.6682	1.27938
$653$	−35.9988	−1.40874	−0.704372	−	0.709831i	$-0.748771\pi$
$653$	−35.9988	−1.40874	−0.704372	+	0.709831i	$0.748771\pi$
$654$	−8.46226	−0.330901
$655$	−21.5249	−0.841046
$656$	−16.5839	−0.647492
$657$	9.36471	0.365352
$658$	45.3107	1.76640
$659$	−37.4880	−1.46033	−0.730163	−	0.683273i	$-0.760556\pi$
$659$	−37.4880	−1.46033	−0.730163	+	0.683273i	$0.760556\pi$
$660$	18.2197	0.709202
$661$	−14.2469	−0.554139	−0.277069	−	0.960850i	$-0.589363\pi$
$661$	−14.2469	−0.554139	−0.277069	+	0.960850i	$0.589363\pi$
$662$	54.3068	2.11069
$663$	17.8191	0.692036
$664$	3.40783	0.132250
$665$	2.93632	0.113866
$666$	8.26141	0.320123
$667$	−37.8897	−1.46709
$668$	−13.0490	−0.504880
$669$	−2.44210	−0.0944171
$670$	−7.74216	−0.299106
$671$	−30.8946	−1.19267
$672$	−70.9937	−2.73864
$673$	43.9984	1.69601	0.848006	−	0.529987i	$-0.177803\pi$
$673$	43.9984	1.69601	0.848006	+	0.529987i	$0.177803\pi$
$674$	35.3596	1.36200
$675$	−1.60463	−0.0617621
$676$	29.9425	1.15164
$677$	19.5369	0.750866	0.375433	−	0.926850i	$-0.377494\pi$
$677$	19.5369	0.750866	0.375433	+	0.926850i	$0.377494\pi$
$678$	58.7356	2.25573
$679$	21.3703	0.820118
$680$	0.459155	0.0176078
$681$	31.7233	1.21564
$682$	−33.4275	−1.28001
$683$	−1.50215	−0.0574781	−0.0287390	−	0.999587i	$-0.509149\pi$
$683$	−1.50215	−0.0574781	−0.0287390	+	0.999587i	$0.509149\pi$
$684$	3.82388	0.146210
$685$	−1.34368	−0.0513393
$686$	3.67436	0.140288
$687$	5.73068	0.218639
$688$	−8.54429	−0.325748
$689$	−9.55270	−0.363929
$690$	−19.6789	−0.749164
$691$	−18.7485	−0.713225	−0.356612	−	0.934252i	$-0.616068\pi$
$691$	−18.7485	−0.713225	−0.356612	+	0.934252i	$0.616068\pi$
$692$	37.2929	1.41766
$693$	32.2283	1.22425
$694$	52.2958	1.98512
$695$	6.05929	0.229842
$696$	−6.40497	−0.242780
$697$	6.73005	0.254919
$698$	27.6710	1.04736
$699$	36.9774	1.39861
$700$	−8.18385	−0.309320
$701$	30.5586	1.15418	0.577091	−	0.816680i	$-0.304188\pi$
$701$	30.5586	1.15418	0.577091	+	0.816680i	$0.304188\pi$
$702$	16.9625	0.640209
$703$	−1.35878	−0.0512473
$704$	33.6895	1.26972
$705$	−13.4615	−0.506989
$706$	−16.2861	−0.612934
$707$	−29.9537	−1.12652
$708$	−17.9285	−0.673796
$709$	23.4516	0.880744	0.440372	−	0.897816i	$-0.354847\pi$
$709$	23.4516	0.880744	0.440372	+	0.897816i	$0.354847\pi$
$710$	−18.2653	−0.685485
$711$	10.1691	0.381369
$712$	−0.130751	−0.00490009
$713$	18.7095	0.700676
$714$	26.6212	0.996275
$715$	19.0829	0.713659
$716$	42.8301	1.60064
$717$	−52.5512	−1.96256
$718$	62.3436	2.32664
$719$	−8.76914	−0.327034	−0.163517	−	0.986541i	$-0.552284\pi$
$719$	−8.76914	−0.327034	−0.163517	+	0.986541i	$0.552284\pi$
$720$	8.46415	0.315440
$721$	−1.27113	−0.0473392
$722$	37.4975	1.39551
$723$	−35.2044	−1.30927
$724$	−22.8414	−0.848893
$725$	9.03381	0.335507
$726$	−11.8592	−0.440135
$727$	−19.4667	−0.721980	−0.360990	−	0.932570i	$-0.617561\pi$
$727$	−19.4667	−0.721980	−0.360990	+	0.932570i	$0.617561\pi$
$728$	6.07723	0.225237
$729$	−13.3394	−0.494053
$730$	8.28407	0.306607
$731$	3.46743	0.128248
$732$	41.6107	1.53797
$733$	−31.5251	−1.16441	−0.582203	−	0.813043i	$-0.697809\pi$
$733$	−31.5251	−1.16441	−0.582203	+	0.813043i	$0.697809\pi$
$734$	66.0957	2.43964
$735$	−17.2117	−0.634863
$736$	−33.9865	−1.25276
$737$	13.9762	0.514821
$738$	−21.1763	−0.779511
$739$	−47.1367	−1.73395	−0.866975	−	0.498351i	$-0.833939\pi$
$739$	−47.1367	−1.73395	−0.866975	+	0.498351i	$0.833939\pi$
$740$	3.78706	0.139215
$741$	9.22172	0.338768
$742$	−14.2715	−0.523923
$743$	−0.860082	−0.0315533	−0.0157767	−	0.999876i	$-0.505022\pi$
$743$	−0.860082	−0.0315533	−0.0157767	+	0.999876i	$0.505022\pi$
$744$	3.16270	0.115950
$745$	12.3594	0.452812
$746$	−40.0694	−1.46705
$747$	−25.4938	−0.932768
$748$	−11.7993	−0.431424
$749$	−41.9746	−1.53372
$750$	4.69193	0.171325
$751$	−11.0469	−0.403108	−0.201554	−	0.979477i	$-0.564599\pi$
$751$	−11.0469	−0.403108	−0.201554	+	0.979477i	$0.564599\pi$
$752$	−21.4820	−0.783369
$753$	−66.3987	−2.41970
$754$	−95.4964	−3.47777
$755$	−0.523389	−0.0190481
$756$	13.1320	0.477607
$757$	43.2224	1.57094	0.785472	−	0.618897i	$-0.212420\pi$
$757$	43.2224	1.57094	0.785472	+	0.618897i	$0.212420\pi$
$758$	−32.1564	−1.16797
$759$	35.5246	1.28946
$760$	0.237622	0.00861944
$761$	−12.7375	−0.461735	−0.230868	−	0.972985i	$-0.574156\pi$
$761$	−12.7375	−0.461735	−0.230868	+	0.972985i	$0.574156\pi$
$762$	63.4609	2.29895
$763$	−6.86167	−0.248409
$764$	2.54147	0.0919472
$765$	−3.43491	−0.124189
$766$	−45.1000	−1.62953
$767$	−18.7779	−0.678030
$768$	30.6642	1.10650
$769$	21.7556	0.784527	0.392263	−	0.919853i	$-0.371692\pi$
$769$	21.7556	0.784527	0.392263	+	0.919853i	$0.371692\pi$
$770$	28.5093	1.02740
$771$	31.9637	1.15114
$772$	−9.37841	−0.337537
$773$	13.8450	0.497969	0.248984	−	0.968508i	$-0.419903\pi$
$773$	13.8450	0.497969	0.248984	+	0.968508i	$0.419903\pi$
$774$	−10.9104	−0.392166
$775$	−4.46079	−0.160236
$776$	1.72939	0.0620815
$777$	15.4242	0.553341
$778$	−21.5330	−0.771995
$779$	3.48293	0.124789
$780$	−25.7019	−0.920277
$781$	32.9727	1.17986
$782$	12.7443	0.455734
$783$	−14.4959	−0.518041
$784$	−27.4666	−0.980951
$785$	12.7726	0.455875
$786$	−100.993	−3.60231
$787$	−5.33265	−0.190088	−0.0950442	−	0.995473i	$-0.530299\pi$
$787$	−5.33265	−0.190088	−0.0950442	+	0.995473i	$0.530299\pi$
$788$	−25.9831	−0.925610
$789$	−26.0472	−0.927304
$790$	8.99560	0.320049
$791$	47.6260	1.69339
$792$	2.60807	0.0926738
$793$	43.5819	1.54764
$794$	13.9439	0.494850
$795$	4.23996	0.150376
$796$	−43.8143	−1.55296
$797$	46.2567	1.63850	0.819248	−	0.573440i	$-0.194391\pi$
$797$	46.2567	1.63850	0.819248	+	0.573440i	$0.194391\pi$
$798$	13.7770	0.487701
$799$	8.71779	0.308413
$800$	8.10319	0.286491
$801$	0.978136	0.0345607
$802$	11.5462	0.407711
$803$	−14.9545	−0.527733
$804$	−18.8240	−0.663871
$805$	−15.9568	−0.562402
$806$	47.1550	1.66097
$807$	29.6568	1.04397
$808$	−2.42400	−0.0852760
$809$	22.3379	0.785357	0.392679	−	0.919676i	$-0.371548\pi$
$809$	22.3379	0.785357	0.392679	+	0.919676i	$0.371548\pi$
$810$	−21.6066	−0.759180
$811$	24.5791	0.863088	0.431544	−	0.902092i	$-0.357969\pi$
$811$	24.5791	0.863088	0.431544	+	0.902092i	$0.357969\pi$
$812$	−73.9313	−2.59448
$813$	−10.9122	−0.382706
$814$	−13.1926	−0.462402
$815$	−15.1867	−0.531965
$816$	−12.6212	−0.441832
$817$	1.79446	0.0627802
$818$	−15.6467	−0.547074
$819$	−45.4633	−1.58862
$820$	−9.70730	−0.338994
$821$	13.1221	0.457963	0.228982	−	0.973431i	$-0.426460\pi$
$821$	13.1221	0.457963	0.228982	+	0.973431i	$0.426460\pi$
$822$	−6.30444	−0.219892
$823$	−32.9743	−1.14941	−0.574706	−	0.818360i	$-0.694884\pi$
$823$	−32.9743	−1.14941	−0.574706	+	0.818360i	$0.694884\pi$
$824$	−0.102866	−0.00358350
$825$	−8.46992	−0.294885
$826$	−28.0537	−0.976112
$827$	28.8322	1.00259	0.501297	−	0.865275i	$-0.332856\pi$
$827$	28.8322	1.00259	0.501297	+	0.865275i	$0.332856\pi$
$828$	−20.7800	−0.722155
$829$	−54.1724	−1.88149	−0.940743	−	0.339119i	$-0.889871\pi$
$829$	−54.1724	−1.88149	−0.940743	+	0.339119i	$0.889871\pi$
$830$	−22.5519	−0.782789
$831$	57.8817	2.00790
$832$	−47.5246	−1.64762
$833$	11.1465	0.386202
$834$	28.4297	0.984442
$835$	6.06616	0.209928
$836$	−6.10635	−0.211192
$837$	7.15791	0.247414
$838$	−15.6786	−0.541607
$839$	−14.1015	−0.486840	−0.243420	−	0.969921i	$-0.578269\pi$
$839$	−14.1015	−0.486840	−0.243420	+	0.969921i	$0.578269\pi$
$840$	−2.69737	−0.0930682
$841$	52.6097	1.81413
$842$	59.9359	2.06553
$843$	−34.6344	−1.19287
$844$	−59.2990	−2.04116
$845$	−13.9196	−0.478847
$846$	−27.4308	−0.943092
$847$	−9.61607	−0.330412
$848$	6.76617	0.232351
$849$	3.76675	0.129275
$850$	−3.03854	−0.104221
$851$	7.38397	0.253119
$852$	−44.4096	−1.52145
$853$	−12.6113	−0.431803	−0.215902	−	0.976415i	$-0.569269\pi$
$853$	−12.6113	−0.431803	−0.215902	+	0.976415i	$0.569269\pi$
$854$	65.1103	2.22803
$855$	−1.77763	−0.0607937
$856$	−3.39679	−0.116100
$857$	−10.9779	−0.374996	−0.187498	−	0.982265i	$-0.560038\pi$
$857$	−10.9779	−0.374996	−0.187498	+	0.982265i	$0.560038\pi$
$858$	89.5355	3.05669
$859$	34.9234	1.19157	0.595786	−	0.803143i	$-0.296841\pi$
$859$	34.9234	1.19157	0.595786	+	0.803143i	$0.296841\pi$
$860$	−5.00136	−0.170545
$861$	−39.5366	−1.34741
$862$	−22.0112	−0.749705
$863$	24.3303	0.828212	0.414106	−	0.910229i	$-0.364094\pi$
$863$	24.3303	0.828212	0.414106	+	0.910229i	$0.364094\pi$
$864$	−13.0026	−0.442357
$865$	−17.3366	−0.589461
$866$	−12.8844	−0.437831
$867$	−34.0268	−1.15561
$868$	36.5064	1.23911
$869$	−16.2390	−0.550869
$870$	42.3860	1.43702
$871$	−19.7158	−0.668044
$872$	−0.555280	−0.0188042
$873$	−12.9375	−0.437867
$874$	6.59540	0.223093
$875$	3.80447	0.128615
$876$	20.1416	0.680521
$877$	−36.5859	−1.23542	−0.617709	−	0.786406i	$-0.711939\pi$
$877$	−36.5859	−1.23542	−0.617709	+	0.786406i	$0.711939\pi$
$878$	−5.75922	−0.194364
$879$	−73.4847	−2.47858
$880$	−13.5164	−0.455637
$881$	−50.4572	−1.69995	−0.849973	−	0.526826i	$-0.823382\pi$
$881$	−50.4572	−1.69995	−0.849973	+	0.526826i	$0.823382\pi$
$882$	−35.0727	−1.18096
$883$	−41.6989	−1.40328	−0.701641	−	0.712531i	$-0.747549\pi$
$883$	−41.6989	−1.40328	−0.701641	+	0.712531i	$0.747549\pi$
$884$	16.6448	0.559826
$885$	8.33455	0.280163
$886$	−30.7536	−1.03319
$887$	26.7926	0.899608	0.449804	−	0.893127i	$-0.351494\pi$
$887$	26.7926	0.899608	0.449804	+	0.893127i	$0.351494\pi$
$888$	1.24820	0.0418870
$889$	51.4576	1.72583
$890$	0.865265	0.0290037
$891$	39.0045	1.30670
$892$	−2.28117	−0.0763791
$893$	4.51163	0.150976
$894$	57.9893	1.93945
$895$	−19.9107	−0.665541
$896$	−9.34373	−0.312152
$897$	−50.1134	−1.67324
$898$	37.5999	1.25472
$899$	−40.2979	−1.34401
$900$	4.95445	0.165148
$901$	−2.74584	−0.0914771
$902$	33.8164	1.12596
$903$	−20.3699	−0.677868
$904$	3.85413	0.128187
$905$	10.6184	0.352968
$906$	−2.45570	−0.0815853
$907$	−38.5958	−1.28155	−0.640776	−	0.767728i	$-0.721387\pi$
$907$	−38.5958	−1.28155	−0.640776	+	0.767728i	$0.721387\pi$
$908$	29.6327	0.983396
$909$	18.1338	0.601459
$910$	−40.2171	−1.33318
$911$	−12.3003	−0.407527	−0.203763	−	0.979020i	$-0.565317\pi$
$911$	−12.3003	−0.407527	−0.203763	+	0.979020i	$0.565317\pi$
$912$	−6.53174	−0.216287
$913$	40.7109	1.34734
$914$	25.0106	0.827279
$915$	−19.3438	−0.639486
$916$	5.35303	0.176869
$917$	−81.8908	−2.70427
$918$	4.87572	0.160923
$919$	8.47428	0.279541	0.139770	−	0.990184i	$-0.455364\pi$
$919$	8.47428	0.279541	0.139770	+	0.990184i	$0.455364\pi$
$920$	−1.29130	−0.0425729
$921$	39.7394	1.30946
$922$	9.06440	0.298520
$923$	−46.5135	−1.53101
$924$	69.3165	2.28034
$925$	−1.76051	−0.0578854
$926$	0.0134102	0.000440687 0
$927$	0.769532	0.0252748
$928$	73.2026	2.40299
$929$	−44.4457	−1.45822	−0.729108	−	0.684399i	$-0.760065\pi$
$929$	−44.4457	−1.45822	−0.729108	+	0.684399i	$0.760065\pi$
$930$	−20.9297	−0.686312
$931$	5.76851	0.189055
$932$	34.5406	1.13141
$933$	24.4227	0.799563
$934$	−4.65460	−0.152303
$935$	5.48520	0.179385
$936$	−3.67912	−0.120256
$937$	−8.08964	−0.264277	−0.132138	−	0.991231i	$-0.542184\pi$
$937$	−8.08964	−0.264277	−0.132138	+	0.991231i	$0.542184\pi$
$938$	−29.4548	−0.961735
$939$	−64.6304	−2.10913
$940$	−12.5744	−0.410131
$941$	55.7564	1.81761	0.908804	−	0.417224i	$-0.136997\pi$
$941$	55.7564	1.81761	0.908804	+	0.417224i	$0.136997\pi$
$942$	59.9283	1.95257
$943$	−18.9272	−0.616354
$944$	13.3004	0.432890
$945$	−6.10477	−0.198588
$946$	17.4228	0.566463
$947$	0.888628	0.0288765	0.0144383	−	0.999896i	$-0.495404\pi$
$947$	0.888628	0.0288765	0.0144383	+	0.999896i	$0.495404\pi$
$948$	21.8716	0.710356
$949$	21.0958	0.684798
$950$	−1.57250	−0.0510187
$951$	−37.4928	−1.21579
$952$	1.74684	0.0566155
$953$	32.8024	1.06257	0.531286	−	0.847192i	$-0.321709\pi$
$953$	32.8024	1.06257	0.531286	+	0.847192i	$0.321709\pi$
$954$	8.63987	0.279726
$955$	−1.18147	−0.0382314
$956$	−49.0881	−1.58762
$957$	−76.5156	−2.47340
$958$	18.0809	0.584167
$959$	−5.11199	−0.165075
$960$	21.0937	0.680797
$961$	−11.1013	−0.358108
$962$	18.6104	0.600023
$963$	25.4112	0.818863
$964$	−32.8845	−1.05914
$965$	4.35980	0.140347
$966$	−74.8680	−2.40884
$967$	−25.0132	−0.804371	−0.402186	−	0.915558i	$-0.631749\pi$
$967$	−25.0132	−0.804371	−0.402186	+	0.915558i	$0.631749\pi$
$968$	−0.778180	−0.0250116
$969$	2.65070	0.0851527
$970$	−11.4445	−0.367462
$971$	−41.3794	−1.32793	−0.663963	−	0.747765i	$-0.731127\pi$
$971$	−41.3794	−1.32793	−0.663963	+	0.747765i	$0.731127\pi$
$972$	−42.1784	−1.35287
$973$	23.0524	0.739026
$974$	−42.2607	−1.35412
$975$	11.9482	0.382649
$976$	−30.8691	−0.988095
$977$	−33.7595	−1.08006	−0.540031	−	0.841645i	$-0.681588\pi$
$977$	−33.7595	−1.08006	−0.540031	+	0.841645i	$0.681588\pi$
$978$	−71.2547	−2.27847
$979$	−1.56198	−0.0499212
$980$	−16.0775	−0.513576
$981$	4.15401	0.132627
$982$	34.3033	1.09466
$983$	−9.22471	−0.294223	−0.147111	−	0.989120i	$-0.546998\pi$
$983$	−9.22471	−0.294223	−0.147111	+	0.989120i	$0.546998\pi$
$984$	−3.19950	−0.101996
$985$	12.0789	0.384867
$986$	−27.4496	−0.874172
$987$	−51.2139	−1.63016
$988$	8.61401	0.274048
$989$	−9.75159	−0.310083
$990$	−17.2594	−0.548538
$991$	−6.75421	−0.214555	−0.107277	−	0.994229i	$-0.534213\pi$
$991$	−6.75421	−0.214555	−0.107277	+	0.994229i	$0.534213\pi$
$992$	−36.1466	−1.14766
$993$	−61.3820	−1.94790
$994$	−69.4899	−2.20409
$995$	20.3682	0.645716
$996$	−54.8319	−1.73742
$997$	−9.89360	−0.313333	−0.156667	−	0.987652i	$-0.550075\pi$
$997$	−9.89360	−0.313333	−0.156667	+	0.987652i	$0.550075\pi$
$998$	58.5305	1.85275
$999$	2.82497	0.0893781

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8045.2.a.b.1.19	✓	126

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8045.2.a.b.1.19	✓	126	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:	\( N \)	\(=\)	\( 8045 = 5 \cdot 1609 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8045.a (trivial)

\(f(q)\)	\(=\)	\(q-2.03743 q^{2} +2.30287 q^{3} +2.15111 q^{4} -1.00000 q^{5} -4.69193 q^{6} -3.80447 q^{7} -0.307877 q^{8} +2.30320 q^{9} +O(q^{10})\)	\(q-2.03743 q^{2} +2.30287 q^{3} +2.15111 q^{4} -1.00000 q^{5} -4.69193 q^{6} -3.80447 q^{7} -0.307877 q^{8} +2.30320 q^{9} +2.03743 q^{10} -3.67798 q^{11} +4.95373 q^{12} +5.18841 q^{13} +7.75134 q^{14} -2.30287 q^{15} -3.67494 q^{16} +1.49136 q^{17} -4.69261 q^{18} +0.771807 q^{19} -2.15111 q^{20} -8.76121 q^{21} +7.49363 q^{22} -4.19421 q^{23} -0.709000 q^{24} +1.00000 q^{25} -10.5710 q^{26} -1.60463 q^{27} -8.18385 q^{28} +9.03381 q^{29} +4.69193 q^{30} -4.46079 q^{31} +8.10319 q^{32} -8.46992 q^{33} -3.03854 q^{34} +3.80447 q^{35} +4.95445 q^{36} -1.76051 q^{37} -1.57250 q^{38} +11.9482 q^{39} +0.307877 q^{40} +4.51269 q^{41} +17.8503 q^{42} +2.32501 q^{43} -7.91175 q^{44} -2.30320 q^{45} +8.54540 q^{46} +5.84554 q^{47} -8.46291 q^{48} +7.47403 q^{49} -2.03743 q^{50} +3.43441 q^{51} +11.1608 q^{52} -1.84116 q^{53} +3.26931 q^{54} +3.67798 q^{55} +1.17131 q^{56} +1.77737 q^{57} -18.4057 q^{58} -3.61920 q^{59} -4.95373 q^{60} +8.39987 q^{61} +9.08854 q^{62} -8.76249 q^{63} -9.15977 q^{64} -5.18841 q^{65} +17.2568 q^{66} -3.79997 q^{67} +3.20808 q^{68} -9.65872 q^{69} -7.75134 q^{70} -8.96489 q^{71} -0.709104 q^{72} +4.06595 q^{73} +3.58692 q^{74} +2.30287 q^{75} +1.66024 q^{76} +13.9928 q^{77} -24.3436 q^{78} +4.41518 q^{79} +3.67494 q^{80} -10.6049 q^{81} -9.19429 q^{82} -11.0688 q^{83} -18.8463 q^{84} -1.49136 q^{85} -4.73704 q^{86} +20.8037 q^{87} +1.13237 q^{88} +0.424685 q^{89} +4.69261 q^{90} -19.7392 q^{91} -9.02221 q^{92} -10.2726 q^{93} -11.9099 q^{94} -0.771807 q^{95} +18.6606 q^{96} -5.61715 q^{97} -15.2278 q^{98} -8.47115 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 126 q + 5 q^{2} - 9 q^{3} + 109 q^{4} - 126 q^{5} - 21 q^{6} - 23 q^{7} + 12 q^{8} + 109 q^{9}+O(q^{10}) \) 126 * q + 5 * q^2 - 9 * q^3 + 109 * q^4 - 126 * q^5 - 21 * q^6 - 23 * q^7 + 12 * q^8 + 109 * q^9	\( 126 q + 5 q^{2} - 9 q^{3} + 109 q^{4} - 126 q^{5} - 21 q^{6} - 23 q^{7} + 12 q^{8} + 109 q^{9} - 5 q^{10} - 44 q^{11} - 11 q^{12} - 35 q^{13} - 14 q^{14} + 9 q^{15} + 75 q^{16} + 11 q^{17} - 15 q^{18} - 130 q^{19} - 109 q^{20} - 44 q^{21} - 14 q^{22} + 75 q^{23} - 63 q^{24} + 126 q^{25} - 43 q^{26} - 42 q^{27} - 77 q^{28} - 24 q^{29} + 21 q^{30} - 78 q^{31} + 24 q^{32} - 29 q^{33} - 57 q^{34} + 23 q^{35} + 50 q^{36} - 31 q^{37} - 3 q^{38} - 57 q^{39} - 12 q^{40} - 38 q^{41} - 10 q^{42} - 100 q^{43} - 90 q^{44} - 109 q^{45} - 96 q^{46} + 12 q^{47} - 22 q^{48} + 65 q^{49} + 5 q^{50} - 74 q^{51} - 112 q^{52} + 20 q^{53} - 90 q^{54} + 44 q^{55} - 57 q^{56} + 6 q^{57} - 35 q^{58} - 97 q^{59} + 11 q^{60} - 102 q^{61} - 16 q^{62} - 15 q^{63} + 4 q^{64} + 35 q^{65} - 83 q^{66} - 121 q^{67} + 41 q^{68} - 71 q^{69} + 14 q^{70} - 32 q^{71} - 32 q^{72} - 85 q^{73} - 42 q^{74} - 9 q^{75} - 275 q^{76} + 13 q^{77} + 10 q^{78} - 97 q^{79} - 75 q^{80} + 86 q^{81} - 55 q^{82} - 73 q^{83} - 111 q^{84} - 11 q^{85} - 56 q^{86} - q^{87} - 37 q^{88} - 67 q^{89} + 15 q^{90} - 180 q^{91} + 98 q^{92} - 44 q^{93} - 86 q^{94} + 130 q^{95} - 179 q^{96} - 50 q^{97} + 18 q^{98} - 217 q^{99}+O(q^{100}) \) 126 * q + 5 * q^2 - 9 * q^3 + 109 * q^4 - 126 * q^5 - 21 * q^6 - 23 * q^7 + 12 * q^8 + 109 * q^9 - 5 * q^10 - 44 * q^11 - 11 * q^12 - 35 * q^13 - 14 * q^14 + 9 * q^15 + 75 * q^16 + 11 * q^17 - 15 * q^18 - 130 * q^19 - 109 * q^20 - 44 * q^21 - 14 * q^22 + 75 * q^23 - 63 * q^24 + 126 * q^25 - 43 * q^26 - 42 * q^27 - 77 * q^28 - 24 * q^29 + 21 * q^30 - 78 * q^31 + 24 * q^32 - 29 * q^33 - 57 * q^34 + 23 * q^35 + 50 * q^36 - 31 * q^37 - 3 * q^38 - 57 * q^39 - 12 * q^40 - 38 * q^41 - 10 * q^42 - 100 * q^43 - 90 * q^44 - 109 * q^45 - 96 * q^46 + 12 * q^47 - 22 * q^48 + 65 * q^49 + 5 * q^50 - 74 * q^51 - 112 * q^52 + 20 * q^53 - 90 * q^54 + 44 * q^55 - 57 * q^56 + 6 * q^57 - 35 * q^58 - 97 * q^59 + 11 * q^60 - 102 * q^61 - 16 * q^62 - 15 * q^63 + 4 * q^64 + 35 * q^65 - 83 * q^66 - 121 * q^67 + 41 * q^68 - 71 * q^69 + 14 * q^70 - 32 * q^71 - 32 * q^72 - 85 * q^73 - 42 * q^74 - 9 * q^75 - 275 * q^76 + 13 * q^77 + 10 * q^78 - 97 * q^79 - 75 * q^80 + 86 * q^81 - 55 * q^82 - 73 * q^83 - 111 * q^84 - 11 * q^85 - 56 * q^86 - q^87 - 37 * q^88 - 67 * q^89 + 15 * q^90 - 180 * q^91 + 98 * q^92 - 44 * q^93 - 86 * q^94 + 130 * q^95 - 179 * q^96 - 50 * q^97 + 18 * q^98 - 217 * q^99

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