LMFDB - Embedded newform 2005.2.a.d.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2005,2,Mod(1,2005)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2005, 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("2005.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2005 = 5 \cdot 401 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2005.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:	$16.0100056053$
Analytic rank:	$1$
Dimension:	$25$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Character	$\chi$	$=$	2005.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.32427 q^{2} +0.0359659 q^{3} +3.40223 q^{4} +1.00000 q^{5} -0.0835945 q^{6} -0.619979 q^{7} -3.25915 q^{8} -2.99871 q^{9} +O(q^{10})$	\(q-2.32427 q^{2} +0.0359659 q^{3} +3.40223 q^{4} +1.00000 q^{5} -0.0835945 q^{6} -0.619979 q^{7} -3.25915 q^{8} -2.99871 q^{9} -2.32427 q^{10} -6.00326 q^{11} +0.122364 q^{12} +1.64839 q^{13} +1.44100 q^{14} +0.0359659 q^{15} +0.770686 q^{16} +5.62233 q^{17} +6.96980 q^{18} +6.41603 q^{19} +3.40223 q^{20} -0.0222981 q^{21} +13.9532 q^{22} -7.70949 q^{23} -0.117218 q^{24} +1.00000 q^{25} -3.83130 q^{26} -0.215749 q^{27} -2.10931 q^{28} +8.71839 q^{29} -0.0835945 q^{30} +8.93060 q^{31} +4.72701 q^{32} -0.215913 q^{33} -13.0678 q^{34} -0.619979 q^{35} -10.2023 q^{36} -8.41809 q^{37} -14.9126 q^{38} +0.0592859 q^{39} -3.25915 q^{40} -4.35543 q^{41} +0.0518268 q^{42} -0.607943 q^{43} -20.4245 q^{44} -2.99871 q^{45} +17.9189 q^{46} -9.01943 q^{47} +0.0277184 q^{48} -6.61563 q^{49} -2.32427 q^{50} +0.202212 q^{51} +5.60820 q^{52} -3.67148 q^{53} +0.501459 q^{54} -6.00326 q^{55} +2.02060 q^{56} +0.230758 q^{57} -20.2639 q^{58} +9.64268 q^{59} +0.122364 q^{60} +14.6300 q^{61} -20.7571 q^{62} +1.85914 q^{63} -12.5282 q^{64} +1.64839 q^{65} +0.501840 q^{66} -2.18280 q^{67} +19.1284 q^{68} -0.277279 q^{69} +1.44100 q^{70} -14.9463 q^{71} +9.77323 q^{72} -8.95822 q^{73} +19.5659 q^{74} +0.0359659 q^{75} +21.8288 q^{76} +3.72190 q^{77} -0.137796 q^{78} -10.4064 q^{79} +0.770686 q^{80} +8.98836 q^{81} +10.1232 q^{82} +4.33080 q^{83} -0.0758632 q^{84} +5.62233 q^{85} +1.41302 q^{86} +0.313565 q^{87} +19.5655 q^{88} -10.8715 q^{89} +6.96980 q^{90} -1.02197 q^{91} -26.2294 q^{92} +0.321197 q^{93} +20.9636 q^{94} +6.41603 q^{95} +0.170011 q^{96} +1.17931 q^{97} +15.3765 q^{98} +18.0020 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q - 5 q^{2} - 10 q^{3} + 25 q^{4} + 25 q^{5} + 2 q^{6} - 31 q^{7} - 30 q^{8} + 17 q^{9}+O(q^{10}) $ 25 * q - 5 * q^2 - 10 * q^3 + 25 * q^4 + 25 * q^5 + 2 * q^6 - 31 * q^7 - 30 * q^8 + 17 * q^9	\( 25 q - 5 q^{2} - 10 q^{3} + 25 q^{4} + 25 q^{5} + 2 q^{6} - 31 q^{7} - 30 q^{8} + 17 q^{9} - 5 q^{10} - 30 q^{11} - 29 q^{12} - 18 q^{13} + 6 q^{14} - 10 q^{15} + 21 q^{16} - 18 q^{17} - 30 q^{18} - 17 q^{19} + 25 q^{20} + 6 q^{21} - 2 q^{22} - 44 q^{23} + 11 q^{24} + 25 q^{25} - 14 q^{26} - 25 q^{27} - 50 q^{28} - 9 q^{29} + 2 q^{30} - 13 q^{31} - 45 q^{32} - 21 q^{33} - 21 q^{34} - 31 q^{35} + 5 q^{36} - 28 q^{37} - 32 q^{38} + 9 q^{39} - 30 q^{40} + 28 q^{41} - 67 q^{42} - 61 q^{43} - 49 q^{44} + 17 q^{45} + 18 q^{46} - 53 q^{47} - 44 q^{48} + 28 q^{49} - 5 q^{50} - 30 q^{51} - 3 q^{52} - 36 q^{53} + 17 q^{54} - 30 q^{55} - 3 q^{56} - 13 q^{57} + 2 q^{58} - 39 q^{59} - 29 q^{60} + 10 q^{61} - 30 q^{62} - 44 q^{63} - 4 q^{64} - 18 q^{65} + 33 q^{66} - 10 q^{67} - 18 q^{68} - 6 q^{69} + 6 q^{70} - 7 q^{71} - q^{72} - 26 q^{73} - 3 q^{74} - 10 q^{75} + 12 q^{76} + 29 q^{77} - 5 q^{78} - 6 q^{79} + 21 q^{80} + 13 q^{81} - 30 q^{82} - 35 q^{83} + 117 q^{84} - 18 q^{85} + 14 q^{86} - 104 q^{87} + 53 q^{88} + 7 q^{89} - 30 q^{90} - 25 q^{91} - 31 q^{92} + 2 q^{93} + 68 q^{94} - 17 q^{95} + 92 q^{96} + 6 q^{97} + 15 q^{98} - 36 q^{99}+O(q^{100}) \) 25 * q - 5 * q^2 - 10 * q^3 + 25 * q^4 + 25 * q^5 + 2 * q^6 - 31 * q^7 - 30 * q^8 + 17 * q^9 - 5 * q^10 - 30 * q^11 - 29 * q^12 - 18 * q^13 + 6 * q^14 - 10 * q^15 + 21 * q^16 - 18 * q^17 - 30 * q^18 - 17 * q^19 + 25 * q^20 + 6 * q^21 - 2 * q^22 - 44 * q^23 + 11 * q^24 + 25 * q^25 - 14 * q^26 - 25 * q^27 - 50 * q^28 - 9 * q^29 + 2 * q^30 - 13 * q^31 - 45 * q^32 - 21 * q^33 - 21 * q^34 - 31 * q^35 + 5 * q^36 - 28 * q^37 - 32 * q^38 + 9 * q^39 - 30 * q^40 + 28 * q^41 - 67 * q^42 - 61 * q^43 - 49 * q^44 + 17 * q^45 + 18 * q^46 - 53 * q^47 - 44 * q^48 + 28 * q^49 - 5 * q^50 - 30 * q^51 - 3 * q^52 - 36 * q^53 + 17 * q^54 - 30 * q^55 - 3 * q^56 - 13 * q^57 + 2 * q^58 - 39 * q^59 - 29 * q^60 + 10 * q^61 - 30 * q^62 - 44 * q^63 - 4 * q^64 - 18 * q^65 + 33 * q^66 - 10 * q^67 - 18 * q^68 - 6 * q^69 + 6 * q^70 - 7 * q^71 - q^72 - 26 * q^73 - 3 * q^74 - 10 * q^75 + 12 * q^76 + 29 * q^77 - 5 * q^78 - 6 * q^79 + 21 * q^80 + 13 * q^81 - 30 * q^82 - 35 * q^83 + 117 * q^84 - 18 * q^85 + 14 * q^86 - 104 * q^87 + 53 * q^88 + 7 * q^89 - 30 * q^90 - 25 * q^91 - 31 * q^92 + 2 * q^93 + 68 * q^94 - 17 * q^95 + 92 * q^96 + 6 * q^97 + 15 * q^98 - 36 * 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 $

$2$	−2.32427	−1.64351	−0.821753	−	0.569844i	$-0.807004\pi$
$2$	−2.32427	−1.64351	−0.821753	+	0.569844i	$0.807004\pi$
$3$	0.0359659	0.0207649	0.0103825	−	0.999946i	$-0.496695\pi$
$3$	0.0359659	0.0207649	0.0103825	+	0.999946i	$0.496695\pi$
$4$	3.40223	1.70111
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−0.0835945	−0.0341273
$7$	−0.619979	−0.234330	−0.117165	−	0.993112i	$-0.537381\pi$
$7$	−0.619979	−0.234330	−0.117165	+	0.993112i	$0.537381\pi$
$8$	−3.25915	−1.15228
$9$	−2.99871	−0.999569
$10$	−2.32427	−0.734998
$11$	−6.00326	−1.81005	−0.905026	−	0.425356i	$-0.860149\pi$
$11$	−6.00326	−1.81005	−0.905026	+	0.425356i	$0.860149\pi$
$12$	0.122364	0.0353235
$13$	1.64839	0.457181	0.228591	−	0.973523i	$-0.426588\pi$
$13$	1.64839	0.457181	0.228591	+	0.973523i	$0.426588\pi$
$14$	1.44100	0.385123
$15$	0.0359659	0.00928636
$16$	0.770686	0.192672
$17$	5.62233	1.36361	0.681807	−	0.731532i	$-0.261194\pi$
$17$	5.62233	1.36361	0.681807	+	0.731532i	$0.261194\pi$
$18$	6.96980	1.64280
$19$	6.41603	1.47194	0.735969	−	0.677015i	$-0.236727\pi$
$19$	6.41603	1.47194	0.735969	+	0.677015i	$0.236727\pi$
$20$	3.40223	0.760761
$21$	−0.0222981	−0.00486585
$22$	13.9532	2.97483
$23$	−7.70949	−1.60754	−0.803770	−	0.594940i	$-0.797176\pi$
$23$	−7.70949	−1.60754	−0.803770	+	0.594940i	$0.797176\pi$
$24$	−0.117218	−0.0239271
$25$	1.00000	0.200000
$26$	−3.83130	−0.751381
$27$	−0.215749	−0.0415209
$28$	−2.10931	−0.398622
$29$	8.71839	1.61896	0.809482	−	0.587144i	$-0.199748\pi$
$29$	8.71839	1.61896	0.809482	+	0.587144i	$0.199748\pi$
$30$	−0.0835945	−0.0152622
$31$	8.93060	1.60398	0.801991	−	0.597336i	$-0.203774\pi$
$31$	8.93060	1.60398	0.801991	+	0.597336i	$0.203774\pi$
$32$	4.72701	0.835626
$33$	−0.215913	−0.0375856
$34$	−13.0678	−2.24111
$35$	−0.619979	−0.104796
$36$	−10.2023	−1.70038
$37$	−8.41809	−1.38393	−0.691963	−	0.721933i	$-0.743254\pi$
$37$	−8.41809	−1.38393	−0.691963	+	0.721933i	$0.743254\pi$
$38$	−14.9126	−2.41914
$39$	0.0592859	0.00949334
$40$	−3.25915	−0.515317
$41$	−4.35543	−0.680204	−0.340102	−	0.940389i	$-0.610462\pi$
$41$	−4.35543	−0.680204	−0.340102	+	0.940389i	$0.610462\pi$
$42$	0.0518268	0.00799705
$43$	−0.607943	−0.0927105	−0.0463553	−	0.998925i	$-0.514761\pi$
$43$	−0.607943	−0.0927105	−0.0463553	+	0.998925i	$0.514761\pi$
$44$	−20.4245	−3.07910
$45$	−2.99871	−0.447021
$46$	17.9189	2.64200
$47$	−9.01943	−1.31562	−0.657810	−	0.753184i	$-0.728517\pi$
$47$	−9.01943	−1.31562	−0.657810	+	0.753184i	$0.728517\pi$
$48$	0.0277184	0.00400081
$49$	−6.61563	−0.945089
$50$	−2.32427	−0.328701
$51$	0.202212	0.0283154
$52$	5.60820	0.777717
$53$	−3.67148	−0.504316	−0.252158	−	0.967686i	$-0.581140\pi$
$53$	−3.67148	−0.504316	−0.252158	+	0.967686i	$0.581140\pi$
$54$	0.501459	0.0682399
$55$	−6.00326	−0.809480
$56$	2.02060	0.270015
$57$	0.230758	0.0305647
$58$	−20.2639	−2.66078
$59$	9.64268	1.25537	0.627685	−	0.778468i	$-0.284003\pi$
$59$	9.64268	1.25537	0.627685	+	0.778468i	$0.284003\pi$
$60$	0.122364	0.0157971
$61$	14.6300	1.87318	0.936590	−	0.350427i	$-0.113963\pi$
$61$	14.6300	1.87318	0.936590	+	0.350427i	$0.113963\pi$
$62$	−20.7571	−2.63616
$63$	1.85914	0.234229
$64$	−12.5282	−1.56603
$65$	1.64839	0.204458
$66$	0.501840	0.0617722
$67$	−2.18280	−0.266671	−0.133336	−	0.991071i	$-0.542569\pi$
$67$	−2.18280	−0.266671	−0.133336	+	0.991071i	$0.542569\pi$
$68$	19.1284	2.31966
$69$	−0.277279	−0.0333805
$70$	1.44100	0.172232
$71$	−14.9463	−1.77380	−0.886899	−	0.461963i	$-0.847145\pi$
$71$	−14.9463	−1.77380	−0.886899	+	0.461963i	$0.847145\pi$
$72$	9.77323	1.15179
$73$	−8.95822	−1.04848	−0.524240	−	0.851570i	$-0.675651\pi$
$73$	−8.95822	−1.04848	−0.524240	+	0.851570i	$0.675651\pi$
$74$	19.5659	2.27449
$75$	0.0359659	0.00415299
$76$	21.8288	2.50393
$77$	3.72190	0.424150
$78$	−0.137796	−0.0156024
$79$	−10.4064	−1.17081	−0.585403	−	0.810742i	$-0.699064\pi$
$79$	−10.4064	−1.17081	−0.585403	+	0.810742i	$0.699064\pi$
$80$	0.770686	0.0861654
$81$	8.98836	0.998707
$82$	10.1232	1.11792
$83$	4.33080	0.475367	0.237683	−	0.971343i	$-0.423612\pi$
$83$	4.33080	0.475367	0.237683	+	0.971343i	$0.423612\pi$
$84$	−0.0758632	−0.00827736
$85$	5.62233	0.609827
$86$	1.41302	0.152370
$87$	0.313565	0.0336177
$88$	19.5655	2.08569
$89$	−10.8715	−1.15237	−0.576187	−	0.817318i	$-0.695460\pi$
$89$	−10.8715	−1.15237	−0.576187	+	0.817318i	$0.695460\pi$
$90$	6.96980	0.734681
$91$	−1.02197	−0.107131
$92$	−26.2294	−2.73461
$93$	0.321197	0.0333066
$94$	20.9636	2.16223
$95$	6.41603	0.658271
$96$	0.170011	0.0173517
$97$	1.17931	0.119741	0.0598706	−	0.998206i	$-0.480931\pi$
$97$	1.17931	0.119741	0.0598706	+	0.998206i	$0.480931\pi$
$98$	15.3765	1.55326
$99$	18.0020	1.80927
$100$	3.40223	0.340223
$101$	−1.20539	−0.119940	−0.0599702	−	0.998200i	$-0.519101\pi$
$101$	−1.20539	−0.119940	−0.0599702	+	0.998200i	$0.519101\pi$
$102$	−0.469995	−0.0465365
$103$	0.973636	0.0959352	0.0479676	−	0.998849i	$-0.484726\pi$
$103$	0.973636	0.0959352	0.0479676	+	0.998849i	$0.484726\pi$
$104$	−5.37235	−0.526802
$105$	−0.0222981	−0.00217607
$106$	8.53350	0.828847
$107$	−16.9346	−1.63713	−0.818563	−	0.574417i	$-0.805229\pi$
$107$	−16.9346	−1.63713	−0.818563	+	0.574417i	$0.805229\pi$
$108$	−0.734027	−0.0706318
$109$	9.25848	0.886801	0.443401	−	0.896324i	$-0.353772\pi$
$109$	9.25848	0.886801	0.443401	+	0.896324i	$0.353772\pi$
$110$	13.9532	1.33039
$111$	−0.302764	−0.0287371
$112$	−0.477809	−0.0451487
$113$	−10.1020	−0.950314	−0.475157	−	0.879901i	$-0.657609\pi$
$113$	−10.1020	−0.950314	−0.475157	+	0.879901i	$0.657609\pi$
$114$	−0.536345	−0.0502333
$115$	−7.70949	−0.718914
$116$	29.6619	2.75404
$117$	−4.94304	−0.456984
$118$	−22.4122	−2.06321
$119$	−3.48572	−0.319536
$120$	−0.117218	−0.0107005
$121$	25.0392	2.27629
$122$	−34.0041	−3.07858
$123$	−0.156647	−0.0141244
$124$	30.3839	2.72855
$125$	1.00000	0.0894427
$126$	−4.32113	−0.384957
$127$	−17.8461	−1.58358	−0.791792	−	0.610791i	$-0.790852\pi$
$127$	−17.8461	−1.58358	−0.791792	+	0.610791i	$0.790852\pi$
$128$	19.6649	1.73815
$129$	−0.0218652	−0.00192513
$130$	−3.83130	−0.336028
$131$	−4.00413	−0.349843	−0.174921	−	0.984582i	$-0.555967\pi$
$131$	−4.00413	−0.349843	−0.174921	+	0.984582i	$0.555967\pi$
$132$	−0.734584	−0.0639374
$133$	−3.97780	−0.344919
$134$	5.07341	0.438276
$135$	−0.215749	−0.0185687
$136$	−18.3240	−1.57127
$137$	−4.61943	−0.394665	−0.197332	−	0.980337i	$-0.563228\pi$
$137$	−4.61943	−0.394665	−0.197332	+	0.980337i	$0.563228\pi$
$138$	0.644471	0.0548610
$139$	−15.2630	−1.29459	−0.647297	−	0.762238i	$-0.724101\pi$
$139$	−15.2630	−1.29459	−0.647297	+	0.762238i	$0.724101\pi$
$140$	−2.10931	−0.178269
$141$	−0.324392	−0.0273188
$142$	34.7392	2.91525
$143$	−9.89573	−0.827522
$144$	−2.31106	−0.192589
$145$	8.71839	0.724023
$146$	20.8213	1.72318
$147$	−0.237937	−0.0196247
$148$	−28.6402	−2.35421
$149$	5.02888	0.411982	0.205991	−	0.978554i	$-0.433958\pi$
$149$	5.02888	0.411982	0.205991	+	0.978554i	$0.433958\pi$
$150$	−0.0835945	−0.00682546
$151$	12.7380	1.03660	0.518302	−	0.855198i	$-0.326564\pi$
$151$	12.7380	1.03660	0.518302	+	0.855198i	$0.326564\pi$
$152$	−20.9108	−1.69609
$153$	−16.8597	−1.36303
$154$	−8.65069	−0.697092
$155$	8.93060	0.717323
$156$	0.201704	0.0161492
$157$	1.08191	0.0863458	0.0431729	−	0.999068i	$-0.486253\pi$
$157$	1.08191	0.0863458	0.0431729	+	0.999068i	$0.486253\pi$
$158$	24.1872	1.92423
$159$	−0.132048	−0.0104721
$160$	4.72701	0.373703
$161$	4.77972	0.376695
$162$	−20.8914	−1.64138
$163$	3.31057	0.259304	0.129652	−	0.991560i	$-0.458614\pi$
$163$	3.31057	0.259304	0.129652	+	0.991560i	$0.458614\pi$
$164$	−14.8182	−1.15710
$165$	−0.215913	−0.0168088
$166$	−10.0659	−0.781268
$167$	−1.77521	−0.137370	−0.0686848	−	0.997638i	$-0.521880\pi$
$167$	−1.77521	−0.137370	−0.0686848	+	0.997638i	$0.521880\pi$
$168$	0.0726729	0.00560683
$169$	−10.2828	−0.790985
$170$	−13.0678	−1.00225
$171$	−19.2398	−1.47130
$172$	−2.06836	−0.157711
$173$	−1.89932	−0.144403	−0.0722014	−	0.997390i	$-0.523002\pi$
$173$	−1.89932	−0.144403	−0.0722014	+	0.997390i	$0.523002\pi$
$174$	−0.728809	−0.0552509
$175$	−0.619979	−0.0468660
$176$	−4.62663	−0.348746
$177$	0.346808	0.0260677
$178$	25.2682	1.89393
$179$	−24.3871	−1.82277	−0.911387	−	0.411550i	$-0.864988\pi$
$179$	−24.3871	−1.82277	−0.911387	+	0.411550i	$0.864988\pi$
$180$	−10.2023	−0.760433
$181$	1.93671	0.143954	0.0719772	−	0.997406i	$-0.477069\pi$
$181$	1.93671	0.143954	0.0719772	+	0.997406i	$0.477069\pi$
$182$	2.37533	0.176071
$183$	0.526182	0.0388965
$184$	25.1264	1.85234
$185$	−8.41809	−0.618910
$186$	−0.746549	−0.0547396
$187$	−33.7523	−2.46821
$188$	−30.6861	−2.23802
$189$	0.133760	0.00972960
$190$	−14.9126	−1.08187
$191$	−2.25419	−0.163107	−0.0815536	−	0.996669i	$-0.525988\pi$
$191$	−2.25419	−0.163107	−0.0815536	+	0.996669i	$0.525988\pi$
$192$	−0.450589	−0.0325185
$193$	−10.1467	−0.730379	−0.365190	−	0.930933i	$-0.618996\pi$
$193$	−10.1467	−0.730379	−0.365190	+	0.930933i	$0.618996\pi$
$194$	−2.74104	−0.196795
$195$	0.0592859	0.00424555
$196$	−22.5079	−1.60770
$197$	5.70010	0.406115	0.203058	−	0.979167i	$-0.434912\pi$
$197$	5.70010	0.406115	0.203058	+	0.979167i	$0.434912\pi$
$198$	−41.8415	−2.97355
$199$	−19.3023	−1.36831	−0.684153	−	0.729338i	$-0.739828\pi$
$199$	−19.3023	−1.36831	−0.684153	+	0.729338i	$0.739828\pi$
$200$	−3.25915	−0.230457
$201$	−0.0785064	−0.00553741
$202$	2.80164	0.197123
$203$	−5.40522	−0.379372
$204$	0.687971	0.0481676
$205$	−4.35543	−0.304196
$206$	−2.26299	−0.157670
$207$	23.1185	1.60685
$208$	1.27039	0.0880859
$209$	−38.5171	−2.66428
$210$	0.0518268	0.00357639
$211$	−9.72951	−0.669807	−0.334904	−	0.942252i	$-0.608704\pi$
$211$	−9.72951	−0.669807	−0.334904	+	0.942252i	$0.608704\pi$
$212$	−12.4912	−0.857899
$213$	−0.537557	−0.0368328
$214$	39.3605	2.69063
$215$	−0.607943	−0.0414614
$216$	0.703158	0.0478438
$217$	−5.53678	−0.375861
$218$	−21.5192	−1.45746
$219$	−0.322191	−0.0217716
$220$	−20.4245	−1.37702
$221$	9.26779	0.623419
$222$	0.703706	0.0472296
$223$	−4.72933	−0.316699	−0.158350	−	0.987383i	$-0.550617\pi$
$223$	−4.72933	−0.316699	−0.158350	+	0.987383i	$0.550617\pi$
$224$	−2.93065	−0.195812
$225$	−2.99871	−0.199914
$226$	23.4797	1.56185
$227$	18.0444	1.19765	0.598823	−	0.800881i	$-0.295635\pi$
$227$	18.0444	1.19765	0.598823	+	0.800881i	$0.295635\pi$
$228$	0.785092	0.0519940
$229$	1.80921	0.119556	0.0597780	−	0.998212i	$-0.480961\pi$
$229$	1.80921	0.119556	0.0597780	+	0.998212i	$0.480961\pi$
$230$	17.9189	1.18154
$231$	0.133861	0.00880744
$232$	−28.4145	−1.86551
$233$	6.12146	0.401030	0.200515	−	0.979691i	$-0.435738\pi$
$233$	6.12146	0.401030	0.200515	+	0.979691i	$0.435738\pi$
$234$	11.4890	0.751057
$235$	−9.01943	−0.588363
$236$	32.8066	2.13552
$237$	−0.374274	−0.0243117
$238$	8.10176	0.525159
$239$	−14.2725	−0.923212	−0.461606	−	0.887085i	$-0.652727\pi$
$239$	−14.2725	−0.923212	−0.461606	+	0.887085i	$0.652727\pi$
$240$	0.0277184	0.00178922
$241$	7.70504	0.496326	0.248163	−	0.968718i	$-0.420173\pi$
$241$	7.70504	0.496326	0.248163	+	0.968718i	$0.420173\pi$
$242$	−58.1977	−3.74109
$243$	0.970522	0.0622590
$244$	49.7746	3.18649
$245$	−6.61563	−0.422657
$246$	0.364090	0.0232135
$247$	10.5761	0.672943
$248$	−29.1061	−1.84824
$249$	0.155761	0.00987096
$250$	−2.32427	−0.147000
$251$	13.4298	0.847683	0.423842	−	0.905736i	$-0.360681\pi$
$251$	13.4298	0.847683	0.423842	+	0.905736i	$0.360681\pi$
$252$	6.32520	0.398450
$253$	46.2821	2.90973
$254$	41.4791	2.60263
$255$	0.202212	0.0126630
$256$	−20.6501	−1.29063
$257$	5.52269	0.344496	0.172248	−	0.985054i	$-0.444897\pi$
$257$	5.52269	0.344496	0.172248	+	0.985054i	$0.444897\pi$
$258$	0.0508207	0.00316396
$259$	5.21904	0.324295
$260$	5.60820	0.347806
$261$	−26.1439	−1.61827
$262$	9.30669	0.574969
$263$	−20.6932	−1.27600	−0.638000	−	0.770036i	$-0.720238\pi$
$263$	−20.6932	−1.27600	−0.638000	+	0.770036i	$0.720238\pi$
$264$	0.703692	0.0433093
$265$	−3.67148	−0.225537
$266$	9.24548	0.566877
$267$	−0.391002	−0.0239290
$268$	−7.42637	−0.453638
$269$	−17.8891	−1.09072	−0.545358	−	0.838204i	$-0.683606\pi$
$269$	−17.8891	−1.09072	−0.545358	+	0.838204i	$0.683606\pi$
$270$	0.501459	0.0305178
$271$	16.7821	1.01944	0.509719	−	0.860341i	$-0.329749\pi$
$271$	16.7821	1.01944	0.509719	+	0.860341i	$0.329749\pi$
$272$	4.33305	0.262730
$273$	−0.0367560	−0.00222458
$274$	10.7368	0.648634
$275$	−6.00326	−0.362010
$276$	−0.943366	−0.0567840
$277$	17.9458	1.07826	0.539129	−	0.842223i	$-0.318754\pi$
$277$	17.9458	1.07826	0.539129	+	0.842223i	$0.318754\pi$
$278$	35.4754	2.12767
$279$	−26.7802	−1.60329
$280$	2.02060	0.120754
$281$	−3.13769	−0.187179	−0.0935894	−	0.995611i	$-0.529834\pi$
$281$	−3.13769	−0.187179	−0.0935894	+	0.995611i	$0.529834\pi$
$282$	0.753975	0.0448986
$283$	15.6874	0.932521	0.466261	−	0.884647i	$-0.345601\pi$
$283$	15.6874	0.932521	0.466261	+	0.884647i	$0.345601\pi$
$284$	−50.8506	−3.01743
$285$	0.230758	0.0136689
$286$	23.0003	1.36004
$287$	2.70027	0.159392
$288$	−14.1749	−0.835266
$289$	14.6105	0.859444
$290$	−20.2639	−1.18994
$291$	0.0424151	0.00248642
$292$	−30.4779	−1.78358
$293$	−29.8505	−1.74388	−0.871942	−	0.489609i	$-0.837140\pi$
$293$	−29.8505	−1.74388	−0.871942	+	0.489609i	$0.837140\pi$
$294$	0.553030	0.0322534
$295$	9.64268	0.561418
$296$	27.4358	1.59467
$297$	1.29520	0.0751550
$298$	−11.6885	−0.677095
$299$	−12.7083	−0.734938
$300$	0.122364	0.00706470
$301$	0.376912	0.0217249
$302$	−29.6066	−1.70367
$303$	−0.0433528	−0.00249055
$304$	4.94475	0.283601
$305$	14.6300	0.837712
$306$	39.1865	2.24014
$307$	−4.02035	−0.229454	−0.114727	−	0.993397i	$-0.536599\pi$
$307$	−4.02035	−0.229454	−0.114727	+	0.993397i	$0.536599\pi$
$308$	12.6627	0.721526
$309$	0.0350177	0.00199209
$310$	−20.7571	−1.17892
$311$	4.60654	0.261213	0.130607	−	0.991434i	$-0.458308\pi$
$311$	4.60654	0.261213	0.130607	+	0.991434i	$0.458308\pi$
$312$	−0.193222	−0.0109390
$313$	14.5999	0.825238	0.412619	−	0.910904i	$-0.364614\pi$
$313$	14.5999	0.825238	0.412619	+	0.910904i	$0.364614\pi$
$314$	−2.51465	−0.141910
$315$	1.85914	0.104750
$316$	−35.4047	−1.99167
$317$	−34.2544	−1.92392	−0.961958	−	0.273197i	$-0.911919\pi$
$317$	−34.2544	−1.92392	−0.961958	+	0.273197i	$0.911919\pi$
$318$	0.306915	0.0172109
$319$	−52.3388	−2.93041
$320$	−12.5282	−0.700349
$321$	−0.609067	−0.0339948
$322$	−11.1094	−0.619101
$323$	36.0730	2.00716
$324$	30.5804	1.69891
$325$	1.64839	0.0914363
$326$	−7.69465	−0.426167
$327$	0.332990	0.0184144
$328$	14.1950	0.783787
$329$	5.59186	0.308289
$330$	0.501840	0.0276254
$331$	−1.47164	−0.0808888	−0.0404444	−	0.999182i	$-0.512877\pi$
$331$	−1.47164	−0.0808888	−0.0404444	+	0.999182i	$0.512877\pi$
$332$	14.7343	0.808652
$333$	25.2434	1.38333
$334$	4.12606	0.225768
$335$	−2.18280	−0.119259
$336$	−0.0171849	−0.000937511 0
$337$	5.65788	0.308204	0.154102	−	0.988055i	$-0.450751\pi$
$337$	5.65788	0.308204	0.154102	+	0.988055i	$0.450751\pi$
$338$	23.9000	1.29999
$339$	−0.363327	−0.0197332
$340$	19.1284	1.03738
$341$	−53.6127	−2.90329
$342$	44.7184	2.41810
$343$	8.44140	0.455793
$344$	1.98138	0.106829
$345$	−0.277279	−0.0149282
$346$	4.41454	0.237327
$347$	−8.17656	−0.438941	−0.219470	−	0.975619i	$-0.570433\pi$
$347$	−8.17656	−0.438941	−0.219470	+	0.975619i	$0.570433\pi$
$348$	1.06682	0.0571875
$349$	−8.78478	−0.470239	−0.235119	−	0.971967i	$-0.575548\pi$
$349$	−8.78478	−0.470239	−0.235119	+	0.971967i	$0.575548\pi$
$350$	1.44100	0.0770246
$351$	−0.355639	−0.0189826
$352$	−28.3775	−1.51253
$353$	−22.2416	−1.18380	−0.591901	−	0.806011i	$-0.701622\pi$
$353$	−22.2416	−1.18380	−0.591901	+	0.806011i	$0.701622\pi$
$354$	−0.806074	−0.0428424
$355$	−14.9463	−0.793267
$356$	−36.9872	−1.96032
$357$	−0.125367	−0.00663514
$358$	56.6821	2.99574
$359$	−31.5111	−1.66309	−0.831546	−	0.555456i	$-0.812544\pi$
$359$	−31.5111	−1.66309	−0.831546	+	0.555456i	$0.812544\pi$
$360$	9.77323	0.515094
$361$	22.1654	1.16660
$362$	−4.50143	−0.236590
$363$	0.900557	0.0472670
$364$	−3.47697	−0.182242
$365$	−8.95822	−0.468895
$366$	−1.22299	−0.0639266
$367$	−5.81216	−0.303392	−0.151696	−	0.988427i	$-0.548474\pi$
$367$	−5.81216	−0.303392	−0.151696	+	0.988427i	$0.548474\pi$
$368$	−5.94160	−0.309727
$369$	13.0607	0.679911
$370$	19.5659	1.01718
$371$	2.27624	0.118176
$372$	1.09279	0.0566583
$373$	−35.2125	−1.82324	−0.911618	−	0.411038i	$-0.865166\pi$
$373$	−35.2125	−1.82324	−0.911618	+	0.411038i	$0.865166\pi$
$374$	78.4494	4.05652
$375$	0.0359659	0.00185727
$376$	29.3957	1.51597
$377$	14.3713	0.740161
$378$	−0.310894	−0.0159907
$379$	4.80405	0.246767	0.123384	−	0.992359i	$-0.460625\pi$
$379$	4.80405	0.246767	0.123384	+	0.992359i	$0.460625\pi$
$380$	21.8288	1.11979
$381$	−0.641851	−0.0328830
$382$	5.23933	0.268068
$383$	33.3128	1.70221	0.851103	−	0.524999i	$-0.175934\pi$
$383$	33.3128	1.70221	0.851103	+	0.524999i	$0.175934\pi$
$384$	0.707267	0.0360926
$385$	3.72190	0.189685
$386$	23.5838	1.20038
$387$	1.82304	0.0926705
$388$	4.01229	0.203693
$389$	7.36890	0.373618	0.186809	−	0.982396i	$-0.440185\pi$
$389$	7.36890	0.373618	0.186809	+	0.982396i	$0.440185\pi$
$390$	−0.137796	−0.00697759
$391$	−43.3453	−2.19207
$392$	21.5613	1.08901
$393$	−0.144012	−0.00726446
$394$	−13.2486	−0.667453
$395$	−10.4064	−0.523600
$396$	61.2469	3.07777
$397$	−11.9268	−0.598588	−0.299294	−	0.954161i	$-0.596751\pi$
$397$	−11.9268	−0.598588	−0.299294	+	0.954161i	$0.596751\pi$
$398$	44.8638	2.24882
$399$	−0.143065	−0.00716223
$400$	0.770686	0.0385343
$401$	1.00000	0.0499376
$401$	1.00000	0.0499376
$402$	0.182470	0.00910077
$403$	14.7211	0.733311
$404$	−4.10100	−0.204032
$405$	8.98836	0.446635
$406$	12.5632	0.623500
$407$	50.5360	2.50498
$408$	−0.659039	−0.0326273
$409$	−0.513799	−0.0254057	−0.0127029	−	0.999919i	$-0.504044\pi$
$409$	−0.513799	−0.0254057	−0.0127029	+	0.999919i	$0.504044\pi$
$410$	10.1232	0.499949
$411$	−0.166142	−0.00819519
$412$	3.31253	0.163197
$413$	−5.97826	−0.294171
$414$	−53.7336	−2.64086
$415$	4.33080	0.212590
$416$	7.79197	0.382033
$417$	−0.548949	−0.0268822
$418$	89.5241	4.37877
$419$	5.38589	0.263118	0.131559	−	0.991308i	$-0.458002\pi$
$419$	5.38589	0.263118	0.131559	+	0.991308i	$0.458002\pi$
$420$	−0.0758632	−0.00370175
$421$	10.4642	0.509994	0.254997	−	0.966942i	$-0.417925\pi$
$421$	10.4642	0.509994	0.254997	+	0.966942i	$0.417925\pi$
$422$	22.6140	1.10083
$423$	27.0466	1.31505
$424$	11.9659	0.581115
$425$	5.62233	0.272723
$426$	1.24943	0.0605349
$427$	−9.07030	−0.438942
$428$	−57.6152	−2.78494
$429$	−0.355909	−0.0171834
$430$	1.41302	0.0681421
$431$	−5.67570	−0.273389	−0.136694	−	0.990613i	$-0.543648\pi$
$431$	−5.67570	−0.273389	−0.136694	+	0.990613i	$0.543648\pi$
$432$	−0.166275	−0.00799990
$433$	−23.0342	−1.10695	−0.553477	−	0.832864i	$-0.686699\pi$
$433$	−23.0342	−1.10695	−0.553477	+	0.832864i	$0.686699\pi$
$434$	12.8690	0.617730
$435$	0.313565	0.0150343
$436$	31.4994	1.50855
$437$	−49.4643	−2.36620
$438$	0.748858	0.0357818
$439$	22.8719	1.09162	0.545809	−	0.837910i	$-0.316223\pi$
$439$	22.8719	1.09162	0.545809	+	0.837910i	$0.316223\pi$
$440$	19.5655	0.932750
$441$	19.8383	0.944682
$442$	−21.5408	−1.02459
$443$	2.17975	0.103563	0.0517816	−	0.998658i	$-0.483510\pi$
$443$	2.17975	0.103563	0.0517816	+	0.998658i	$0.483510\pi$
$444$	−1.03007	−0.0488851
$445$	−10.8715	−0.515357
$446$	10.9922	0.520497
$447$	0.180868	0.00855478
$448$	7.76724	0.366967
$449$	16.4820	0.777833	0.388916	−	0.921273i	$-0.372850\pi$
$449$	16.4820	0.777833	0.388916	+	0.921273i	$0.372850\pi$
$450$	6.96980	0.328560
$451$	26.1468	1.23120
$452$	−34.3692	−1.61659
$453$	0.458134	0.0215250
$454$	−41.9399	−1.96834
$455$	−1.02197	−0.0479106
$456$	−0.752076	−0.0352192
$457$	16.4253	0.768343	0.384171	−	0.923262i	$-0.374487\pi$
$457$	16.4253	0.768343	0.384171	+	0.923262i	$0.374487\pi$
$458$	−4.20509	−0.196491
$459$	−1.21301	−0.0566185
$460$	−26.2294	−1.22295
$461$	24.5497	1.14339	0.571697	−	0.820465i	$-0.306285\pi$
$461$	24.5497	1.14339	0.571697	+	0.820465i	$0.306285\pi$
$462$	−0.311130	−0.0144751
$463$	5.42360	0.252056	0.126028	−	0.992027i	$-0.459777\pi$
$463$	5.42360	0.252056	0.126028	+	0.992027i	$0.459777\pi$
$464$	6.71914	0.311928
$465$	0.321197	0.0148952
$466$	−14.2279	−0.659095
$467$	−8.23405	−0.381026	−0.190513	−	0.981685i	$-0.561015\pi$
$467$	−8.23405	−0.381026	−0.190513	+	0.981685i	$0.561015\pi$
$468$	−16.8173	−0.777382
$469$	1.35329	0.0624891
$470$	20.9636	0.966978
$471$	0.0389119	0.00179296
$472$	−31.4269	−1.44654
$473$	3.64964	0.167811
$474$	0.869913	0.0399564
$475$	6.41603	0.294388
$476$	−11.8592	−0.543566
$477$	11.0097	0.504099
$478$	33.1732	1.51731
$479$	−2.58818	−0.118257	−0.0591284	−	0.998250i	$-0.518832\pi$
$479$	−2.58818	−0.118257	−0.0591284	+	0.998250i	$0.518832\pi$
$480$	0.170011	0.00775993
$481$	−13.8763	−0.632705
$482$	−17.9086	−0.815714
$483$	0.171907	0.00782205
$484$	85.1889	3.87222
$485$	1.17931	0.0535499
$486$	−2.25575	−0.102323
$487$	−26.6269	−1.20658	−0.603291	−	0.797521i	$-0.706144\pi$
$487$	−26.6269	−1.20658	−0.603291	+	0.797521i	$0.706144\pi$
$488$	−47.6814	−2.15843
$489$	0.119068	0.00538442
$490$	15.3765	0.694639
$491$	−11.4964	−0.518827	−0.259414	−	0.965766i	$-0.583529\pi$
$491$	−11.4964	−0.518827	−0.259414	+	0.965766i	$0.583529\pi$
$492$	−0.532949	−0.0240272
$493$	49.0176	2.20764
$494$	−24.5818	−1.10599
$495$	18.0020	0.809131
$496$	6.88269	0.309042
$497$	9.26638	0.415654
$498$	−0.362031	−0.0162230
$499$	27.5804	1.23467	0.617334	−	0.786701i	$-0.288213\pi$
$499$	27.5804	1.23467	0.617334	+	0.786701i	$0.288213\pi$
$500$	3.40223	0.152152
$501$	−0.0638469	−0.00285247
$502$	−31.2145	−1.39317
$503$	36.9225	1.64629	0.823147	−	0.567828i	$-0.192216\pi$
$503$	36.9225	1.64629	0.823147	+	0.567828i	$0.192216\pi$
$504$	−6.05920	−0.269898
$505$	−1.20539	−0.0536390
$506$	−107.572	−4.78216
$507$	−0.369831	−0.0164248
$508$	−60.7164	−2.69386
$509$	−13.0821	−0.579852	−0.289926	−	0.957049i	$-0.593631\pi$
$509$	−13.0821	−0.579852	−0.289926	+	0.957049i	$0.593631\pi$
$510$	−0.469995	−0.0208117
$511$	5.55391	0.245691
$512$	8.66661	0.383014
$513$	−1.38425	−0.0611162
$514$	−12.8362	−0.566181
$515$	0.973636	0.0429035
$516$	−0.0743905	−0.00327486
$517$	54.1460	2.38134
$518$	−12.1305	−0.532981
$519$	−0.0683109	−0.00299852
$520$	−5.37235	−0.235593
$521$	−9.05205	−0.396577	−0.198289	−	0.980144i	$-0.563538\pi$
$521$	−9.05205	−0.396577	−0.198289	+	0.980144i	$0.563538\pi$
$522$	60.7654	2.65963
$523$	−25.0376	−1.09482	−0.547409	−	0.836865i	$-0.684386\pi$
$523$	−25.0376	−1.09482	−0.547409	+	0.836865i	$0.684386\pi$
$524$	−13.6230	−0.595122
$525$	−0.0222981	−0.000973170 0
$526$	48.0967	2.09711
$527$	50.2107	2.18721
$528$	−0.166401	−0.00724168
$529$	36.4363	1.58419
$530$	8.53350	0.370671
$531$	−28.9156	−1.25483
$532$	−13.5334	−0.586747
$533$	−7.17945	−0.310977
$534$	0.908794	0.0393274
$535$	−16.9346	−0.732145
$536$	7.11406	0.307281
$537$	−0.877103	−0.0378498
$538$	41.5790	1.79260
$539$	39.7153	1.71066
$540$	−0.734027	−0.0315875
$541$	0.792296	0.0340635	0.0170317	−	0.999855i	$-0.494578\pi$
$541$	0.792296	0.0340635	0.0170317	+	0.999855i	$0.494578\pi$
$542$	−39.0061	−1.67545
$543$	0.0696555	0.00298920
$544$	26.5768	1.13947
$545$	9.25848	0.396590
$546$	0.0854309	0.00365610
$547$	40.2450	1.72075	0.860376	−	0.509660i	$-0.170229\pi$
$547$	40.2450	1.72075	0.860376	+	0.509660i	$0.170229\pi$
$548$	−15.7164	−0.671369
$549$	−43.8711	−1.87237
$550$	13.9532	0.594966
$551$	55.9375	2.38302
$552$	0.903694	0.0384638
$553$	6.45172	0.274355
$554$	−41.7108	−1.77212
$555$	−0.302764	−0.0128516
$556$	−51.9283	−2.20225
$557$	5.44892	0.230878	0.115439	−	0.993315i	$-0.463172\pi$
$557$	5.44892	0.230878	0.115439	+	0.993315i	$0.463172\pi$
$558$	62.2445	2.63502
$559$	−1.00213	−0.0423855
$560$	−0.477809	−0.0201911
$561$	−1.21393	−0.0512523
$562$	7.29283	0.307630
$563$	−31.6769	−1.33502	−0.667511	−	0.744599i	$-0.732641\pi$
$563$	−31.6769	−1.33502	−0.667511	+	0.744599i	$0.732641\pi$
$564$	−1.10366	−0.0464723
$565$	−10.1020	−0.424993
$566$	−36.4618	−1.53260
$567$	−5.57259	−0.234027
$568$	48.7122	2.04392
$569$	28.4572	1.19299	0.596495	−	0.802617i	$-0.296560\pi$
$569$	28.4572	1.19299	0.596495	+	0.802617i	$0.296560\pi$
$570$	−0.536345	−0.0224650
$571$	35.5773	1.48886	0.744431	−	0.667700i	$-0.232721\pi$
$571$	35.5773	1.48886	0.744431	+	0.667700i	$0.232721\pi$
$572$	−33.6675	−1.40771
$573$	−0.0810739	−0.00338691
$574$	−6.27616	−0.261962
$575$	−7.70949	−0.321508
$576$	37.5685	1.56535
$577$	12.5055	0.520610	0.260305	−	0.965526i	$-0.416177\pi$
$577$	12.5055	0.520610	0.260305	+	0.965526i	$0.416177\pi$
$578$	−33.9588	−1.41250
$579$	−0.364937	−0.0151663
$580$	29.6619	1.23164
$581$	−2.68500	−0.111393
$582$	−0.0985841	−0.00408644
$583$	22.0408	0.912838
$584$	29.1962	1.20815
$585$	−4.94304	−0.204370
$586$	69.3806	2.86608
$587$	−21.0961	−0.870729	−0.435364	−	0.900254i	$-0.643380\pi$
$587$	−21.0961	−0.870729	−0.435364	+	0.900254i	$0.643380\pi$
$588$	−0.809516	−0.0333839
$589$	57.2990	2.36096
$590$	−22.4122	−0.922694
$591$	0.205009	0.00843296
$592$	−6.48771	−0.266643
$593$	−0.879797	−0.0361289	−0.0180645	−	0.999837i	$-0.505750\pi$
$593$	−0.879797	−0.0361289	−0.0180645	+	0.999837i	$0.505750\pi$
$594$	−3.01039	−0.123518
$595$	−3.48572	−0.142901
$596$	17.1094	0.700827
$597$	−0.694226	−0.0284128
$598$	29.5374	1.20787
$599$	32.8279	1.34131	0.670655	−	0.741769i	$-0.266013\pi$
$599$	32.8279	1.34131	0.670655	+	0.741769i	$0.266013\pi$
$600$	−0.117218	−0.00478542
$601$	−24.8017	−1.01168	−0.505842	−	0.862626i	$-0.668818\pi$
$601$	−24.8017	−1.01168	−0.505842	+	0.862626i	$0.668818\pi$
$602$	−0.876045	−0.0357049
$603$	6.54557	0.266556
$604$	43.3376	1.76338
$605$	25.0392	1.01799
$606$	0.100764	0.00409324
$607$	−24.4589	−0.992757	−0.496379	−	0.868106i	$-0.665337\pi$
$607$	−24.4589	−0.992757	−0.496379	+	0.868106i	$0.665337\pi$
$608$	30.3287	1.22999
$609$	−0.194404	−0.00787764
$610$	−34.0041	−1.37678
$611$	−14.8676	−0.601477
$612$	−57.3605	−2.31866
$613$	−11.9455	−0.482476	−0.241238	−	0.970466i	$-0.577553\pi$
$613$	−11.9455	−0.482476	−0.241238	+	0.970466i	$0.577553\pi$
$614$	9.34438	0.377108
$615$	−0.156647	−0.00631662
$616$	−12.1302	−0.488740
$617$	15.9939	0.643892	0.321946	−	0.946758i	$-0.395663\pi$
$617$	15.9939	0.643892	0.321946	+	0.946758i	$0.395663\pi$
$618$	−0.0813906	−0.00327401
$619$	2.70406	0.108686	0.0543428	−	0.998522i	$-0.482694\pi$
$619$	2.70406	0.108686	0.0543428	+	0.998522i	$0.482694\pi$
$620$	30.3839	1.22025
$621$	1.66332	0.0667466
$622$	−10.7068	−0.429305
$623$	6.74008	0.270036
$624$	0.0456908	0.00182910
$625$	1.00000	0.0400000
$626$	−33.9342	−1.35628
$627$	−1.38530	−0.0553237
$628$	3.68090	0.146884
$629$	−47.3292	−1.88714
$630$	−4.32113	−0.172158
$631$	−45.2691	−1.80213	−0.901067	−	0.433680i	$-0.857215\pi$
$631$	−45.2691	−1.80213	−0.901067	+	0.433680i	$0.857215\pi$
$632$	33.9158	1.34910
$633$	−0.349931	−0.0139085
$634$	79.6163	3.16197
$635$	−17.8461	−0.708200
$636$	−0.449257	−0.0178142
$637$	−10.9051	−0.432077
$638$	121.649	4.81615
$639$	44.8195	1.77303
$640$	19.6649	0.777325
$641$	−9.21668	−0.364037	−0.182019	−	0.983295i	$-0.558263\pi$
$641$	−9.21668	−0.364037	−0.182019	+	0.983295i	$0.558263\pi$
$642$	1.41564	0.0558707
$643$	−45.1064	−1.77882	−0.889410	−	0.457110i	$-0.848885\pi$
$643$	−45.1064	−1.77882	−0.889410	+	0.457110i	$0.848885\pi$
$644$	16.2617	0.640801
$645$	−0.0218652	−0.000860943 0
$646$	−83.8434	−3.29877
$647$	−23.2750	−0.915035	−0.457517	−	0.889201i	$-0.651261\pi$
$647$	−23.2750	−0.915035	−0.457517	+	0.889201i	$0.651261\pi$
$648$	−29.2944	−1.15079
$649$	−57.8875	−2.27228
$650$	−3.83130	−0.150276
$651$	−0.199135	−0.00780474
$652$	11.2633	0.441105
$653$	13.8125	0.540526	0.270263	−	0.962787i	$-0.412889\pi$
$653$	13.8125	0.540526	0.270263	+	0.962787i	$0.412889\pi$
$654$	−0.773957	−0.0302641
$655$	−4.00413	−0.156454
$656$	−3.35667	−0.131056
$657$	26.8631	1.04803
$658$	−12.9970	−0.506675
$659$	31.8823	1.24196	0.620980	−	0.783827i	$-0.286735\pi$
$659$	31.8823	1.24196	0.620980	+	0.783827i	$0.286735\pi$
$660$	−0.734584	−0.0285937
$661$	40.1125	1.56020	0.780098	−	0.625657i	$-0.215169\pi$
$661$	40.1125	1.56020	0.780098	+	0.625657i	$0.215169\pi$
$662$	3.42049	0.132941
$663$	0.333325	0.0129453
$664$	−14.1147	−0.547757
$665$	−3.97780	−0.154253
$666$	−58.6724	−2.27351
$667$	−67.2144	−2.60255
$668$	−6.03965	−0.233681
$669$	−0.170095	−0.00657624
$670$	5.07341	0.196003
$671$	−87.8278	−3.39055
$672$	−0.105404	−0.00406603
$673$	31.2189	1.20340	0.601700	−	0.798722i	$-0.294490\pi$
$673$	31.2189	1.20340	0.601700	+	0.798722i	$0.294490\pi$
$674$	−13.1504	−0.506536
$675$	−0.215749	−0.00830418
$676$	−34.9844	−1.34555
$677$	−24.9982	−0.960757	−0.480379	−	0.877061i	$-0.659501\pi$
$677$	−24.9982	−0.960757	−0.480379	+	0.877061i	$0.659501\pi$
$678$	0.844470	0.0324317
$679$	−0.731150	−0.0280590
$680$	−18.3240	−0.702693
$681$	0.648982	0.0248690
$682$	124.610	4.77158
$683$	−35.9647	−1.37615	−0.688076	−	0.725639i	$-0.741544\pi$
$683$	−35.9647	−1.37615	−0.688076	+	0.725639i	$0.741544\pi$
$684$	−65.4581	−2.50285
$685$	−4.61943	−0.176500
$686$	−19.6201	−0.749098
$687$	0.0650700	0.00248257
$688$	−0.468534	−0.0178627
$689$	−6.05203	−0.230564
$690$	0.644471	0.0245346
$691$	−16.7979	−0.639021	−0.319511	−	0.947583i	$-0.603518\pi$
$691$	−16.7979	−0.639021	−0.319511	+	0.947583i	$0.603518\pi$
$692$	−6.46192	−0.245646
$693$	−11.1609	−0.423967
$694$	19.0045	0.721402
$695$	−15.2630	−0.578960
$696$	−1.02195	−0.0387371
$697$	−24.4876	−0.927536
$698$	20.4182	0.772840
$699$	0.220164	0.00832736
$700$	−2.10931	−0.0797244
$701$	−5.00330	−0.188972	−0.0944859	−	0.995526i	$-0.530121\pi$
$701$	−5.00330	−0.188972	−0.0944859	+	0.995526i	$0.530121\pi$
$702$	0.826600	0.0311980
$703$	−54.0107	−2.03705
$704$	75.2102	2.83459
$705$	−0.324392	−0.0122173
$706$	51.6955	1.94559
$707$	0.747314	0.0281056
$708$	1.17992	0.0443440
$709$	39.3952	1.47952	0.739759	−	0.672872i	$-0.234939\pi$
$709$	39.3952	1.47952	0.739759	+	0.672872i	$0.234939\pi$
$710$	34.7392	1.30374
$711$	31.2056	1.17030
$712$	35.4317	1.32786
$713$	−68.8504	−2.57847
$714$	0.291387	0.0109049
$715$	−9.89573	−0.370079
$716$	−82.9703	−3.10074
$717$	−0.513324	−0.0191704
$718$	73.2403	2.73330
$719$	−24.7660	−0.923616	−0.461808	−	0.886980i	$-0.652799\pi$
$719$	−24.7660	−0.923616	−0.461808	+	0.886980i	$0.652799\pi$
$720$	−2.31106	−0.0861282
$721$	−0.603634	−0.0224805
$722$	−51.5184	−1.91732
$723$	0.277119	0.0103062
$724$	6.58912	0.244883
$725$	8.71839	0.323793
$726$	−2.09314	−0.0776836
$727$	−34.7945	−1.29045	−0.645227	−	0.763991i	$-0.723237\pi$
$727$	−34.7945	−1.29045	−0.645227	+	0.763991i	$0.723237\pi$
$728$	3.33075	0.123446
$729$	−26.9302	−0.997414
$730$	20.8213	0.770632
$731$	−3.41806	−0.126421
$732$	1.79019	0.0661673
$733$	−19.0850	−0.704919	−0.352459	−	0.935827i	$-0.614655\pi$
$733$	−19.0850	−0.704919	−0.352459	+	0.935827i	$0.614655\pi$
$734$	13.5090	0.498627
$735$	−0.237937	−0.00877644
$736$	−36.4429	−1.34330
$737$	13.1039	0.482689
$738$	−30.3565	−1.11744
$739$	−17.2872	−0.635920	−0.317960	−	0.948104i	$-0.602998\pi$
$739$	−17.2872	−0.635920	−0.317960	+	0.948104i	$0.602998\pi$
$740$	−28.6402	−1.05284
$741$	0.380380	0.0139736
$742$	−5.29059	−0.194224
$743$	45.5726	1.67190	0.835948	−	0.548809i	$-0.184919\pi$
$743$	45.5726	1.67190	0.835948	+	0.548809i	$0.184919\pi$
$744$	−1.04683	−0.0383786
$745$	5.02888	0.184244
$746$	81.8434	2.99650
$747$	−12.9868	−0.475162
$748$	−114.833	−4.19871
$749$	10.4991	0.383628
$750$	−0.0835945	−0.00305244
$751$	0.342305	0.0124909	0.00624543	−	0.999980i	$-0.498012\pi$
$751$	0.342305	0.0124909	0.00624543	+	0.999980i	$0.498012\pi$
$752$	−6.95116	−0.253483
$753$	0.483016	0.0176021
$754$	−33.4028	−1.21646
$755$	12.7380	0.463584
$756$	0.455081	0.0165511
$757$	−6.30052	−0.228996	−0.114498	−	0.993423i	$-0.536526\pi$
$757$	−6.30052	−0.228996	−0.114498	+	0.993423i	$0.536526\pi$
$758$	−11.1659	−0.405564
$759$	1.66458	0.0604204
$760$	−20.9108	−0.758514
$761$	12.5766	0.455901	0.227951	−	0.973673i	$-0.426797\pi$
$761$	12.5766	0.455901	0.227951	+	0.973673i	$0.426797\pi$
$762$	1.49183	0.0540435
$763$	−5.74006	−0.207804
$764$	−7.66925	−0.277464
$765$	−16.8597	−0.609564
$766$	−77.4279	−2.79759
$767$	15.8949	0.573932
$768$	−0.742701	−0.0267999
$769$	−0.909203	−0.0327867	−0.0163934	−	0.999866i	$-0.505218\pi$
$769$	−0.909203	−0.0327867	−0.0163934	+	0.999866i	$0.505218\pi$
$770$	−8.65069	−0.311749
$771$	0.198629	0.00715344
$772$	−34.5215	−1.24246
$773$	52.5952	1.89172	0.945858	−	0.324580i	$-0.105223\pi$
$773$	52.5952	1.89172	0.945858	+	0.324580i	$0.105223\pi$
$774$	−4.23724	−0.152305
$775$	8.93060	0.320796
$776$	−3.84356	−0.137976
$777$	0.187708	0.00673397
$778$	−17.1273	−0.614043
$779$	−27.9446	−1.00122
$780$	0.201704	0.00722216
$781$	89.7265	3.21067
$782$	100.746	3.60267
$783$	−1.88098	−0.0672209
$784$	−5.09857	−0.182092
$785$	1.08191	0.0386150
$786$	0.334724	0.0119392
$787$	28.0859	1.00115	0.500577	−	0.865692i	$-0.333121\pi$
$787$	28.0859	1.00115	0.500577	+	0.865692i	$0.333121\pi$
$788$	19.3930	0.690848
$789$	−0.744252	−0.0264961
$790$	24.1872	0.860540
$791$	6.26301	0.222687
$792$	−58.6713	−2.08479
$793$	24.1160	0.856383
$794$	27.7210	0.983783
$795$	−0.132048	−0.00468326
$796$	−65.6709	−2.32764
$797$	7.46219	0.264324	0.132162	−	0.991228i	$-0.457808\pi$
$797$	7.46219	0.264324	0.132162	+	0.991228i	$0.457808\pi$
$798$	0.332522	0.0117712
$799$	−50.7102	−1.79400
$800$	4.72701	0.167125
$801$	32.6003	1.15188
$802$	−2.32427	−0.0820728
$803$	53.7786	1.89780
$804$	−0.267096	−0.00941976
$805$	4.77972	0.168463
$806$	−34.2158	−1.20520
$807$	−0.643396	−0.0226486
$808$	3.92853	0.138205
$809$	10.2533	0.360485	0.180243	−	0.983622i	$-0.442312\pi$
$809$	10.2533	0.360485	0.180243	+	0.983622i	$0.442312\pi$
$810$	−20.8914	−0.734048
$811$	23.5211	0.825937	0.412968	−	0.910745i	$-0.364492\pi$
$811$	23.5211	0.825937	0.412968	+	0.910745i	$0.364492\pi$
$812$	−18.3898	−0.645355
$813$	0.603583	0.0211686
$814$	−117.459	−4.11695
$815$	3.31057	0.115964
$816$	0.155842	0.00545557
$817$	−3.90058	−0.136464
$818$	1.19421	0.0417545
$819$	3.06458	0.107085
$820$	−14.8182	−0.517472
$821$	−9.01383	−0.314585	−0.157292	−	0.987552i	$-0.550277\pi$
$821$	−9.01383	−0.314585	−0.157292	+	0.987552i	$0.550277\pi$
$822$	0.386159	0.0134688
$823$	−35.2327	−1.22813	−0.614067	−	0.789254i	$-0.710468\pi$
$823$	−35.2327	−1.22813	−0.614067	+	0.789254i	$0.710468\pi$
$824$	−3.17322	−0.110544
$825$	−0.215913	−0.00751712
$826$	13.8951	0.483471
$827$	41.0642	1.42794	0.713971	−	0.700175i	$-0.246895\pi$
$827$	41.0642	1.42794	0.713971	+	0.700175i	$0.246895\pi$
$828$	78.6544	2.73343
$829$	−37.0633	−1.28726	−0.643631	−	0.765336i	$-0.722573\pi$
$829$	−37.0633	−1.28726	−0.643631	+	0.765336i	$0.722573\pi$
$830$	−10.0659	−0.349394
$831$	0.645437	0.0223900
$832$	−20.6514	−0.715959
$833$	−37.1952	−1.28874
$834$	1.27591	0.0441810
$835$	−1.77521	−0.0614335
$836$	−131.044	−4.53225
$837$	−1.92677	−0.0665988
$838$	−12.5183	−0.432436
$839$	−40.6205	−1.40237	−0.701187	−	0.712977i	$-0.747346\pi$
$839$	−40.6205	−1.40237	−0.701187	+	0.712977i	$0.747346\pi$
$840$	0.0726729	0.00250745
$841$	47.0103	1.62105
$842$	−24.3216	−0.838179
$843$	−0.112850	−0.00388676
$844$	−33.1020	−1.13942
$845$	−10.2828	−0.353739
$846$	−62.8637	−2.16130
$847$	−15.5238	−0.533403
$848$	−2.82956	−0.0971674
$849$	0.564213	0.0193637
$850$	−13.0678	−0.448222
$851$	64.8992	2.22472
$852$	−1.82889	−0.0626567
$853$	−15.6354	−0.535345	−0.267673	−	0.963510i	$-0.586255\pi$
$853$	−15.6354	−0.535345	−0.267673	+	0.963510i	$0.586255\pi$
$854$	21.0818	0.721405
$855$	−19.2398	−0.657987
$856$	55.1923	1.88643
$857$	40.3552	1.37851	0.689253	−	0.724520i	$-0.257939\pi$
$857$	40.3552	1.37851	0.689253	+	0.724520i	$0.257939\pi$
$858$	0.827228	0.0282411
$859$	6.23212	0.212637	0.106319	−	0.994332i	$-0.466094\pi$
$859$	6.23212	0.212637	0.106319	+	0.994332i	$0.466094\pi$
$860$	−2.06836	−0.0705305
$861$	0.0971179	0.00330977
$862$	13.1919	0.449316
$863$	−6.14294	−0.209108	−0.104554	−	0.994519i	$-0.533342\pi$
$863$	−6.14294	−0.209108	−0.104554	+	0.994519i	$0.533342\pi$
$864$	−1.01985	−0.0346960
$865$	−1.89932	−0.0645789
$866$	53.5378	1.81929
$867$	0.525482	0.0178463
$868$	−18.8374	−0.639382
$869$	62.4721	2.11922
$870$	−0.728809	−0.0247089
$871$	−3.59811	−0.121917
$872$	−30.1748	−1.02185
$873$	−3.53642	−0.119690
$874$	114.968	3.88887
$875$	−0.619979	−0.0209591
$876$	−1.09617	−0.0370360
$877$	12.5789	0.424760	0.212380	−	0.977187i	$-0.431879\pi$
$877$	12.5789	0.424760	0.212380	+	0.977187i	$0.431879\pi$
$878$	−53.1605	−1.79408
$879$	−1.07360	−0.0362116
$880$	−4.62663	−0.155964
$881$	14.2007	0.478433	0.239217	−	0.970966i	$-0.423109\pi$
$881$	14.2007	0.478433	0.239217	+	0.970966i	$0.423109\pi$
$882$	−46.1096	−1.55259
$883$	−42.3187	−1.42414	−0.712068	−	0.702110i	$-0.752241\pi$
$883$	−42.3187	−1.42414	−0.712068	+	0.702110i	$0.752241\pi$
$884$	31.5311	1.06051
$885$	0.346808	0.0116578
$886$	−5.06633	−0.170207
$887$	−20.4854	−0.687834	−0.343917	−	0.939000i	$-0.611754\pi$
$887$	−20.4854	−0.687834	−0.343917	+	0.939000i	$0.611754\pi$
$888$	0.986754	0.0331133
$889$	11.0642	0.371081
$890$	25.2682	0.846992
$891$	−53.9595	−1.80771
$892$	−16.0902	−0.538741
$893$	−57.8690	−1.93651
$894$	−0.420386	−0.0140598
$895$	−24.3871	−0.815170
$896$	−12.1918	−0.407301
$897$	−0.457064	−0.0152609
$898$	−38.3085	−1.27837
$899$	77.8604	2.59679
$900$	−10.2023	−0.340076
$901$	−20.6422	−0.687693
$902$	−60.7722	−2.02349
$903$	0.0135560	0.000451115 0
$904$	32.9238	1.09503
$905$	1.93671	0.0643784
$906$	−1.06483	−0.0353765
$907$	−44.0601	−1.46299	−0.731496	−	0.681846i	$-0.761177\pi$
$907$	−44.0601	−1.46299	−0.731496	+	0.681846i	$0.761177\pi$
$908$	61.3910	2.03733
$909$	3.61460	0.119889
$910$	2.37533	0.0787414
$911$	−11.0915	−0.367477	−0.183739	−	0.982975i	$-0.558820\pi$
$911$	−11.0915	−0.367477	−0.183739	+	0.982975i	$0.558820\pi$
$912$	0.177842	0.00588895
$913$	−25.9989	−0.860438
$914$	−38.1768	−1.26278
$915$	0.526182	0.0173950
$916$	6.15535	0.203378
$917$	2.48248	0.0819787
$918$	2.81936	0.0930529
$919$	−42.2615	−1.39408	−0.697039	−	0.717033i	$-0.745500\pi$
$919$	−42.2615	−1.39408	−0.697039	+	0.717033i	$0.745500\pi$
$920$	25.1264	0.828392
$921$	−0.144596	−0.00476459
$922$	−57.0601	−1.87917
$923$	−24.6373	−0.810948
$924$	0.455427	0.0149824
$925$	−8.41809	−0.276785
$926$	−12.6059	−0.414256
$927$	−2.91965	−0.0958938
$928$	41.2120	1.35285
$929$	−5.16525	−0.169466	−0.0847331	−	0.996404i	$-0.527004\pi$
$929$	−5.16525	−0.169466	−0.0847331	+	0.996404i	$0.527004\pi$
$930$	−0.746549	−0.0244803
$931$	−42.4461	−1.39111
$932$	20.8266	0.682197
$933$	0.165679	0.00542407
$934$	19.1381	0.626219
$935$	−33.7523	−1.10382
$936$	16.1101	0.526575
$937$	−2.66681	−0.0871209	−0.0435604	−	0.999051i	$-0.513870\pi$
$937$	−2.66681	−0.0871209	−0.0435604	+	0.999051i	$0.513870\pi$
$938$	−3.14541	−0.102701
$939$	0.525101	0.0171360
$940$	−30.6861	−1.00087
$941$	−20.6734	−0.673935	−0.336968	−	0.941516i	$-0.609401\pi$
$941$	−20.6734	−0.673935	−0.336968	+	0.941516i	$0.609401\pi$
$942$	−0.0904416	−0.00294675
$943$	33.5782	1.09346
$944$	7.43148	0.241874
$945$	0.133760	0.00435121
$946$	−8.48275	−0.275798
$947$	2.93723	0.0954473	0.0477237	−	0.998861i	$-0.484803\pi$
$947$	2.93723	0.0954473	0.0477237	+	0.998861i	$0.484803\pi$
$948$	−1.27336	−0.0413570
$949$	−14.7667	−0.479346
$950$	−14.9126	−0.483828
$951$	−1.23199	−0.0399500
$952$	11.3605	0.368196
$953$	−30.2636	−0.980335	−0.490168	−	0.871628i	$-0.663064\pi$
$953$	−30.2636	−0.980335	−0.490168	+	0.871628i	$0.663064\pi$
$954$	−25.5895	−0.828489
$955$	−2.25419	−0.0729438
$956$	−48.5583	−1.57049
$957$	−1.88241	−0.0608498
$958$	6.01562	0.194356
$959$	2.86395	0.0924818
$960$	−0.450589	−0.0145427
$961$	48.7556	1.57276
$962$	32.2523	1.03985
$963$	50.7818	1.63642
$964$	26.2143	0.844306
$965$	−10.1467	−0.326635
$966$	−0.399559	−0.0128556
$967$	−7.63368	−0.245483	−0.122741	−	0.992439i	$-0.539169\pi$
$967$	−7.63368	−0.245483	−0.122741	+	0.992439i	$0.539169\pi$
$968$	−81.6064	−2.62293
$969$	1.29740	0.0416785
$970$	−2.74104	−0.0880095
$971$	26.7518	0.858505	0.429252	−	0.903185i	$-0.358777\pi$
$971$	26.7518	0.858505	0.429252	+	0.903185i	$0.358777\pi$
$972$	3.30193	0.105910
$973$	9.46277	0.303362
$974$	61.8882	1.98302
$975$	0.0592859	0.00189867
$976$	11.2751	0.360909
$977$	32.9087	1.05284	0.526421	−	0.850224i	$-0.323534\pi$
$977$	32.9087	1.05284	0.526421	+	0.850224i	$0.323534\pi$
$978$	−0.276745	−0.00884933
$979$	65.2643	2.08586
$980$	−22.5079	−0.718987
$981$	−27.7635	−0.886419
$982$	26.7208	0.852696
$983$	37.5292	1.19700	0.598498	−	0.801124i	$-0.295765\pi$
$983$	37.5292	1.19700	0.598498	+	0.801124i	$0.295765\pi$
$984$	0.510536	0.0162753
$985$	5.70010	0.181620
$986$	−113.930	−3.62828
$987$	0.201116	0.00640161
$988$	35.9824	1.14475
$989$	4.68694	0.149036
$990$	−41.8415	−1.32981
$991$	58.7309	1.86565	0.932824	−	0.360333i	$-0.117337\pi$
$991$	58.7309	1.86565	0.932824	+	0.360333i	$0.117337\pi$
$992$	42.2151	1.34033
$993$	−0.0529290	−0.00167965
$994$	−21.5376	−0.683130
$995$	−19.3023	−0.611925
$996$	0.529934	0.0167916
$997$	−32.8606	−1.04071	−0.520353	−	0.853951i	$-0.674200\pi$
$997$	−32.8606	−1.04071	−0.520353	+	0.853951i	$0.674200\pi$
$998$	−64.1042	−2.02918
$999$	1.81619	0.0574619

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2005.2.a.d.1.4	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2005.2.a.d.1.4	✓	25	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:	\( N \)	\(=\)	\( 2005 = 5 \cdot 401 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2005.a (trivial)

\(f(q)\)	\(=\)	\(q-2.32427 q^{2} +0.0359659 q^{3} +3.40223 q^{4} +1.00000 q^{5} -0.0835945 q^{6} -0.619979 q^{7} -3.25915 q^{8} -2.99871 q^{9} +O(q^{10})\)	\(q-2.32427 q^{2} +0.0359659 q^{3} +3.40223 q^{4} +1.00000 q^{5} -0.0835945 q^{6} -0.619979 q^{7} -3.25915 q^{8} -2.99871 q^{9} -2.32427 q^{10} -6.00326 q^{11} +0.122364 q^{12} +1.64839 q^{13} +1.44100 q^{14} +0.0359659 q^{15} +0.770686 q^{16} +5.62233 q^{17} +6.96980 q^{18} +6.41603 q^{19} +3.40223 q^{20} -0.0222981 q^{21} +13.9532 q^{22} -7.70949 q^{23} -0.117218 q^{24} +1.00000 q^{25} -3.83130 q^{26} -0.215749 q^{27} -2.10931 q^{28} +8.71839 q^{29} -0.0835945 q^{30} +8.93060 q^{31} +4.72701 q^{32} -0.215913 q^{33} -13.0678 q^{34} -0.619979 q^{35} -10.2023 q^{36} -8.41809 q^{37} -14.9126 q^{38} +0.0592859 q^{39} -3.25915 q^{40} -4.35543 q^{41} +0.0518268 q^{42} -0.607943 q^{43} -20.4245 q^{44} -2.99871 q^{45} +17.9189 q^{46} -9.01943 q^{47} +0.0277184 q^{48} -6.61563 q^{49} -2.32427 q^{50} +0.202212 q^{51} +5.60820 q^{52} -3.67148 q^{53} +0.501459 q^{54} -6.00326 q^{55} +2.02060 q^{56} +0.230758 q^{57} -20.2639 q^{58} +9.64268 q^{59} +0.122364 q^{60} +14.6300 q^{61} -20.7571 q^{62} +1.85914 q^{63} -12.5282 q^{64} +1.64839 q^{65} +0.501840 q^{66} -2.18280 q^{67} +19.1284 q^{68} -0.277279 q^{69} +1.44100 q^{70} -14.9463 q^{71} +9.77323 q^{72} -8.95822 q^{73} +19.5659 q^{74} +0.0359659 q^{75} +21.8288 q^{76} +3.72190 q^{77} -0.137796 q^{78} -10.4064 q^{79} +0.770686 q^{80} +8.98836 q^{81} +10.1232 q^{82} +4.33080 q^{83} -0.0758632 q^{84} +5.62233 q^{85} +1.41302 q^{86} +0.313565 q^{87} +19.5655 q^{88} -10.8715 q^{89} +6.96980 q^{90} -1.02197 q^{91} -26.2294 q^{92} +0.321197 q^{93} +20.9636 q^{94} +6.41603 q^{95} +0.170011 q^{96} +1.17931 q^{97} +15.3765 q^{98} +18.0020 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q - 5 q^{2} - 10 q^{3} + 25 q^{4} + 25 q^{5} + 2 q^{6} - 31 q^{7} - 30 q^{8} + 17 q^{9}+O(q^{10}) \) 25 * q - 5 * q^2 - 10 * q^3 + 25 * q^4 + 25 * q^5 + 2 * q^6 - 31 * q^7 - 30 * q^8 + 17 * q^9	\( 25 q - 5 q^{2} - 10 q^{3} + 25 q^{4} + 25 q^{5} + 2 q^{6} - 31 q^{7} - 30 q^{8} + 17 q^{9} - 5 q^{10} - 30 q^{11} - 29 q^{12} - 18 q^{13} + 6 q^{14} - 10 q^{15} + 21 q^{16} - 18 q^{17} - 30 q^{18} - 17 q^{19} + 25 q^{20} + 6 q^{21} - 2 q^{22} - 44 q^{23} + 11 q^{24} + 25 q^{25} - 14 q^{26} - 25 q^{27} - 50 q^{28} - 9 q^{29} + 2 q^{30} - 13 q^{31} - 45 q^{32} - 21 q^{33} - 21 q^{34} - 31 q^{35} + 5 q^{36} - 28 q^{37} - 32 q^{38} + 9 q^{39} - 30 q^{40} + 28 q^{41} - 67 q^{42} - 61 q^{43} - 49 q^{44} + 17 q^{45} + 18 q^{46} - 53 q^{47} - 44 q^{48} + 28 q^{49} - 5 q^{50} - 30 q^{51} - 3 q^{52} - 36 q^{53} + 17 q^{54} - 30 q^{55} - 3 q^{56} - 13 q^{57} + 2 q^{58} - 39 q^{59} - 29 q^{60} + 10 q^{61} - 30 q^{62} - 44 q^{63} - 4 q^{64} - 18 q^{65} + 33 q^{66} - 10 q^{67} - 18 q^{68} - 6 q^{69} + 6 q^{70} - 7 q^{71} - q^{72} - 26 q^{73} - 3 q^{74} - 10 q^{75} + 12 q^{76} + 29 q^{77} - 5 q^{78} - 6 q^{79} + 21 q^{80} + 13 q^{81} - 30 q^{82} - 35 q^{83} + 117 q^{84} - 18 q^{85} + 14 q^{86} - 104 q^{87} + 53 q^{88} + 7 q^{89} - 30 q^{90} - 25 q^{91} - 31 q^{92} + 2 q^{93} + 68 q^{94} - 17 q^{95} + 92 q^{96} + 6 q^{97} + 15 q^{98} - 36 q^{99}+O(q^{100}) \) 25 * q - 5 * q^2 - 10 * q^3 + 25 * q^4 + 25 * q^5 + 2 * q^6 - 31 * q^7 - 30 * q^8 + 17 * q^9 - 5 * q^10 - 30 * q^11 - 29 * q^12 - 18 * q^13 + 6 * q^14 - 10 * q^15 + 21 * q^16 - 18 * q^17 - 30 * q^18 - 17 * q^19 + 25 * q^20 + 6 * q^21 - 2 * q^22 - 44 * q^23 + 11 * q^24 + 25 * q^25 - 14 * q^26 - 25 * q^27 - 50 * q^28 - 9 * q^29 + 2 * q^30 - 13 * q^31 - 45 * q^32 - 21 * q^33 - 21 * q^34 - 31 * q^35 + 5 * q^36 - 28 * q^37 - 32 * q^38 + 9 * q^39 - 30 * q^40 + 28 * q^41 - 67 * q^42 - 61 * q^43 - 49 * q^44 + 17 * q^45 + 18 * q^46 - 53 * q^47 - 44 * q^48 + 28 * q^49 - 5 * q^50 - 30 * q^51 - 3 * q^52 - 36 * q^53 + 17 * q^54 - 30 * q^55 - 3 * q^56 - 13 * q^57 + 2 * q^58 - 39 * q^59 - 29 * q^60 + 10 * q^61 - 30 * q^62 - 44 * q^63 - 4 * q^64 - 18 * q^65 + 33 * q^66 - 10 * q^67 - 18 * q^68 - 6 * q^69 + 6 * q^70 - 7 * q^71 - q^72 - 26 * q^73 - 3 * q^74 - 10 * q^75 + 12 * q^76 + 29 * q^77 - 5 * q^78 - 6 * q^79 + 21 * q^80 + 13 * q^81 - 30 * q^82 - 35 * q^83 + 117 * q^84 - 18 * q^85 + 14 * q^86 - 104 * q^87 + 53 * q^88 + 7 * q^89 - 30 * q^90 - 25 * q^91 - 31 * q^92 + 2 * q^93 + 68 * q^94 - 17 * q^95 + 92 * q^96 + 6 * q^97 + 15 * q^98 - 36 * q^99

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