LMFDB - Embedded newform 800.2.o.g.207.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [800,2,Mod(143,800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(800, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 2, 3]))

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

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

chi := DirichletCharacter("800.143");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 800 = 2^{5} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	800.o (of order $4$, degree $2$, not minimal)

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:	no
Analytic conductor:	$6.38803216170$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{20})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{6} + x^{4} - x^{2} + 1 $ x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			207.2
Root			$0.951057 + 0.309017i$ of defining polynomial
Character	$\chi$	$=$	800.207
Dual form			800.2.o.g.143.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.61803 + 1.61803i) q^{3} +(1.17557 - 1.17557i) q^{7} -2.23607i q^{9} +O(q^{10})$	$q+(-1.61803 + 1.61803i) q^{3} +(1.17557 - 1.17557i) q^{7} -2.23607i q^{9} -1.23607 q^{11} +(3.07768 + 3.07768i) q^{13} +(1.00000 + 1.00000i) q^{17} +2.00000i q^{19} +3.80423i q^{21} +(2.62866 + 2.62866i) q^{23} +(-1.23607 - 1.23607i) q^{27} -1.45309 q^{29} +5.25731i q^{31} +(2.00000 - 2.00000i) q^{33} +(-3.07768 + 3.07768i) q^{37} -9.95959 q^{39} -7.70820 q^{41} +(2.38197 - 2.38197i) q^{43} +(-7.33094 + 7.33094i) q^{47} +4.23607i q^{49} -3.23607 q^{51} +(0.726543 + 0.726543i) q^{53} +(-3.23607 - 3.23607i) q^{57} -8.47214i q^{59} +9.95959i q^{61} +(-2.62866 - 2.62866i) q^{63} +(-2.38197 - 2.38197i) q^{67} -8.50651 q^{69} +7.05342i q^{71} +(-8.70820 + 8.70820i) q^{73} +(-1.45309 + 1.45309i) q^{77} +12.3107 q^{79} +10.7082 q^{81} +(-4.38197 + 4.38197i) q^{83} +(2.35114 - 2.35114i) q^{87} -6.47214i q^{89} +7.23607 q^{91} +(-8.50651 - 8.50651i) q^{93} +(0.236068 + 0.236068i) q^{97} +2.76393i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{3}+O(q^{10}) $ 8 * q - 4 * q^3	$ 8 q - 4 q^{3} + 8 q^{11} + 8 q^{17} + 8 q^{27} + 16 q^{33} - 8 q^{41} + 28 q^{43} - 8 q^{51} - 8 q^{57} - 28 q^{67} - 16 q^{73} + 32 q^{81} - 44 q^{83} + 40 q^{91} - 16 q^{97}+O(q^{100}) $ 8 * q - 4 * q^3 + 8 * q^11 + 8 * q^17 + 8 * q^27 + 16 * q^33 - 8 * q^41 + 28 * q^43 - 8 * q^51 - 8 * q^57 - 28 * q^67 - 16 * q^73 + 32 * q^81 - 44 * q^83 + 40 * q^91 - 16 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/800\mathbb{Z}\right)^\times$.

$n$	$101$	$351$	$577$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{1}{4}\right)$

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 $

$2$		0			0
$2$		0			0
$3$	−1.61803	+	1.61803i	−0.934172	+	0.934172i	−0.997963	−	0.0637909i	$-0.979681\pi$
$3$	−1.61803	+	1.61803i	−0.934172	+	0.934172i	0.0637909	+	0.997963i	$0.479681\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	1.17557	−	1.17557i	0.444324	−	0.444324i	−0.449138	−	0.893462i	$-0.648269\pi$
$7$	1.17557	−	1.17557i	0.444324	−	0.444324i	0.893462	+	0.449138i	$0.148269\pi$
$8$		0			0
$9$		−	2.23607i		−	0.745356i
$10$		0			0
$11$	−1.23607			−0.372689			−0.186344	−	0.982485i	$-0.559664\pi$
$11$	−1.23607			−0.372689			−0.186344	+	0.982485i	$0.559664\pi$
$12$		0			0
$13$	3.07768	+	3.07768i	0.853596	+	0.853596i	0.990574	−	0.136978i	$-0.0437390\pi$
$13$	3.07768	+	3.07768i	0.853596	+	0.853596i	−0.136978	+	0.990574i	$0.543739\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.00000	+	1.00000i	0.242536	+	0.242536i	0.817898	−	0.575363i	$-0.195139\pi$
$17$	1.00000	+	1.00000i	0.242536	+	0.242536i	−0.575363	+	0.817898i	$0.695139\pi$
$18$		0			0
$19$			2.00000i			0.458831i	0.973329	+	0.229416i	$0.0736815\pi$
$19$			2.00000i			0.458831i	−0.973329	+	0.229416i	$0.926318\pi$
$20$		0			0
$21$			3.80423i			0.830150i
$22$		0			0
$23$	2.62866	+	2.62866i	0.548113	+	0.548113i	0.925895	−	0.377782i	$-0.123313\pi$
$23$	2.62866	+	2.62866i	0.548113	+	0.548113i	−0.377782	+	0.925895i	$0.623313\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−1.23607	−	1.23607i	−0.237881	−	0.237881i
$28$		0			0
$29$	−1.45309			−0.269831			−0.134916	−	0.990857i	$-0.543076\pi$
$29$	−1.45309			−0.269831			−0.134916	+	0.990857i	$0.543076\pi$
$30$		0			0
$31$			5.25731i			0.944241i	0.881534	+	0.472120i	$0.156511\pi$
$31$			5.25731i			0.944241i	−0.881534	+	0.472120i	$0.843489\pi$
$32$		0			0
$33$	2.00000	−	2.00000i	0.348155	−	0.348155i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−3.07768	+	3.07768i	−0.505968	+	0.505968i	−0.913286	−	0.407318i	$-0.866464\pi$
$37$	−3.07768	+	3.07768i	−0.505968	+	0.505968i	0.407318	+	0.913286i	$0.366464\pi$
$38$		0			0
$39$	−9.95959			−1.59481
$40$		0			0
$41$	−7.70820			−1.20382			−0.601910	−	0.798564i	$-0.705593\pi$
$41$	−7.70820			−1.20382			−0.601910	+	0.798564i	$0.705593\pi$
$42$		0			0
$43$	2.38197	−	2.38197i	0.363246	−	0.363246i	−0.501760	−	0.865007i	$-0.667314\pi$
$43$	2.38197	−	2.38197i	0.363246	−	0.363246i	0.865007	+	0.501760i	$0.167314\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−7.33094	+	7.33094i	−1.06933	+	1.06933i	−0.0719165	+	0.997411i	$0.522912\pi$
$47$	−7.33094	+	7.33094i	−1.06933	+	1.06933i	−0.997411	+	0.0719165i	$0.977088\pi$
$48$		0			0
$49$			4.23607i			0.605153i
$50$		0			0
$51$	−3.23607			−0.453140
$52$		0			0
$53$	0.726543	+	0.726543i	0.0997983	+	0.0997983i	0.755243	−	0.655445i	$-0.227519\pi$
$53$	0.726543	+	0.726543i	0.0997983	+	0.0997983i	−0.655445	+	0.755243i	$0.727519\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−3.23607	−	3.23607i	−0.428628	−	0.428628i
$58$		0			0
$59$		−	8.47214i		−	1.10298i	−0.834182	−	0.551489i	$-0.814060\pi$
$59$		−	8.47214i		−	1.10298i	0.834182	−	0.551489i	$-0.185940\pi$
$60$		0			0
$61$			9.95959i			1.27520i	0.770370	+	0.637598i	$0.220072\pi$
$61$			9.95959i			1.27520i	−0.770370	+	0.637598i	$0.779928\pi$
$62$		0			0
$63$	−2.62866	−	2.62866i	−0.331179	−	0.331179i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−2.38197	−	2.38197i	−0.291003	−	0.291003i	0.546473	−	0.837477i	$-0.315970\pi$
$67$	−2.38197	−	2.38197i	−0.291003	−	0.291003i	−0.837477	+	0.546473i	$0.815970\pi$
$68$		0			0
$69$	−8.50651			−1.02406
$70$		0			0
$71$			7.05342i			0.837087i	0.908197	+	0.418544i	$0.137459\pi$
$71$			7.05342i			0.837087i	−0.908197	+	0.418544i	$0.862541\pi$
$72$		0			0
$73$	−8.70820	+	8.70820i	−1.01922	+	1.01922i	−0.0194065	+	0.999812i	$0.506178\pi$
$73$	−8.70820	+	8.70820i	−1.01922	+	1.01922i	−0.999812	+	0.0194065i	$0.993822\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−1.45309	+	1.45309i	−0.165594	+	0.165594i
$78$		0			0
$79$	12.3107			1.38507			0.692533	−	0.721386i	$-0.256495\pi$
$79$	12.3107			1.38507			0.692533	+	0.721386i	$0.256495\pi$
$80$		0			0
$81$	10.7082			1.18980
$82$		0			0
$83$	−4.38197	+	4.38197i	−0.480983	+	0.480983i	−0.905446	−	0.424462i	$-0.860463\pi$
$83$	−4.38197	+	4.38197i	−0.480983	+	0.480983i	0.424462	+	0.905446i	$0.360463\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	2.35114	−	2.35114i	0.252069	−	0.252069i
$88$		0			0
$89$		−	6.47214i		−	0.686045i	−0.939327	−	0.343023i	$-0.888549\pi$
$89$		−	6.47214i		−	0.686045i	0.939327	−	0.343023i	$-0.111451\pi$
$90$		0			0
$91$	7.23607			0.758546
$92$		0			0
$93$	−8.50651	−	8.50651i	−0.882084	−	0.882084i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	0.236068	+	0.236068i	0.0239691	+	0.0239691i	0.718990	−	0.695021i	$-0.244605\pi$
$97$	0.236068	+	0.236068i	0.0239691	+	0.0239691i	−0.695021	+	0.718990i	$0.744605\pi$
$98$		0			0
$99$			2.76393i			0.277786i
$100$		0			0
$101$		−	12.3107i		−	1.22496i	−0.790485	−	0.612482i	$-0.790171\pi$
$101$		−	12.3107i		−	1.22496i	0.790485	−	0.612482i	$-0.209829\pi$
$102$		0			0
$103$	7.33094	+	7.33094i	0.722339	+	0.722339i	0.969081	−	0.246742i	$-0.0793601\pi$
$103$	7.33094	+	7.33094i	0.722339	+	0.722339i	−0.246742	+	0.969081i	$0.579360\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	12.0902	+	12.0902i	1.16880	+	1.16880i	0.982491	+	0.186310i	$0.0596528\pi$
$107$	12.0902	+	12.0902i	1.16880	+	1.16880i	0.186310	+	0.982491i	$0.440347\pi$
$108$		0			0
$109$	6.71040			0.642739			0.321370	−	0.946954i	$-0.395857\pi$
$109$	6.71040			0.642739			0.321370	+	0.946954i	$0.395857\pi$
$110$		0			0
$111$		−	9.95959i		−	0.945323i
$112$		0			0
$113$	4.70820	−	4.70820i	0.442911	−	0.442911i	−0.450078	−	0.892989i	$-0.648604\pi$
$113$	4.70820	−	4.70820i	0.442911	−	0.442911i	0.892989	+	0.450078i	$0.148604\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	6.88191	−	6.88191i	0.636233	−	0.636233i
$118$		0			0
$119$	2.35114			0.215529
$120$		0			0
$121$	−9.47214			−0.861103
$122$		0			0
$123$	12.4721	−	12.4721i	1.12457	−	1.12457i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	8.78402	−	8.78402i	0.779456	−	0.779456i	−0.200282	−	0.979738i	$-0.564186\pi$
$127$	8.78402	−	8.78402i	0.779456	−	0.779456i	0.979738	+	0.200282i	$0.0641859\pi$
$128$		0			0
$129$			7.70820i			0.678670i
$130$		0			0
$131$	−0.291796			−0.0254943			−0.0127472	−	0.999919i	$-0.504058\pi$
$131$	−0.291796			−0.0254943			−0.0127472	+	0.999919i	$0.504058\pi$
$132$		0			0
$133$	2.35114	+	2.35114i	0.203870	+	0.203870i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−3.47214	−	3.47214i	−0.296645	−	0.296645i	0.543054	−	0.839698i	$-0.317268\pi$
$137$	−3.47214	−	3.47214i	−0.296645	−	0.296645i	−0.839698	+	0.543054i	$0.817268\pi$
$138$		0			0
$139$			5.41641i			0.459414i	0.973260	+	0.229707i	$0.0737767\pi$
$139$			5.41641i			0.459414i	−0.973260	+	0.229707i	$0.926223\pi$
$140$		0			0
$141$		−	23.7234i		−	1.99787i
$142$		0			0
$143$	−3.80423	−	3.80423i	−0.318125	−	0.318125i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−6.85410	−	6.85410i	−0.565317	−	0.565317i
$148$		0			0
$149$	13.2088			1.08211			0.541053	−	0.840988i	$-0.318026\pi$
$149$	13.2088			1.08211			0.541053	+	0.840988i	$0.318026\pi$
$150$		0			0
$151$		−	14.6619i		−	1.19317i	−0.802551	−	0.596583i	$-0.796525\pi$
$151$		−	14.6619i		−	1.19317i	0.802551	−	0.596583i	$-0.203475\pi$
$152$		0			0
$153$	2.23607	−	2.23607i	0.180775	−	0.180775i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	9.78808	−	9.78808i	0.781174	−	0.781174i	−0.198855	−	0.980029i	$-0.563722\pi$
$157$	9.78808	−	9.78808i	0.781174	−	0.781174i	0.980029	+	0.198855i	$0.0637223\pi$
$158$		0			0
$159$	−2.35114			−0.186458
$160$		0			0
$161$	6.18034			0.487079
$162$		0			0
$163$	−7.14590	+	7.14590i	−0.559710	+	0.559710i	−0.929225	−	0.369515i	$-0.879524\pi$
$163$	−7.14590	+	7.14590i	−0.559710	+	0.559710i	0.369515	+	0.929225i	$0.379524\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	0.277515	−	0.277515i	0.0214747	−	0.0214747i	−0.696288	−	0.717763i	$-0.745166\pi$
$167$	0.277515	−	0.277515i	0.0214747	−	0.0214747i	0.717763	+	0.696288i	$0.245166\pi$
$168$		0			0
$169$			5.94427i			0.457252i
$170$		0			0
$171$	4.47214			0.341993
$172$		0			0
$173$	6.32688	+	6.32688i	0.481024	+	0.481024i	0.905459	−	0.424435i	$-0.139527\pi$
$173$	6.32688	+	6.32688i	0.481024	+	0.481024i	−0.424435	+	0.905459i	$0.639527\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	13.7082	+	13.7082i	1.03037	+	1.03037i
$178$		0			0
$179$			16.4721i			1.23119i	0.788065	+	0.615593i	$0.211083\pi$
$179$			16.4721i			1.23119i	−0.788065	+	0.615593i	$0.788917\pi$
$180$		0			0
$181$			9.40456i			0.699036i	0.936930	+	0.349518i	$0.113655\pi$
$181$			9.40456i			0.699036i	−0.936930	+	0.349518i	$0.886345\pi$
$182$		0			0
$183$	−16.1150	−	16.1150i	−1.19125	−	1.19125i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−1.23607	−	1.23607i	−0.0903902	−	0.0903902i
$188$		0			0
$189$	−2.90617			−0.211393
$190$		0			0
$191$		−	12.8658i		−	0.930934i	−0.885065	−	0.465467i	$-0.845886\pi$
$191$		−	12.8658i		−	0.930934i	0.885065	−	0.465467i	$-0.154114\pi$
$192$		0			0
$193$	7.47214	−	7.47214i	0.537856	−	0.537856i	−0.385043	−	0.922899i	$-0.625813\pi$
$193$	7.47214	−	7.47214i	0.537856	−	0.537856i	0.922899	+	0.385043i	$0.125813\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−2.17963	+	2.17963i	−0.155292	+	0.155292i	−0.780477	−	0.625185i	$-0.785024\pi$
$197$	−2.17963	+	2.17963i	−0.155292	+	0.155292i	0.625185	+	0.780477i	$0.285024\pi$
$198$		0			0
$199$	18.1231			1.28471			0.642355	−	0.766407i	$-0.277957\pi$
$199$	18.1231			1.28471			0.642355	+	0.766407i	$0.277957\pi$
$200$		0			0
$201$	7.70820			0.543695
$202$		0			0
$203$	−1.70820	+	1.70820i	−0.119892	+	0.119892i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	5.87785	−	5.87785i	0.408539	−	0.408539i
$208$		0			0
$209$		−	2.47214i		−	0.171001i
$210$		0			0
$211$	−15.7082			−1.08140			−0.540699	−	0.841216i	$-0.681840\pi$
$211$	−15.7082			−1.08140			−0.540699	+	0.841216i	$0.681840\pi$
$212$		0			0
$213$	−11.4127	−	11.4127i	−0.781984	−	0.781984i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	6.18034	+	6.18034i	0.419549	+	0.419549i
$218$		0			0
$219$		−	28.1803i		−	1.90425i
$220$		0			0
$221$			6.15537i			0.414055i
$222$		0			0
$223$	1.73060	+	1.73060i	0.115890	+	0.115890i	0.762673	−	0.646784i	$-0.223886\pi$
$223$	1.73060	+	1.73060i	0.115890	+	0.115890i	−0.646784	+	0.762673i	$0.723886\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−11.6180	−	11.6180i	−0.771116	−	0.771116i	0.207186	−	0.978302i	$-0.433570\pi$
$227$	−11.6180	−	11.6180i	−0.771116	−	0.771116i	−0.978302	+	0.207186i	$0.933570\pi$
$228$		0			0
$229$	21.3723			1.41232			0.706160	−	0.708053i	$-0.250426\pi$
$229$	21.3723			1.41232			0.706160	+	0.708053i	$0.250426\pi$
$230$		0			0
$231$		−	4.70228i		−	0.309387i
$232$		0			0
$233$	3.47214	−	3.47214i	0.227467	−	0.227467i	−0.584167	−	0.811634i	$-0.698579\pi$
$233$	3.47214	−	3.47214i	0.227467	−	0.227467i	0.811634	+	0.584167i	$0.198579\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−19.9192	+	19.9192i	−1.29389	+	1.29389i
$238$		0			0
$239$	−29.3238			−1.89680			−0.948398	−	0.317083i	$-0.897297\pi$
$239$	−29.3238			−1.89680			−0.948398	+	0.317083i	$0.897297\pi$
$240$		0			0
$241$	6.76393			0.435703			0.217852	−	0.975982i	$-0.430095\pi$
$241$	6.76393			0.435703			0.217852	+	0.975982i	$0.430095\pi$
$242$		0			0
$243$	−13.6180	+	13.6180i	−0.873597	+	0.873597i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−6.15537	+	6.15537i	−0.391657	+	0.391657i
$248$		0			0
$249$		−	14.1803i		−	0.898643i
$250$		0			0
$251$	22.1803			1.40001			0.700005	−	0.714138i	$-0.253181\pi$
$251$	22.1803			1.40001			0.700005	+	0.714138i	$0.253181\pi$
$252$		0			0
$253$	−3.24920	−	3.24920i	−0.204275	−	0.204275i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−18.7082	−	18.7082i	−1.16699	−	1.16699i	−0.982913	−	0.184073i	$-0.941072\pi$
$257$	−18.7082	−	18.7082i	−1.16699	−	1.16699i	−0.184073	−	0.982913i	$-0.558928\pi$
$258$		0			0
$259$			7.23607i			0.449627i
$260$		0			0
$261$			3.24920i			0.201120i
$262$		0			0
$263$	−16.3925	−	16.3925i	−1.01080	−	1.01080i	−0.999941	−	0.0108623i	$-0.996542\pi$
$263$	−16.3925	−	16.3925i	−1.01080	−	1.01080i	−0.0108623	−	0.999941i	$-0.503458\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	10.4721	+	10.4721i	0.640884	+	0.640884i
$268$		0			0
$269$	−17.9111			−1.09206			−0.546029	−	0.837766i	$-0.683861\pi$
$269$	−17.9111			−1.09206			−0.546029	+	0.837766i	$0.683861\pi$
$270$		0			0
$271$			31.6749i			1.92411i	0.272851	+	0.962056i	$0.412033\pi$
$271$			31.6749i			1.92411i	−0.272851	+	0.962056i	$0.587967\pi$
$272$		0			0
$273$	−11.7082	+	11.7082i	−0.708613	+	0.708613i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	2.17963	−	2.17963i	0.130961	−	0.130961i	−0.638588	−	0.769549i	$-0.720481\pi$
$277$	2.17963	−	2.17963i	0.130961	−	0.130961i	0.769549	+	0.638588i	$0.220481\pi$
$278$		0			0
$279$	11.7557			0.703796
$280$		0			0
$281$	3.70820			0.221213			0.110606	−	0.993864i	$-0.464721\pi$
$281$	3.70820			0.221213			0.110606	+	0.993864i	$0.464721\pi$
$282$		0			0
$283$	15.6180	−	15.6180i	0.928396	−	0.928396i	−0.0692066	−	0.997602i	$-0.522047\pi$
$283$	15.6180	−	15.6180i	0.928396	−	0.928396i	0.997602	+	0.0692066i	$0.0220468\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−9.06154	+	9.06154i	−0.534886	+	0.534886i
$288$		0			0
$289$		−	15.0000i		−	0.882353i
$290$		0			0
$291$	−0.763932			−0.0447825
$292$		0			0
$293$	−0.726543	−	0.726543i	−0.0424451	−	0.0424451i	0.685566	−	0.728011i	$-0.259555\pi$
$293$	−0.726543	−	0.726543i	−0.0424451	−	0.0424451i	−0.728011	+	0.685566i	$0.759555\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	1.52786	+	1.52786i	0.0886557	+	0.0886557i
$298$		0			0
$299$			16.1803i			0.935733i
$300$		0			0
$301$		−	5.60034i		−	0.322798i
$302$		0			0
$303$	19.9192	+	19.9192i	1.14433	+	1.14433i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	6.56231	+	6.56231i	0.374531	+	0.374531i	0.869124	−	0.494594i	$-0.164683\pi$
$307$	6.56231	+	6.56231i	0.374531	+	0.374531i	−0.494594	+	0.869124i	$0.664683\pi$
$308$		0			0
$309$	−23.7234			−1.34958
$310$		0			0
$311$			8.16348i			0.462909i	0.972846	+	0.231454i	$0.0743484\pi$
$311$			8.16348i			0.462909i	−0.972846	+	0.231454i	$0.925652\pi$
$312$		0			0
$313$	6.23607	−	6.23607i	0.352483	−	0.352483i	−0.508549	−	0.861033i	$-0.669818\pi$
$313$	6.23607	−	6.23607i	0.352483	−	0.352483i	0.861033	+	0.508549i	$0.169818\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	10.6861	−	10.6861i	0.600193	−	0.600193i	−0.340171	−	0.940364i	$-0.610485\pi$
$317$	10.6861	−	10.6861i	0.600193	−	0.600193i	0.940364	+	0.340171i	$0.110485\pi$
$318$		0			0
$319$	1.79611			0.100563
$320$		0			0
$321$	−39.1246			−2.18372
$322$		0			0
$323$	−2.00000	+	2.00000i	−0.111283	+	0.111283i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−10.8576	+	10.8576i	−0.600429	+	0.600429i
$328$		0			0
$329$			17.2361i			0.950255i
$330$		0			0
$331$	−28.0689			−1.54281			−0.771403	−	0.636347i	$-0.780445\pi$
$331$	−28.0689			−1.54281			−0.771403	+	0.636347i	$0.780445\pi$
$332$		0			0
$333$	6.88191	+	6.88191i	0.377126	+	0.377126i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	2.05573	+	2.05573i	0.111983	+	0.111983i	0.760878	−	0.648895i	$-0.224769\pi$
$337$	2.05573	+	2.05573i	0.111983	+	0.111983i	−0.648895	+	0.760878i	$0.724769\pi$
$338$		0			0
$339$			15.2361i			0.827510i
$340$		0			0
$341$		−	6.49839i		−	0.351908i
$342$		0			0
$343$	13.2088	+	13.2088i	0.713208	+	0.713208i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−3.03444	−	3.03444i	−0.162897	−	0.162897i	0.620952	−	0.783849i	$-0.286746\pi$
$347$	−3.03444	−	3.03444i	−0.162897	−	0.162897i	−0.783849	+	0.620952i	$0.786746\pi$
$348$		0			0
$349$	15.5599			0.832904			0.416452	−	0.909158i	$-0.363273\pi$
$349$	15.5599			0.832904			0.416452	+	0.909158i	$0.363273\pi$
$350$		0			0
$351$		−	7.60845i		−	0.406109i
$352$		0			0
$353$	−4.41641	+	4.41641i	−0.235062	+	0.235062i	−0.814802	−	0.579740i	$-0.803154\pi$
$353$	−4.41641	+	4.41641i	−0.235062	+	0.235062i	0.579740	+	0.814802i	$0.303154\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−3.80423	+	3.80423i	−0.201341	+	0.201341i
$358$		0			0
$359$	1.79611			0.0947952			0.0473976	−	0.998876i	$-0.484907\pi$
$359$	1.79611			0.0947952			0.0473976	+	0.998876i	$0.484907\pi$
$360$		0			0
$361$	15.0000			0.789474
$362$		0			0
$363$	15.3262	−	15.3262i	0.804419	−	0.804419i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	24.0009	−	24.0009i	1.25284	−	1.25284i	0.298396	−	0.954442i	$-0.403548\pi$
$367$	24.0009	−	24.0009i	1.25284	−	1.25284i	0.954442	−	0.298396i	$-0.0964516\pi$
$368$		0			0
$369$			17.2361i			0.897274i
$370$		0			0
$371$	1.70820			0.0886855
$372$		0			0
$373$	15.0454	+	15.0454i	0.779021	+	0.779021i	0.979664	−	0.200644i	$-0.0643033\pi$
$373$	15.0454	+	15.0454i	0.779021	+	0.779021i	−0.200644	+	0.979664i	$0.564303\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−4.47214	−	4.47214i	−0.230327	−	0.230327i
$378$		0			0
$379$		−	35.8885i		−	1.84347i	−0.387820	−	0.921735i	$-0.626772\pi$
$379$		−	35.8885i		−	1.84347i	0.387820	−	0.921735i	$-0.373228\pi$
$380$		0			0
$381$			28.4257i			1.45629i
$382$		0			0
$383$	11.1352	+	11.1352i	0.568980	+	0.568980i	0.931843	−	0.362862i	$-0.118201\pi$
$383$	11.1352	+	11.1352i	0.568980	+	0.568980i	−0.362862	+	0.931843i	$0.618201\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−5.32624	−	5.32624i	−0.270748	−	0.270748i
$388$		0			0
$389$	−23.7234			−1.20282			−0.601412	−	0.798939i	$-0.705395\pi$
$389$	−23.7234			−1.20282			−0.601412	+	0.798939i	$0.705395\pi$
$390$		0			0
$391$			5.25731i			0.265874i
$392$		0			0
$393$	0.472136	−	0.472136i	0.0238161	−	0.0238161i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	7.22494	−	7.22494i	0.362609	−	0.362609i	−0.502164	−	0.864773i	$-0.667462\pi$
$397$	7.22494	−	7.22494i	0.362609	−	0.362609i	0.864773	+	0.502164i	$0.167462\pi$
$398$		0			0
$399$	−7.60845			−0.380899
$400$		0			0
$401$	−3.88854			−0.194185			−0.0970923	−	0.995275i	$-0.530954\pi$
$401$	−3.88854			−0.194185			−0.0970923	+	0.995275i	$0.530954\pi$
$402$		0			0
$403$	−16.1803	+	16.1803i	−0.806000	+	0.806000i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	3.80423	−	3.80423i	0.188568	−	0.188568i
$408$		0			0
$409$			27.5967i			1.36457i	0.731086	+	0.682286i	$0.239014\pi$
$409$			27.5967i			1.36457i	−0.731086	+	0.682286i	$0.760986\pi$
$410$		0			0
$411$	11.2361			0.554234
$412$		0			0
$413$	−9.95959	−	9.95959i	−0.490080	−	0.490080i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−8.76393	−	8.76393i	−0.429172	−	0.429172i
$418$		0			0
$419$		−	24.8328i		−	1.21316i	−0.795022	−	0.606581i	$-0.792540\pi$
$419$		−	24.8328i		−	1.21316i	0.795022	−	0.606581i	$-0.207460\pi$
$420$		0			0
$421$			3.46120i			0.168689i	0.996437	+	0.0843443i	$0.0268795\pi$
$421$			3.46120i			0.168689i	−0.996437	+	0.0843443i	$0.973120\pi$
$422$		0			0
$423$	16.3925	+	16.3925i	0.797029	+	0.797029i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	11.7082	+	11.7082i	0.566600	+	0.566600i
$428$		0			0
$429$	12.3107			0.594368
$430$		0			0
$431$		−	11.7557i		−	0.566252i	−0.959083	−	0.283126i	$-0.908628\pi$
$431$		−	11.7557i		−	0.566252i	0.959083	−	0.283126i	$-0.0913715\pi$
$432$		0			0
$433$	−23.1803	+	23.1803i	−1.11398	+	1.11398i	−0.121369	+	0.992608i	$0.538728\pi$
$433$	−23.1803	+	23.1803i	−1.11398	+	1.11398i	−0.992608	+	0.121369i	$0.961272\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−5.25731	+	5.25731i	−0.251491	+	0.251491i
$438$		0			0
$439$	−11.2007			−0.534579			−0.267290	−	0.963616i	$-0.586128\pi$
$439$	−11.2007			−0.534579			−0.267290	+	0.963616i	$0.586128\pi$
$440$		0			0
$441$	9.47214			0.451054
$442$		0			0
$443$	10.0902	−	10.0902i	0.479398	−	0.479398i	−0.425541	−	0.904939i	$-0.639916\pi$
$443$	10.0902	−	10.0902i	0.479398	−	0.479398i	0.904939	+	0.425541i	$0.139916\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−21.3723	+	21.3723i	−1.01087	+	1.01087i
$448$		0			0
$449$		−	31.5967i		−	1.49114i	−0.666426	−	0.745571i	$-0.732177\pi$
$449$		−	31.5967i		−	1.49114i	0.666426	−	0.745571i	$-0.267823\pi$
$450$		0			0
$451$	9.52786			0.448650
$452$		0			0
$453$	23.7234	+	23.7234i	1.11462	+	1.11462i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	21.6525	+	21.6525i	1.01286	+	1.01286i	0.999916	+	0.0129439i	$0.00412028\pi$
$457$	21.6525	+	21.6525i	1.01286	+	1.01286i	0.0129439	+	0.999916i	$0.495880\pi$
$458$		0			0
$459$		−	2.47214i		−	0.115389i
$460$		0			0
$461$			6.49839i			0.302660i	0.988483	+	0.151330i	$0.0483557\pi$
$461$			6.49839i			0.302660i	−0.988483	+	0.151330i	$0.951644\pi$
$462$		0			0
$463$	−17.2905	−	17.2905i	−0.803559	−	0.803559i	0.180091	−	0.983650i	$-0.442361\pi$
$463$	−17.2905	−	17.2905i	−0.803559	−	0.803559i	−0.983650	+	0.180091i	$0.942361\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−2.67376	−	2.67376i	−0.123727	−	0.123727i	0.642532	−	0.766259i	$-0.277884\pi$
$467$	−2.67376	−	2.67376i	−0.123727	−	0.123727i	−0.766259	+	0.642532i	$0.777884\pi$
$468$		0			0
$469$	−5.60034			−0.258600
$470$		0			0
$471$			31.6749i			1.45950i
$472$		0			0
$473$	−2.94427	+	2.94427i	−0.135378	+	0.135378i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	1.62460	−	1.62460i	0.0743853	−	0.0743853i
$478$		0			0
$479$	7.60845			0.347639			0.173820	−	0.984778i	$-0.444389\pi$
$479$	7.60845			0.347639			0.173820	+	0.984778i	$0.444389\pi$
$480$		0			0
$481$	−18.9443			−0.863784
$482$		0			0
$483$	−10.0000	+	10.0000i	−0.455016	+	0.455016i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−9.33905	+	9.33905i	−0.423193	+	0.423193i	−0.886302	−	0.463109i	$-0.846734\pi$
$487$	−9.33905	+	9.33905i	−0.423193	+	0.423193i	0.463109	+	0.886302i	$0.346734\pi$
$488$		0			0
$489$		−	23.1246i		−	1.04573i
$490$		0			0
$491$	10.7639			0.485769			0.242885	−	0.970055i	$-0.421906\pi$
$491$	10.7639			0.485769			0.242885	+	0.970055i	$0.421906\pi$
$492$		0			0
$493$	−1.45309	−	1.45309i	−0.0654437	−	0.0654437i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	8.29180	+	8.29180i	0.371938	+	0.371938i
$498$		0			0
$499$			23.8885i			1.06940i	0.845043	+	0.534699i	$0.179575\pi$
$499$			23.8885i			1.06940i	−0.845043	+	0.534699i	$0.820425\pi$
$500$		0			0
$501$			0.898056i			0.0401222i
$502$		0			0
$503$	5.53483	+	5.53483i	0.246786	+	0.246786i	0.819650	−	0.572864i	$-0.194168\pi$
$503$	5.53483	+	5.53483i	0.246786	+	0.246786i	−0.572864	+	0.819650i	$0.694168\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−9.61803	−	9.61803i	−0.427152	−	0.427152i
$508$		0			0
$509$	−9.06154			−0.401646			−0.200823	−	0.979628i	$-0.564362\pi$
$509$	−9.06154			−0.401646			−0.200823	+	0.979628i	$0.564362\pi$
$510$		0			0
$511$			20.4742i			0.905726i
$512$		0			0
$513$	2.47214	−	2.47214i	0.109147	−	0.109147i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	9.06154	−	9.06154i	0.398526	−	0.398526i
$518$		0			0
$519$	−20.4742			−0.898718
$520$		0			0
$521$	8.47214			0.371171			0.185586	−	0.982628i	$-0.440582\pi$
$521$	8.47214			0.371171			0.185586	+	0.982628i	$0.440582\pi$
$522$		0			0
$523$	−16.7426	+	16.7426i	−0.732105	+	0.732105i	−0.971036	−	0.238932i	$-0.923203\pi$
$523$	−16.7426	+	16.7426i	−0.732105	+	0.732105i	0.238932	+	0.971036i	$0.423203\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−5.25731	+	5.25731i	−0.229012	+	0.229012i
$528$		0			0
$529$		−	9.18034i		−	0.399145i
$530$		0			0
$531$	−18.9443			−0.822111
$532$		0			0
$533$	−23.7234	−	23.7234i	−1.02758	−	1.02758i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−26.6525	−	26.6525i	−1.15014	−	1.15014i
$538$		0			0
$539$		−	5.23607i		−	0.225533i
$540$		0			0
$541$			2.90617i			0.124946i	0.998047	+	0.0624730i	$0.0198987\pi$
$541$			2.90617i			0.124946i	−0.998047	+	0.0624730i	$0.980101\pi$
$542$		0			0
$543$	−15.2169	−	15.2169i	−0.653020	−	0.653020i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−8.56231	−	8.56231i	−0.366098	−	0.366098i	0.499954	−	0.866052i	$-0.333350\pi$
$547$	−8.56231	−	8.56231i	−0.366098	−	0.366098i	−0.866052	+	0.499954i	$0.833350\pi$
$548$		0			0
$549$	22.2703			0.950474
$550$		0			0
$551$		−	2.90617i		−	0.123807i
$552$		0			0
$553$	14.4721	−	14.4721i	0.615418	−	0.615418i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−17.1845	+	17.1845i	−0.728132	+	0.728132i	−0.970247	−	0.242116i	$-0.922159\pi$
$557$	−17.1845	+	17.1845i	−0.728132	+	0.728132i	0.242116	+	0.970247i	$0.422159\pi$
$558$		0			0
$559$	14.6619			0.620131
$560$		0			0
$561$	4.00000			0.168880
$562$		0			0
$563$	11.3262	−	11.3262i	0.477344	−	0.477344i	−0.426937	−	0.904281i	$-0.640407\pi$
$563$	11.3262	−	11.3262i	0.477344	−	0.477344i	0.904281	+	0.426937i	$0.140407\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	12.5882	−	12.5882i	0.528657	−	0.528657i
$568$		0			0
$569$			13.1246i			0.550212i	0.961414	+	0.275106i	$0.0887130\pi$
$569$			13.1246i			0.550212i	−0.961414	+	0.275106i	$0.911287\pi$
$570$		0			0
$571$	8.65248			0.362095			0.181047	−	0.983474i	$-0.442051\pi$
$571$	8.65248			0.362095			0.181047	+	0.983474i	$0.442051\pi$
$572$		0			0
$573$	20.8172	+	20.8172i	0.869653	+	0.869653i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−21.7639	−	21.7639i	−0.906044	−	0.906044i	0.0899059	−	0.995950i	$-0.471343\pi$
$577$	−21.7639	−	21.7639i	−0.906044	−	0.906044i	−0.995950	+	0.0899059i	$0.971343\pi$
$578$		0			0
$579$			24.1803i			1.00490i
$580$		0			0
$581$			10.3026i			0.427425i
$582$		0			0
$583$	−0.898056	−	0.898056i	−0.0371937	−	0.0371937i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	5.90983	+	5.90983i	0.243925	+	0.243925i	0.818472	−	0.574547i	$-0.194822\pi$
$587$	5.90983	+	5.90983i	0.243925	+	0.243925i	−0.574547	+	0.818472i	$0.694822\pi$
$588$		0			0
$589$	−10.5146			−0.433247
$590$		0			0
$591$		−	7.05342i		−	0.290139i
$592$		0			0
$593$	6.41641	−	6.41641i	0.263490	−	0.263490i	−0.562980	−	0.826470i	$-0.690345\pi$
$593$	6.41641	−	6.41641i	0.263490	−	0.263490i	0.826470	+	0.562980i	$0.190345\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−29.3238	+	29.3238i	−1.20014	+	1.20014i
$598$		0			0
$599$	−27.5276			−1.12475			−0.562374	−	0.826883i	$-0.690112\pi$
$599$	−27.5276			−1.12475			−0.562374	+	0.826883i	$0.690112\pi$
$600$		0			0
$601$	−4.29180			−0.175066			−0.0875330	−	0.996162i	$-0.527898\pi$
$601$	−4.29180			−0.175066			−0.0875330	+	0.996162i	$0.527898\pi$
$602$		0			0
$603$	−5.32624	+	5.32624i	−0.216901	+	0.216901i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	6.08985	−	6.08985i	0.247180	−	0.247180i	−0.572633	−	0.819812i	$-0.694078\pi$
$607$	6.08985	−	6.08985i	0.247180	−	0.247180i	0.819812	+	0.572633i	$0.194078\pi$
$608$		0			0
$609$		−	5.52786i		−	0.224000i
$610$		0			0
$611$	−45.1246			−1.82555
$612$		0			0
$613$	−27.3561	−	27.3561i	−1.10490	−	1.10490i	−0.993810	−	0.111094i	$-0.964565\pi$
$613$	−27.3561	−	27.3561i	−1.10490	−	1.10490i	−0.111094	−	0.993810i	$-0.535435\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	30.8885	+	30.8885i	1.24353	+	1.24353i	0.958528	+	0.284998i	$0.0919929\pi$
$617$	30.8885	+	30.8885i	1.24353	+	1.24353i	0.284998	+	0.958528i	$0.408007\pi$
$618$		0			0
$619$			27.3050i			1.09748i	0.835994	+	0.548739i	$0.184892\pi$
$619$			27.3050i			1.09748i	−0.835994	+	0.548739i	$0.815108\pi$
$620$		0			0
$621$		−	6.49839i		−	0.260772i
$622$		0			0
$623$	−7.60845	−	7.60845i	−0.304826	−	0.304826i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	4.00000	+	4.00000i	0.159745	+	0.159745i
$628$		0			0
$629$	−6.15537			−0.245431
$630$		0			0
$631$		−	28.0827i		−	1.11795i	−0.829183	−	0.558977i	$-0.811194\pi$
$631$		−	28.0827i		−	1.11795i	0.829183	−	0.558977i	$-0.188806\pi$
$632$		0			0
$633$	25.4164	−	25.4164i	1.01021	−	1.01021i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−13.0373	+	13.0373i	−0.516556	+	0.516556i
$638$		0			0
$639$	15.7719			0.623928
$640$		0			0
$641$	−7.34752			−0.290210			−0.145105	−	0.989416i	$-0.546352\pi$
$641$	−7.34752			−0.290210			−0.145105	+	0.989416i	$0.546352\pi$
$642$		0			0
$643$	8.56231	−	8.56231i	0.337664	−	0.337664i	−0.517823	−	0.855488i	$-0.673258\pi$
$643$	8.56231	−	8.56231i	0.337664	−	0.337664i	0.855488	+	0.517823i	$0.173258\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	22.8909	−	22.8909i	0.899933	−	0.899933i	−0.0954968	−	0.995430i	$-0.530444\pi$
$647$	22.8909	−	22.8909i	0.899933	−	0.899933i	0.995430	+	0.0954968i	$0.0304440\pi$
$648$		0			0
$649$			10.4721i			0.411067i
$650$		0			0
$651$	−20.0000			−0.783862
$652$		0			0
$653$	11.2412	+	11.2412i	0.439901	+	0.439901i	0.891979	−	0.452078i	$-0.149317\pi$
$653$	11.2412	+	11.2412i	0.439901	+	0.439901i	−0.452078	+	0.891979i	$0.649317\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	19.4721	+	19.4721i	0.759680	+	0.759680i
$658$		0			0
$659$		−	18.0000i		−	0.701180i	−0.936529	−	0.350590i	$-0.885981\pi$
$659$		−	18.0000i		−	0.701180i	0.936529	−	0.350590i	$-0.114019\pi$
$660$		0			0
$661$			2.35114i			0.0914488i	0.998954	+	0.0457244i	$0.0145596\pi$
$661$			2.35114i			0.0914488i	−0.998954	+	0.0457244i	$0.985440\pi$
$662$		0			0
$663$	−9.95959	−	9.95959i	−0.386799	−	0.386799i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−3.81966	−	3.81966i	−0.147898	−	0.147898i
$668$		0			0
$669$	−5.60034			−0.216522
$670$		0			0
$671$		−	12.3107i		−	0.475251i
$672$		0			0
$673$	30.7082	−	30.7082i	1.18371	−	1.18371i	0.204940	−	0.978775i	$-0.434300\pi$
$673$	30.7082	−	30.7082i	1.18371	−	1.18371i	0.978775	−	0.204940i	$-0.0656998\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	8.33499	−	8.33499i	0.320340	−	0.320340i	−0.528558	−	0.848897i	$-0.677267\pi$
$677$	8.33499	−	8.33499i	0.320340	−	0.320340i	0.848897	+	0.528558i	$0.177267\pi$
$678$		0			0
$679$	0.555029			0.0213001
$680$		0			0
$681$	37.5967			1.44071
$682$		0			0
$683$	1.79837	−	1.79837i	0.0688129	−	0.0688129i	−0.671863	−	0.740676i	$-0.734506\pi$
$683$	1.79837	−	1.79837i	0.0688129	−	0.0688129i	0.740676	+	0.671863i	$0.234506\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−34.5811	+	34.5811i	−1.31935	+	1.31935i
$688$		0			0
$689$			4.47214i			0.170375i
$690$		0			0
$691$	31.1246			1.18404			0.592018	−	0.805925i	$-0.298331\pi$
$691$	31.1246			1.18404			0.592018	+	0.805925i	$0.298331\pi$
$692$		0			0
$693$	3.24920	+	3.24920i	0.123427	+	0.123427i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−7.70820	−	7.70820i	−0.291969	−	0.291969i
$698$		0			0
$699$			11.2361i			0.424987i
$700$		0			0
$701$		−	40.3934i		−	1.52564i	−0.646612	−	0.762819i	$-0.723815\pi$
$701$		−	40.3934i		−	1.52564i	0.646612	−	0.762819i	$-0.276185\pi$
$702$		0			0
$703$	−6.15537	−	6.15537i	−0.232154	−	0.232154i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−14.4721	−	14.4721i	−0.544281	−	0.544281i
$708$		0			0
$709$	13.7638			0.516911			0.258456	−	0.966023i	$-0.416786\pi$
$709$	13.7638			0.516911			0.258456	+	0.966023i	$0.416786\pi$
$710$		0			0
$711$		−	27.5276i		−	1.03237i
$712$		0			0
$713$	−13.8197	+	13.8197i	−0.517550	+	0.517550i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	47.4468	−	47.4468i	1.77193	−	1.77193i
$718$		0			0
$719$	44.5407			1.66109			0.830543	−	0.556954i	$-0.188030\pi$
$719$	44.5407			1.66109			0.830543	+	0.556954i	$0.188030\pi$
$720$		0			0
$721$	17.2361			0.641905
$722$		0			0
$723$	−10.9443	+	10.9443i	−0.407022	+	0.407022i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	5.87785	−	5.87785i	0.217997	−	0.217997i	−0.589657	−	0.807654i	$-0.700737\pi$
$727$	5.87785	−	5.87785i	0.217997	−	0.217997i	0.807654	+	0.589657i	$0.200737\pi$
$728$		0			0
$729$		−	11.9443i		−	0.442380i
$730$		0			0
$731$	4.76393			0.176200
$732$		0			0
$733$	−1.28157	−	1.28157i	−0.0473359	−	0.0473359i	0.683043	−	0.730379i	$-0.260656\pi$
$733$	−1.28157	−	1.28157i	−0.0473359	−	0.0473359i	−0.730379	+	0.683043i	$0.760656\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	2.94427	+	2.94427i	0.108454	+	0.108454i
$738$		0			0
$739$		−	17.4164i		−	0.640673i	−0.947304	−	0.320336i	$-0.896204\pi$
$739$		−	17.4164i		−	0.640673i	0.947304	−	0.320336i	$-0.103796\pi$
$740$		0			0
$741$		−	19.9192i		−	0.731750i
$742$		0			0
$743$	−13.4863	−	13.4863i	−0.494765	−	0.494765i	0.415039	−	0.909804i	$-0.363768\pi$
$743$	−13.4863	−	13.4863i	−0.494765	−	0.494765i	−0.909804	+	0.415039i	$0.863768\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	9.79837	+	9.79837i	0.358504	+	0.358504i
$748$		0			0
$749$	28.4257			1.03865
$750$		0			0
$751$		−	7.05342i		−	0.257383i	−0.991685	−	0.128692i	$-0.958922\pi$
$751$		−	7.05342i		−	0.257383i	0.991685	−	0.128692i	$-0.0410777\pi$
$752$		0			0
$753$	−35.8885	+	35.8885i	−1.30785	+	1.30785i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	38.0018	−	38.0018i	1.38120	−	1.38120i	0.538701	−	0.842497i	$-0.318915\pi$
$757$	38.0018	−	38.0018i	1.38120	−	1.38120i	0.842497	−	0.538701i	$-0.181085\pi$
$758$		0			0
$759$	10.5146			0.381657
$760$		0			0
$761$	−14.9443			−0.541729			−0.270865	−	0.962617i	$-0.587310\pi$
$761$	−14.9443			−0.541729			−0.270865	+	0.962617i	$0.587310\pi$
$762$		0			0
$763$	7.88854	−	7.88854i	0.285584	−	0.285584i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	26.0746	−	26.0746i	0.941498	−	0.941498i
$768$		0			0
$769$			2.47214i			0.0891475i	0.999006	+	0.0445738i	$0.0141930\pi$
$769$			2.47214i			0.0891475i	−0.999006	+	0.0445738i	$0.985807\pi$
$770$		0			0
$771$	60.5410			2.18033
$772$		0			0
$773$	4.18774	+	4.18774i	0.150623	+	0.150623i	0.778396	−	0.627773i	$-0.216034\pi$
$773$	4.18774	+	4.18774i	0.150623	+	0.150623i	−0.627773	+	0.778396i	$0.716034\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−11.7082	−	11.7082i	−0.420029	−	0.420029i
$778$		0			0
$779$		−	15.4164i		−	0.552350i
$780$		0			0
$781$		−	8.71851i		−	0.311973i
$782$		0			0
$783$	1.79611	+	1.79611i	0.0641878	+	0.0641878i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	11.1459	+	11.1459i	0.397308	+	0.397308i	0.877283	−	0.479974i	$-0.159354\pi$
$787$	11.1459	+	11.1459i	0.397308	+	0.397308i	−0.479974	+	0.877283i	$0.659354\pi$
$788$		0			0
$789$	53.0472			1.88853
$790$		0			0
$791$		−	11.0697i		−	0.393591i
$792$		0			0
$793$	−30.6525	+	30.6525i	−1.08850	+	1.08850i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−19.7477	+	19.7477i	−0.699498	+	0.699498i	−0.964302	−	0.264804i	$-0.914693\pi$
$797$	−19.7477	+	19.7477i	−0.699498	+	0.699498i	0.264804	+	0.964302i	$0.414693\pi$
$798$		0			0
$799$	−14.6619			−0.518700
$800$		0			0
$801$	−14.4721			−0.511348
$802$		0			0
$803$	10.7639	−	10.7639i	0.379851	−	0.379851i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	28.9807	−	28.9807i	1.02017	−	1.02017i
$808$		0			0
$809$			12.9443i			0.455096i	0.973767	+	0.227548i	$0.0730709\pi$
$809$			12.9443i			0.455096i	−0.973767	+	0.227548i	$0.926929\pi$
$810$		0			0
$811$	32.0689			1.12609			0.563045	−	0.826426i	$-0.309630\pi$
$811$	32.0689			1.12609			0.563045	+	0.826426i	$0.309630\pi$
$812$		0			0
$813$	−51.2511	−	51.2511i	−1.79745	−	1.79745i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	4.76393	+	4.76393i	0.166669	+	0.166669i
$818$		0			0
$819$		−	16.1803i		−	0.565387i
$820$		0			0
$821$		−	20.4742i		−	0.714555i	−0.933998	−	0.357278i	$-0.883705\pi$
$821$		−	20.4742i		−	0.714555i	0.933998	−	0.357278i	$-0.116295\pi$
$822$		0			0
$823$	24.3440	+	24.3440i	0.848577	+	0.848577i	0.989956	−	0.141379i	$-0.0451535\pi$
$823$	24.3440	+	24.3440i	0.848577	+	0.848577i	−0.141379	+	0.989956i	$0.545154\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	14.8541	+	14.8541i	0.516528	+	0.516528i	0.916519	−	0.399991i	$-0.130987\pi$
$827$	14.8541	+	14.8541i	0.516528	+	0.516528i	−0.399991	+	0.916519i	$0.630987\pi$
$828$		0			0
$829$	3.11817			0.108299			0.0541493	−	0.998533i	$-0.482755\pi$
$829$	3.11817			0.108299			0.0541493	+	0.998533i	$0.482755\pi$
$830$		0			0
$831$			7.05342i			0.244681i
$832$		0			0
$833$	−4.23607	+	4.23607i	−0.146771	+	0.146771i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	6.49839	−	6.49839i	0.224617	−	0.224617i
$838$		0			0
$839$	9.40456			0.324682			0.162341	−	0.986735i	$-0.448096\pi$
$839$	9.40456			0.324682			0.162341	+	0.986735i	$0.448096\pi$
$840$		0			0
$841$	−26.8885			−0.927191
$842$		0			0
$843$	−6.00000	+	6.00000i	−0.206651	+	0.206651i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−11.1352	+	11.1352i	−0.382609	+	0.382609i
$848$		0			0
$849$			50.5410i			1.73456i
$850$		0			0
$851$	−16.1803			−0.554655
$852$		0			0
$853$	24.7930	+	24.7930i	0.848896	+	0.848896i	0.989995	−	0.141100i	$-0.0450639\pi$
$853$	24.7930	+	24.7930i	0.848896	+	0.848896i	−0.141100	+	0.989995i	$0.545064\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	17.8328	+	17.8328i	0.609157	+	0.609157i	0.942726	−	0.333568i	$-0.108253\pi$
$857$	17.8328	+	17.8328i	0.609157	+	0.609157i	−0.333568	+	0.942726i	$0.608253\pi$
$858$		0			0
$859$			7.52786i			0.256847i	0.991719	+	0.128424i	$0.0409917\pi$
$859$			7.52786i			0.256847i	−0.991719	+	0.128424i	$0.959008\pi$
$860$		0			0
$861$		−	29.3238i		−	0.999351i
$862$		0			0
$863$	6.22088	+	6.22088i	0.211761	+	0.211761i	0.805015	−	0.593254i	$-0.202157\pi$
$863$	6.22088	+	6.22088i	0.211761	+	0.211761i	−0.593254	+	0.805015i	$0.702157\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	24.2705	+	24.2705i	0.824270	+	0.824270i
$868$		0			0
$869$	−15.2169			−0.516198
$870$		0			0
$871$		−	14.6619i		−	0.496799i
$872$		0			0
$873$	0.527864	−	0.527864i	0.0178655	−	0.0178655i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	5.08580	−	5.08580i	0.171735	−	0.171735i	−0.616006	−	0.787741i	$-0.711250\pi$
$877$	5.08580	−	5.08580i	0.171735	−	0.171735i	0.787741	+	0.616006i	$0.211250\pi$
$878$		0			0
$879$	2.35114			0.0793020
$880$		0			0
$881$	47.1246			1.58767			0.793834	−	0.608134i	$-0.208082\pi$
$881$	47.1246			1.58767			0.793834	+	0.608134i	$0.208082\pi$
$882$		0			0
$883$	21.7984	−	21.7984i	0.733574	−	0.733574i	−0.237752	−	0.971326i	$-0.576411\pi$
$883$	21.7984	−	21.7984i	0.733574	−	0.733574i	0.971326	+	0.237752i	$0.0764106\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−28.1482	+	28.1482i	−0.945123	+	0.945123i	−0.998571	−	0.0534473i	$-0.982979\pi$
$887$	−28.1482	+	28.1482i	−0.945123	+	0.945123i	0.0534473	+	0.998571i	$0.482979\pi$
$888$		0			0
$889$		−	20.6525i		−	0.692662i
$890$		0			0
$891$	−13.2361			−0.443425
$892$		0			0
$893$	−14.6619	−	14.6619i	−0.490641	−	0.490641i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−26.1803	−	26.1803i	−0.874136	−	0.874136i
$898$		0			0
$899$		−	7.63932i		−	0.254786i
$900$		0			0
$901$			1.45309i			0.0484093i
$902$		0			0
$903$	9.06154	+	9.06154i	0.301549	+	0.301549i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−23.3262	−	23.3262i	−0.774535	−	0.774535i	0.204361	−	0.978896i	$-0.434488\pi$
$907$	−23.3262	−	23.3262i	−0.774535	−	0.774535i	−0.978896	+	0.204361i	$0.934488\pi$
$908$		0			0
$909$	−27.5276			−0.913034
$910$		0			0
$911$		−	25.1765i		−	0.834135i	−0.908876	−	0.417067i	$-0.863058\pi$
$911$		−	25.1765i		−	0.834135i	0.908876	−	0.417067i	$-0.136942\pi$
$912$		0			0
$913$	5.41641	−	5.41641i	0.179257	−	0.179257i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−0.343027	+	0.343027i	−0.0113277	+	0.0113277i
$918$		0			0
$919$	21.7153			0.716322			0.358161	−	0.933660i	$-0.383404\pi$
$919$	21.7153			0.716322			0.358161	+	0.933660i	$0.383404\pi$
$920$		0			0
$921$	−21.2361			−0.699752
$922$		0			0
$923$	−21.7082	+	21.7082i	−0.714534	+	0.714534i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	16.3925	−	16.3925i	0.538400	−	0.538400i
$928$		0			0
$929$		−	5.34752i		−	0.175447i	−0.996145	−	0.0877233i	$-0.972041\pi$
$929$		−	5.34752i		−	0.175447i	0.996145	−	0.0877233i	$-0.0279591\pi$
$930$		0			0
$931$	−8.47214			−0.277663
$932$		0			0
$933$	−13.2088	−	13.2088i	−0.432436	−	0.432436i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−20.3050	−	20.3050i	−0.663334	−	0.663334i	0.292831	−	0.956164i	$-0.405403\pi$
$937$	−20.3050	−	20.3050i	−0.663334	−	0.663334i	−0.956164	+	0.292831i	$0.905403\pi$
$938$		0			0
$939$			20.1803i			0.658561i
$940$		0			0
$941$			60.8676i			1.98423i	0.125340	+	0.992114i	$0.459998\pi$
$941$			60.8676i			1.98423i	−0.125340	+	0.992114i	$0.540002\pi$
$942$		0			0
$943$	−20.2622	−	20.2622i	−0.659828	−	0.659828i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−8.85410	−	8.85410i	−0.287720	−	0.287720i	0.548458	−	0.836178i	$-0.315215\pi$
$947$	−8.85410	−	8.85410i	−0.287720	−	0.287720i	−0.836178	+	0.548458i	$0.815215\pi$
$948$		0			0
$949$	−53.6022			−1.74000
$950$		0			0
$951$			34.5811i			1.12137i
$952$		0			0
$953$	6.81966	−	6.81966i	0.220910	−	0.220910i	−0.587971	−	0.808882i	$-0.700073\pi$
$953$	6.81966	−	6.81966i	0.220910	−	0.220910i	0.808882	+	0.587971i	$0.200073\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−2.90617	+	2.90617i	−0.0939431	+	0.0939431i
$958$		0			0
$959$	−8.16348			−0.263613
$960$		0			0
$961$	3.36068			0.108409
$962$		0			0
$963$	27.0344	−	27.0344i	0.871173	−	0.871173i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−12.0332	+	12.0332i	−0.386962	+	0.386962i	−0.873602	−	0.486640i	$-0.838222\pi$
$967$	−12.0332	+	12.0332i	−0.386962	+	0.386962i	0.486640	+	0.873602i	$0.338222\pi$
$968$		0			0
$969$		−	6.47214i		−	0.207915i
$970$		0			0
$971$	−33.5967			−1.07817			−0.539085	−	0.842251i	$-0.681230\pi$
$971$	−33.5967			−1.07817			−0.539085	+	0.842251i	$0.681230\pi$
$972$		0			0
$973$	6.36737	+	6.36737i	0.204128	+	0.204128i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	28.2361	+	28.2361i	0.903352	+	0.903352i	0.995725	−	0.0923727i	$-0.0294451\pi$
$977$	28.2361	+	28.2361i	0.903352	+	0.903352i	−0.0923727	+	0.995725i	$0.529445\pi$
$978$		0			0
$979$			8.00000i			0.255681i
$980$		0			0
$981$		−	15.0049i		−	0.479070i
$982$		0			0
$983$	0.620541	+	0.620541i	0.0197922	+	0.0197922i	0.716934	−	0.697141i	$-0.245545\pi$
$983$	0.620541	+	0.620541i	0.0197922	+	0.0197922i	−0.697141	+	0.716934i	$0.745545\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−27.8885	−	27.8885i	−0.887702	−	0.887702i
$988$		0			0
$989$	12.5227			0.398200
$990$		0			0
$991$			19.3642i			0.615123i	0.951528	+	0.307561i	$0.0995129\pi$
$991$			19.3642i			0.615123i	−0.951528	+	0.307561i	$0.900487\pi$
$992$		0			0
$993$	45.4164	−	45.4164i	1.44125	−	1.44125i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−35.6506	+	35.6506i	−1.12907	+	1.12907i	−0.138738	+	0.990329i	$0.544304\pi$
$997$	−35.6506	+	35.6506i	−1.12907	+	1.12907i	−0.990329	+	0.138738i	$0.955696\pi$
$998$		0			0
$999$	7.60845			0.240721

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	800.2.o.g.207.2		8
4.3	odd	2		200.2.k.h.107.3		8
5.2	odd	4		160.2.o.a.143.3		8
5.3	odd	4	inner	800.2.o.g.143.1		8
5.4	even	2		160.2.o.a.47.4		8
8.3	odd	2	inner	800.2.o.g.207.1		8
8.5	even	2		200.2.k.h.107.1		8
15.2	even	4		1440.2.bi.c.1423.4		8
15.14	odd	2		1440.2.bi.c.847.1		8
20.3	even	4		200.2.k.h.43.1		8
20.7	even	4		40.2.k.a.3.4	yes	8
20.19	odd	2		40.2.k.a.27.2	yes	8
40.3	even	4	inner	800.2.o.g.143.2		8
40.13	odd	4		200.2.k.h.43.3		8
40.19	odd	2		160.2.o.a.47.3		8
40.27	even	4		160.2.o.a.143.4		8
40.29	even	2		40.2.k.a.27.4	yes	8
40.37	odd	4		40.2.k.a.3.2	✓	8
60.47	odd	4		360.2.w.c.163.1		8
60.59	even	2		360.2.w.c.307.3		8
80.19	odd	4		1280.2.n.m.767.2		8
80.27	even	4		1280.2.n.m.1023.1		8
80.29	even	4		1280.2.n.q.767.4		8
80.37	odd	4		1280.2.n.q.1023.3		8
80.59	odd	4		1280.2.n.q.767.3		8
80.67	even	4		1280.2.n.q.1023.4		8
80.69	even	4		1280.2.n.m.767.1		8
80.77	odd	4		1280.2.n.m.1023.2		8
120.29	odd	2		360.2.w.c.307.1		8
120.59	even	2		1440.2.bi.c.847.4		8
120.77	even	4		360.2.w.c.163.3		8
120.107	odd	4		1440.2.bi.c.1423.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.2.k.a.3.2	✓	8	40.37	odd	4
40.2.k.a.3.4	yes	8	20.7	even	4
40.2.k.a.27.2	yes	8	20.19	odd	2
40.2.k.a.27.4	yes	8	40.29	even	2
160.2.o.a.47.3		8	40.19	odd	2
160.2.o.a.47.4		8	5.4	even	2
160.2.o.a.143.3		8	5.2	odd	4
160.2.o.a.143.4		8	40.27	even	4
200.2.k.h.43.1		8	20.3	even	4
200.2.k.h.43.3		8	40.13	odd	4
200.2.k.h.107.1		8	8.5	even	2
200.2.k.h.107.3		8	4.3	odd	2
360.2.w.c.163.1		8	60.47	odd	4
360.2.w.c.163.3		8	120.77	even	4
360.2.w.c.307.1		8	120.29	odd	2
360.2.w.c.307.3		8	60.59	even	2
800.2.o.g.143.1		8	5.3	odd	4	inner
800.2.o.g.143.2		8	40.3	even	4	inner
800.2.o.g.207.1		8	8.3	odd	2	inner
800.2.o.g.207.2		8	1.1	even	1	trivial
1280.2.n.m.767.1		8	80.69	even	4
1280.2.n.m.767.2		8	80.19	odd	4
1280.2.n.m.1023.1		8	80.27	even	4
1280.2.n.m.1023.2		8	80.77	odd	4
1280.2.n.q.767.3		8	80.59	odd	4
1280.2.n.q.767.4		8	80.29	even	4
1280.2.n.q.1023.3		8	80.37	odd	4
1280.2.n.q.1023.4		8	80.67	even	4
1440.2.bi.c.847.1		8	15.14	odd	2
1440.2.bi.c.847.4		8	120.59	even	2
1440.2.bi.c.1423.1		8	120.107	odd	4
1440.2.bi.c.1423.4		8	15.2	even	4

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:	\( N \)	\(=\)	\( 800 = 2^{5} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	800.o (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.61803 + 1.61803i) q^{3} +(1.17557 - 1.17557i) q^{7} -2.23607i q^{9} +O(q^{10})\)	\(q+(-1.61803 + 1.61803i) q^{3} +(1.17557 - 1.17557i) q^{7} -2.23607i q^{9} -1.23607 q^{11} +(3.07768 + 3.07768i) q^{13} +(1.00000 + 1.00000i) q^{17} +2.00000i q^{19} +3.80423i q^{21} +(2.62866 + 2.62866i) q^{23} +(-1.23607 - 1.23607i) q^{27} -1.45309 q^{29} +5.25731i q^{31} +(2.00000 - 2.00000i) q^{33} +(-3.07768 + 3.07768i) q^{37} -9.95959 q^{39} -7.70820 q^{41} +(2.38197 - 2.38197i) q^{43} +(-7.33094 + 7.33094i) q^{47} +4.23607i q^{49} -3.23607 q^{51} +(0.726543 + 0.726543i) q^{53} +(-3.23607 - 3.23607i) q^{57} -8.47214i q^{59} +9.95959i q^{61} +(-2.62866 - 2.62866i) q^{63} +(-2.38197 - 2.38197i) q^{67} -8.50651 q^{69} +7.05342i q^{71} +(-8.70820 + 8.70820i) q^{73} +(-1.45309 + 1.45309i) q^{77} +12.3107 q^{79} +10.7082 q^{81} +(-4.38197 + 4.38197i) q^{83} +(2.35114 - 2.35114i) q^{87} -6.47214i q^{89} +7.23607 q^{91} +(-8.50651 - 8.50651i) q^{93} +(0.236068 + 0.236068i) q^{97} +2.76393i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{3}+O(q^{10}) \) 8 * q - 4 * q^3	\( 8 q - 4 q^{3} + 8 q^{11} + 8 q^{17} + 8 q^{27} + 16 q^{33} - 8 q^{41} + 28 q^{43} - 8 q^{51} - 8 q^{57} - 28 q^{67} - 16 q^{73} + 32 q^{81} - 44 q^{83} + 40 q^{91} - 16 q^{97}+O(q^{100}) \) 8 * q - 4 * q^3 + 8 * q^11 + 8 * q^17 + 8 * q^27 + 16 * q^33 - 8 * q^41 + 28 * q^43 - 8 * q^51 - 8 * q^57 - 28 * q^67 - 16 * q^73 + 32 * q^81 - 44 * q^83 + 40 * q^91 - 16 * q^97

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