LMFDB - Embedded newform 1600.2.j.d.1007.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1600,2,Mod(143,1600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1600, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 1, 3]))

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

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

chi := DirichletCharacter("1600.143");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1600 = 2^{6} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1600.j (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:	$12.7760643234$
Analytic rank:	$0$
Dimension:	$18$
Relative dimension:	$9$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} + 2 x^{16} - 4 x^{15} - 5 x^{14} - 14 x^{13} - 10 x^{12} + 6 x^{11} + 37 x^{10} + 70 x^{9} + \cdots + 512 $ x^18 + 2x^16 - 4x^15 - 5x^14 - 14x^13 - 10x^12 + 6x^11 + 37x^10 + 70x^9 + 74x^8 + 24x^7 - 80x^6 - 224x^5 - 160x^4 - 256x^3 + 256*x^2 + 512
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{15} $
Twist minimal:	no (minimal twist has level 80)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1007.6
Root			$-1.08900 - 0.902261i$ of defining polynomial
Character	$\chi$	$=$	1600.1007
Dual form			1600.2.j.d.143.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.496487i q^{3} +(1.55426 + 1.55426i) q^{7} +2.75350 q^{9} +O(q^{10})$	$q+0.496487i q^{3} +(1.55426 + 1.55426i) q^{7} +2.75350 q^{9} +(4.19607 + 4.19607i) q^{11} -5.09530 q^{13} +(-0.213542 - 0.213542i) q^{17} +(0.844754 + 0.844754i) q^{19} +(-0.771668 + 0.771668i) q^{21} +(1.70744 - 1.70744i) q^{23} +2.85654i q^{27} +(-2.24750 + 2.24750i) q^{29} +0.818209i q^{31} +(-2.08329 + 2.08329i) q^{33} +5.12639 q^{37} -2.52975i q^{39} -3.34727i q^{41} -4.49131 q^{43} +(-4.29355 + 4.29355i) q^{47} -2.16858i q^{49} +(0.106021 - 0.106021i) q^{51} -1.00653i q^{53} +(-0.419410 + 0.419410i) q^{57} +(-7.65005 + 7.65005i) q^{59} +(-1.90291 - 1.90291i) q^{61} +(4.27964 + 4.27964i) q^{63} +11.0221 q^{67} +(0.847724 + 0.847724i) q^{69} +10.5331 q^{71} +(2.70854 + 2.70854i) q^{73} +13.0435i q^{77} -8.32010 q^{79} +6.84226 q^{81} +9.17237i q^{83} +(-1.11585 - 1.11585i) q^{87} +4.25101 q^{89} +(-7.91940 - 7.91940i) q^{91} -0.406230 q^{93} +(7.16000 + 7.16000i) q^{97} +(11.5539 + 11.5539i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 2 q^{7} - 10 q^{9}+O(q^{10}) $ 18 * q + 2 * q^7 - 10 * q^9	$ 18 q + 2 q^{7} - 10 q^{9} + 2 q^{11} + 6 q^{17} - 2 q^{19} - 16 q^{21} - 2 q^{23} - 14 q^{29} + 8 q^{33} - 8 q^{37} - 44 q^{43} - 38 q^{47} - 8 q^{51} - 24 q^{57} + 10 q^{59} + 14 q^{61} + 6 q^{63} + 12 q^{67} + 32 q^{69} - 24 q^{71} - 14 q^{73} - 16 q^{79} + 2 q^{81} + 24 q^{87} - 12 q^{89} - 16 q^{93} - 18 q^{97} + 22 q^{99}+O(q^{100}) $ 18 * q + 2 * q^7 - 10 * q^9 + 2 * q^11 + 6 * q^17 - 2 * q^19 - 16 * q^21 - 2 * q^23 - 14 * q^29 + 8 * q^33 - 8 * q^37 - 44 * q^43 - 38 * q^47 - 8 * q^51 - 24 * q^57 + 10 * q^59 + 14 * q^61 + 6 * q^63 + 12 * q^67 + 32 * q^69 - 24 * q^71 - 14 * q^73 - 16 * q^79 + 2 * q^81 + 24 * q^87 - 12 * q^89 - 16 * q^93 - 18 * q^97 + 22 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1151$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$e\left(\frac{3}{4}\right)$	$-1$

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
$3$			0.496487i			0.286647i	0.989676	+	0.143324i	$0.0457790\pi$
$3$			0.496487i			0.286647i	−0.989676	+	0.143324i	$0.954221\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	1.55426	+	1.55426i	0.587453	+	0.587453i	0.936941	−	0.349488i	$-0.113644\pi$
$7$	1.55426	+	1.55426i	0.587453	+	0.587453i	−0.349488	+	0.936941i	$0.613644\pi$
$8$		0			0
$9$	2.75350			0.917833
$10$		0			0
$11$	4.19607	+	4.19607i	1.26516	+	1.26516i	0.948558	+	0.316604i	$0.102543\pi$
$11$	4.19607	+	4.19607i	1.26516	+	1.26516i	0.316604	+	0.948558i	$0.397457\pi$
$12$		0			0
$13$	−5.09530			−1.41318			−0.706591	−	0.707622i	$-0.749768\pi$
$13$	−5.09530			−1.41318			−0.706591	+	0.707622i	$0.749768\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−0.213542	−	0.213542i	−0.0517916	−	0.0517916i	0.680737	−	0.732528i	$-0.261660\pi$
$17$	−0.213542	−	0.213542i	−0.0517916	−	0.0517916i	−0.732528	+	0.680737i	$0.761660\pi$
$18$		0			0
$19$	0.844754	+	0.844754i	0.193800	+	0.193800i	0.797336	−	0.603536i	$-0.206242\pi$
$19$	0.844754	+	0.844754i	0.193800	+	0.193800i	−0.603536	+	0.797336i	$0.706242\pi$
$20$		0			0
$21$	−0.771668	+	0.771668i	−0.168392	+	0.168392i
$22$		0			0
$23$	1.70744	−	1.70744i	0.356027	−	0.356027i	−0.506319	−	0.862346i	$-0.668994\pi$
$23$	1.70744	−	1.70744i	0.356027	−	0.356027i	0.862346	+	0.506319i	$0.168994\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$			2.85654i			0.549741i
$28$		0			0
$29$	−2.24750	+	2.24750i	−0.417350	+	0.417350i	−0.884289	−	0.466939i	$-0.845357\pi$
$29$	−2.24750	+	2.24750i	−0.417350	+	0.417350i	0.466939	+	0.884289i	$0.345357\pi$
$30$		0			0
$31$			0.818209i			0.146955i	0.997297	+	0.0734773i	$0.0234097\pi$
$31$			0.818209i			0.146955i	−0.997297	+	0.0734773i	$0.976590\pi$
$32$		0			0
$33$	−2.08329	+	2.08329i	−0.362655	+	0.362655i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	5.12639			0.842774			0.421387	−	0.906881i	$-0.361543\pi$
$37$	5.12639			0.842774			0.421387	+	0.906881i	$0.361543\pi$
$38$		0			0
$39$		−	2.52975i		−	0.405084i
$40$		0			0
$41$		−	3.34727i		−	0.522756i	−0.965237	−	0.261378i	$-0.915823\pi$
$41$		−	3.34727i		−	0.522756i	0.965237	−	0.261378i	$-0.0841769\pi$
$42$		0			0
$43$	−4.49131			−0.684919			−0.342460	−	0.939533i	$-0.611260\pi$
$43$	−4.49131			−0.684919			−0.342460	+	0.939533i	$0.611260\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−4.29355	+	4.29355i	−0.626278	+	0.626278i	−0.947130	−	0.320851i	$-0.896031\pi$
$47$	−4.29355	+	4.29355i	−0.626278	+	0.626278i	0.320851	+	0.947130i	$0.396031\pi$
$48$		0			0
$49$		−	2.16858i		−	0.309797i
$50$		0			0
$51$	0.106021	−	0.106021i	0.0148459	−	0.0148459i
$52$		0			0
$53$		−	1.00653i		−	0.138258i	−0.997608	−	0.0691291i	$-0.977978\pi$
$53$		−	1.00653i		−	0.138258i	0.997608	−	0.0691291i	$-0.0220220\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−0.419410	+	0.419410i	−0.0555521	+	0.0555521i
$58$		0			0
$59$	−7.65005	+	7.65005i	−0.995952	+	0.995952i	−0.999992	−	0.00404030i	$-0.998714\pi$
$59$	−7.65005	+	7.65005i	−0.995952	+	0.995952i	0.00404030	+	0.999992i	$0.498714\pi$
$60$		0			0
$61$	−1.90291	−	1.90291i	−0.243643	−	0.243643i	0.574712	−	0.818355i	$-0.305114\pi$
$61$	−1.90291	−	1.90291i	−0.243643	−	0.243643i	−0.818355	+	0.574712i	$0.805114\pi$
$62$		0			0
$63$	4.27964	+	4.27964i	0.539184	+	0.539184i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	11.0221			1.34656			0.673280	−	0.739387i	$-0.264885\pi$
$67$	11.0221			1.34656			0.673280	+	0.739387i	$0.264885\pi$
$68$		0			0
$69$	0.847724	+	0.847724i	0.102054	+	0.102054i
$70$		0			0
$71$	10.5331			1.25005			0.625027	−	0.780604i	$-0.285088\pi$
$71$	10.5331			1.25005			0.625027	+	0.780604i	$0.285088\pi$
$72$		0			0
$73$	2.70854	+	2.70854i	0.317010	+	0.317010i	0.847618	−	0.530607i	$-0.178036\pi$
$73$	2.70854	+	2.70854i	0.317010	+	0.317010i	−0.530607	+	0.847618i	$0.678036\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			13.0435i			1.48645i
$78$		0			0
$79$	−8.32010			−0.936085			−0.468042	−	0.883706i	$-0.655041\pi$
$79$	−8.32010			−0.936085			−0.468042	+	0.883706i	$0.655041\pi$
$80$		0			0
$81$	6.84226			0.760252
$82$		0			0
$83$			9.17237i			1.00680i	0.864054	+	0.503399i	$0.167917\pi$
$83$			9.17237i			1.00680i	−0.864054	+	0.503399i	$0.832083\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−1.11585	−	1.11585i	−0.119632	−	0.119632i
$88$		0			0
$89$	4.25101			0.450606			0.225303	−	0.974289i	$-0.427663\pi$
$89$	4.25101			0.450606			0.225303	+	0.974289i	$0.427663\pi$
$90$		0			0
$91$	−7.91940	−	7.91940i	−0.830178	−	0.830178i
$92$		0			0
$93$	−0.406230			−0.0421241
$94$		0			0
$95$		0			0
$96$		0			0
$97$	7.16000	+	7.16000i	0.726987	+	0.726987i	0.970019	−	0.243031i	$-0.0781417\pi$
$97$	7.16000	+	7.16000i	0.726987	+	0.726987i	−0.243031	+	0.970019i	$0.578142\pi$
$98$		0			0
$99$	11.5539	+	11.5539i	1.16121	+	1.16121i
$100$		0			0
$101$	8.38846	−	8.38846i	0.834683	−	0.834683i	−0.153470	−	0.988153i	$-0.549045\pi$
$101$	8.38846	−	8.38846i	0.834683	−	0.834683i	0.988153	+	0.153470i	$0.0490448\pi$
$102$		0			0
$103$	−5.16478	+	5.16478i	−0.508901	+	0.508901i	−0.914189	−	0.405288i	$-0.867171\pi$
$103$	−5.16478	+	5.16478i	−0.508901	+	0.508901i	0.405288	+	0.914189i	$0.367171\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			8.97973i			0.868103i	0.900888	+	0.434052i	$0.142916\pi$
$107$			8.97973i			0.868103i	−0.900888	+	0.434052i	$0.857084\pi$
$108$		0			0
$109$	−10.9081	+	10.9081i	−1.04481	+	1.04481i	−0.0458592	+	0.998948i	$0.514603\pi$
$109$	−10.9081	+	10.9081i	−1.04481	+	1.04481i	−0.998948	+	0.0458592i	$0.985397\pi$
$110$		0			0
$111$			2.54519i			0.241579i
$112$		0			0
$113$	4.29684	−	4.29684i	0.404212	−	0.404212i	−0.475502	−	0.879715i	$-0.657734\pi$
$113$	4.29684	−	4.29684i	0.404212	−	0.404212i	0.879715	+	0.475502i	$0.157734\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−14.0299			−1.29707
$118$		0			0
$119$		−	0.663798i		−	0.0608503i
$120$		0			0
$121$			24.2140i			2.20127i
$122$		0			0
$123$	1.66188			0.149846
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−0.759686	+	0.759686i	−0.0674112	+	0.0674112i	−0.740009	−	0.672597i	$-0.765179\pi$
$127$	−0.759686	+	0.759686i	−0.0674112	+	0.0674112i	0.672597	+	0.740009i	$0.265179\pi$
$128$		0			0
$129$		−	2.22988i		−	0.196330i
$130$		0			0
$131$	−7.59995	+	7.59995i	−0.664010	+	0.664010i	−0.956323	−	0.292312i	$-0.905575\pi$
$131$	−7.59995	+	7.59995i	−0.664010	+	0.664010i	0.292312	+	0.956323i	$0.405575\pi$
$132$		0			0
$133$			2.62593i			0.227697i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	12.7789	−	12.7789i	1.09178	−	1.09178i	0.0964376	−	0.995339i	$-0.469255\pi$
$137$	12.7789	−	12.7789i	1.09178	−	1.09178i	0.995339	−	0.0964376i	$-0.0307448\pi$
$138$		0			0
$139$	7.74227	−	7.74227i	0.656691	−	0.656691i	−0.297905	−	0.954596i	$-0.596288\pi$
$139$	7.74227	−	7.74227i	0.656691	−	0.656691i	0.954596	+	0.297905i	$0.0962877\pi$
$140$		0			0
$141$	−2.13169	−	2.13169i	−0.179521	−	0.179521i
$142$		0			0
$143$	−21.3802	−	21.3802i	−1.78790	−	1.78790i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	1.07667			0.0888024
$148$		0			0
$149$	−9.57165	−	9.57165i	−0.784140	−	0.784140i	0.196386	−	0.980527i	$-0.437079\pi$
$149$	−9.57165	−	9.57165i	−0.784140	−	0.784140i	−0.980527	+	0.196386i	$0.937079\pi$
$150$		0			0
$151$	9.68791			0.788391			0.394195	−	0.919027i	$-0.371023\pi$
$151$	9.68791			0.788391			0.394195	+	0.919027i	$0.371023\pi$
$152$		0			0
$153$	−0.587989	−	0.587989i	−0.0475361	−	0.0475361i
$154$		0			0
$155$		0			0
$156$		0			0
$157$			9.97637i			0.796201i	0.917342	+	0.398101i	$0.130331\pi$
$157$			9.97637i			0.796201i	−0.917342	+	0.398101i	$0.869669\pi$
$158$		0			0
$159$	0.499732			0.0396313
$160$		0			0
$161$	5.30761			0.418298
$162$		0			0
$163$		−	9.48267i		−	0.742740i	−0.928485	−	0.371370i	$-0.878888\pi$
$163$		−	9.48267i		−	0.742740i	0.928485	−	0.371370i	$-0.121112\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−9.43528	−	9.43528i	−0.730124	−	0.730124i	0.240520	−	0.970644i	$-0.422682\pi$
$167$	−9.43528	−	9.43528i	−0.730124	−	0.730124i	−0.970644	+	0.240520i	$0.922682\pi$
$168$		0			0
$169$	12.9621			0.997082
$170$		0			0
$171$	2.32603	+	2.32603i	0.177876	+	0.177876i
$172$		0			0
$173$	−8.94716			−0.680240			−0.340120	−	0.940382i	$-0.610468\pi$
$173$	−8.94716			−0.680240			−0.340120	+	0.940382i	$0.610468\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−3.79815	−	3.79815i	−0.285487	−	0.285487i
$178$		0			0
$179$	−3.02430	−	3.02430i	−0.226047	−	0.226047i	0.584992	−	0.811039i	$-0.301098\pi$
$179$	−3.02430	−	3.02430i	−0.226047	−	0.226047i	−0.811039	+	0.584992i	$0.801098\pi$
$180$		0			0
$181$	−1.54845	+	1.54845i	−0.115095	+	0.115095i	−0.762309	−	0.647213i	$-0.775934\pi$
$181$	−1.54845	+	1.54845i	−0.115095	+	0.115095i	0.647213	+	0.762309i	$0.275934\pi$
$182$		0			0
$183$	0.944773	−	0.944773i	0.0698396	−	0.0698396i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		−	1.79208i		−	0.131050i
$188$		0			0
$189$	−4.43979	+	4.43979i	−0.322947	+	0.322947i
$190$		0			0
$191$		−	20.1005i		−	1.45442i	−0.686415	−	0.727210i	$-0.740817\pi$
$191$		−	20.1005i		−	1.45442i	0.686415	−	0.727210i	$-0.259183\pi$
$192$		0			0
$193$	−3.82483	+	3.82483i	−0.275317	+	0.275317i	−0.831236	−	0.555919i	$-0.812366\pi$
$193$	−3.82483	+	3.82483i	−0.275317	+	0.275317i	0.555919	+	0.831236i	$0.312366\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	1.11758			0.0796246			0.0398123	−	0.999207i	$-0.487324\pi$
$197$	1.11758			0.0796246			0.0398123	+	0.999207i	$0.487324\pi$
$198$		0			0
$199$		−	25.5830i		−	1.81353i	−0.421635	−	0.906766i	$-0.638544\pi$
$199$		−	25.5830i		−	1.81353i	0.421635	−	0.906766i	$-0.361456\pi$
$200$		0			0
$201$			5.47232i			0.385988i
$202$		0			0
$203$	−6.98637			−0.490347
$204$		0			0
$205$		0			0
$206$		0			0
$207$	4.70145	−	4.70145i	0.326773	−	0.326773i
$208$		0			0
$209$			7.08929i			0.490376i
$210$		0			0
$211$	−0.411613	+	0.411613i	−0.0283366	+	0.0283366i	−0.721133	−	0.692797i	$-0.756378\pi$
$211$	−0.411613	+	0.411613i	−0.0283366	+	0.0283366i	0.692797	+	0.721133i	$0.256378\pi$
$212$		0			0
$213$			5.22957i			0.358324i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−1.27171	+	1.27171i	−0.0863290	+	0.0863290i
$218$		0			0
$219$	−1.34475	+	1.34475i	−0.0908701	+	0.0908701i
$220$		0			0
$221$	1.08806	+	1.08806i	0.0731909	+	0.0731909i
$222$		0			0
$223$	16.7466	+	16.7466i	1.12143	+	1.12143i	0.991526	+	0.129908i	$0.0414682\pi$
$223$	16.7466	+	16.7466i	1.12143	+	1.12143i	0.129908	+	0.991526i	$0.458532\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	13.7807			0.914659			0.457330	−	0.889297i	$-0.348806\pi$
$227$	13.7807			0.914659			0.457330	+	0.889297i	$0.348806\pi$
$228$		0			0
$229$	7.90971	+	7.90971i	0.522688	+	0.522688i	0.918382	−	0.395694i	$-0.129496\pi$
$229$	7.90971	+	7.90971i	0.522688	+	0.522688i	−0.395694	+	0.918382i	$0.629496\pi$
$230$		0			0
$231$	−6.47594			−0.426086
$232$		0			0
$233$	−1.67997	−	1.67997i	−0.110058	−	0.110058i	0.649933	−	0.759991i	$-0.274797\pi$
$233$	−1.67997	−	1.67997i	−0.110058	−	0.110058i	−0.759991	+	0.649933i	$0.774797\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		−	4.13083i		−	0.268326i
$238$		0			0
$239$	11.7685			0.761241			0.380620	−	0.924731i	$-0.375710\pi$
$239$	11.7685			0.761241			0.380620	+	0.924731i	$0.375710\pi$
$240$		0			0
$241$	−13.2730			−0.854991			−0.427495	−	0.904018i	$-0.640604\pi$
$241$	−13.2730			−0.854991			−0.427495	+	0.904018i	$0.640604\pi$
$242$		0			0
$243$			11.9667i			0.767665i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−4.30427	−	4.30427i	−0.273874	−	0.273874i
$248$		0			0
$249$	−4.55396			−0.288596
$250$		0			0
$251$	−10.3795	−	10.3795i	−0.655149	−	0.655149i	0.299079	−	0.954228i	$-0.403321\pi$
$251$	−10.3795	−	10.3795i	−0.655149	−	0.655149i	−0.954228	+	0.299079i	$0.903321\pi$
$252$		0			0
$253$	14.3291			0.900863
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−20.4353	−	20.4353i	−1.27472	−	1.27472i	−0.943582	−	0.331140i	$-0.892567\pi$
$257$	−20.4353	−	20.4353i	−1.27472	−	1.27472i	−0.331140	−	0.943582i	$-0.607433\pi$
$258$		0			0
$259$	7.96772	+	7.96772i	0.495090	+	0.495090i
$260$		0			0
$261$	−6.18848	+	6.18848i	−0.383058	+	0.383058i
$262$		0			0
$263$	14.0611	−	14.0611i	0.867047	−	0.867047i	−0.125098	−	0.992144i	$-0.539924\pi$
$263$	14.0611	−	14.0611i	0.867047	−	0.867047i	0.992144	+	0.125098i	$0.0399244\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$			2.11057i			0.129165i
$268$		0			0
$269$	6.61443	−	6.61443i	0.403289	−	0.403289i	−0.476101	−	0.879390i	$-0.657950\pi$
$269$	6.61443	−	6.61443i	0.403289	−	0.403289i	0.879390	+	0.476101i	$0.157950\pi$
$270$		0			0
$271$			10.6219i			0.645237i	0.946529	+	0.322619i	$0.104563\pi$
$271$			10.6219i			0.645237i	−0.946529	+	0.322619i	$0.895437\pi$
$272$		0			0
$273$	3.93188	−	3.93188i	0.237968	−	0.237968i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	8.28511			0.497804			0.248902	−	0.968529i	$-0.419930\pi$
$277$	8.28511			0.497804			0.248902	+	0.968529i	$0.419930\pi$
$278$		0			0
$279$			2.25294i			0.134880i
$280$		0			0
$281$		−	21.0176i		−	1.25380i	−0.779098	−	0.626902i	$-0.784323\pi$
$281$		−	21.0176i		−	1.25380i	0.779098	−	0.626902i	$-0.215677\pi$
$282$		0			0
$283$	14.4748			0.860436			0.430218	−	0.902725i	$-0.358437\pi$
$283$	14.4748			0.860436			0.430218	+	0.902725i	$0.358437\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	5.20251	−	5.20251i	0.307095	−	0.307095i
$288$		0			0
$289$		−	16.9088i		−	0.994635i
$290$		0			0
$291$	−3.55485	+	3.55485i	−0.208389	+	0.208389i
$292$		0			0
$293$		−	11.9165i		−	0.696171i	−0.937463	−	0.348086i	$-0.886832\pi$
$293$		−	11.9165i		−	0.696171i	0.937463	−	0.348086i	$-0.113168\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−11.9862	+	11.9862i	−0.695512	+	0.695512i
$298$		0			0
$299$	−8.69993	+	8.69993i	−0.503130	+	0.503130i
$300$		0			0
$301$	−6.98065	−	6.98065i	−0.402358	−	0.402358i
$302$		0			0
$303$	4.16477	+	4.16477i	0.239260	+	0.239260i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−25.4511			−1.45257			−0.726287	−	0.687392i	$-0.758755\pi$
$307$	−25.4511			−1.45257			−0.726287	+	0.687392i	$0.758755\pi$
$308$		0			0
$309$	−2.56425	−	2.56425i	−0.145875	−	0.145875i
$310$		0			0
$311$	−21.4775			−1.21788			−0.608939	−	0.793217i	$-0.708404\pi$
$311$	−21.4775			−1.21788			−0.608939	+	0.793217i	$0.708404\pi$
$312$		0			0
$313$	18.7965	+	18.7965i	1.06244	+	1.06244i	0.997916	+	0.0645277i	$0.0205541\pi$
$313$	18.7965	+	18.7965i	1.06244	+	1.06244i	0.0645277	+	0.997916i	$0.479446\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	16.2531i		−	0.912864i	−0.889758	−	0.456432i	$-0.849127\pi$
$317$		−	16.2531i		−	0.912864i	0.889758	−	0.456432i	$-0.150873\pi$
$318$		0			0
$319$	−18.8613			−1.05603
$320$		0			0
$321$	−4.45832			−0.248839
$322$		0			0
$323$		−	0.360781i		−	0.0200744i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−5.41574	−	5.41574i	−0.299491	−	0.299491i
$328$		0			0
$329$	−13.3465			−0.735818
$330$		0			0
$331$	8.71558	+	8.71558i	0.479052	+	0.479052i	0.904828	−	0.425777i	$-0.139999\pi$
$331$	8.71558	+	8.71558i	0.479052	+	0.479052i	−0.425777	+	0.904828i	$0.639999\pi$
$332$		0			0
$333$	14.1155			0.773526
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−0.0406874	−	0.0406874i	−0.00221638	−	0.00221638i	0.705998	−	0.708214i	$-0.250499\pi$
$337$	−0.0406874	−	0.0406874i	−0.00221638	−	0.00221638i	−0.708214	+	0.705998i	$0.750499\pi$
$338$		0			0
$339$	2.13333	+	2.13333i	0.115866	+	0.115866i
$340$		0			0
$341$	−3.43326	+	3.43326i	−0.185921	+	0.185921i
$342$		0			0
$343$	14.2503	−	14.2503i	0.769445	−	0.769445i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	35.7094i		−	1.91698i	−0.285124	−	0.958491i	$-0.592035\pi$
$347$		−	35.7094i		−	1.91698i	0.285124	−	0.958491i	$-0.407965\pi$
$348$		0			0
$349$	0.274452	−	0.274452i	0.0146911	−	0.0146911i	−0.699723	−	0.714414i	$-0.746693\pi$
$349$	0.274452	−	0.274452i	0.0146911	−	0.0146911i	0.714414	+	0.699723i	$0.246693\pi$
$350$		0			0
$351$		−	14.5549i		−	0.776884i
$352$		0			0
$353$	15.6215	−	15.6215i	0.831446	−	0.831446i	−0.156268	−	0.987715i	$-0.549946\pi$
$353$	15.6215	−	15.6215i	0.831446	−	0.831446i	0.987715	+	0.156268i	$0.0499464\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	0.329567			0.0174426
$358$		0			0
$359$			0.768787i			0.0405750i	0.999794	+	0.0202875i	$0.00645816\pi$
$359$			0.768787i			0.0405750i	−0.999794	+	0.0202875i	$0.993542\pi$
$360$		0			0
$361$		−	17.5728i		−	0.924883i
$362$		0			0
$363$	−12.0219			−0.630988
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−13.7849	+	13.7849i	−0.719568	+	0.719568i	−0.968517	−	0.248949i	$-0.919915\pi$
$367$	−13.7849	+	13.7849i	−0.719568	+	0.719568i	0.248949	+	0.968517i	$0.419915\pi$
$368$		0			0
$369$		−	9.21671i		−	0.479803i
$370$		0			0
$371$	1.56441	−	1.56441i	0.0812202	−	0.0812202i
$372$		0			0
$373$		−	21.4003i		−	1.10806i	−0.832496	−	0.554031i	$-0.813089\pi$
$373$		−	21.4003i		−	1.10806i	0.832496	−	0.554031i	$-0.186911\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	11.4517	−	11.4517i	0.589791	−	0.589791i
$378$		0			0
$379$	11.3922	−	11.3922i	0.585180	−	0.585180i	−0.351142	−	0.936322i	$-0.614207\pi$
$379$	11.3922	−	11.3922i	0.585180	−	0.585180i	0.936322	+	0.351142i	$0.114207\pi$
$380$		0			0
$381$	−0.377174	−	0.377174i	−0.0193232	−	0.0193232i
$382$		0			0
$383$	−4.42635	−	4.42635i	−0.226176	−	0.226176i	0.584917	−	0.811093i	$-0.301127\pi$
$383$	−4.42635	−	4.42635i	−0.226176	−	0.226176i	−0.811093	+	0.584917i	$0.801127\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−12.3668			−0.628642
$388$		0			0
$389$	−12.3502	−	12.3502i	−0.626180	−	0.626180i	0.320924	−	0.947105i	$-0.396006\pi$
$389$	−12.3502	−	12.3502i	−0.626180	−	0.626180i	−0.947105	+	0.320924i	$0.896006\pi$
$390$		0			0
$391$	−0.729222			−0.0368784
$392$		0			0
$393$	−3.77328	−	3.77328i	−0.190337	−	0.190337i
$394$		0			0
$395$		0			0
$396$		0			0
$397$		−	17.9832i		−	0.902551i	−0.892385	−	0.451275i	$-0.850969\pi$
$397$		−	17.9832i		−	0.902551i	0.892385	−	0.451275i	$-0.149031\pi$
$398$		0			0
$399$	−1.30374			−0.0652686
$400$		0			0
$401$	9.06570			0.452720			0.226360	−	0.974044i	$-0.427317\pi$
$401$	9.06570			0.452720			0.226360	+	0.974044i	$0.427317\pi$
$402$		0			0
$403$		−	4.16902i		−	0.207674i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	21.5107	+	21.5107i	1.06625	+	1.06625i
$408$		0			0
$409$	30.0616			1.48645			0.743226	−	0.669040i	$-0.233295\pi$
$409$	30.0616			1.48645			0.743226	+	0.669040i	$0.233295\pi$
$410$		0			0
$411$	6.34457	+	6.34457i	0.312955	+	0.312955i
$412$		0			0
$413$	−23.7803			−1.17015
$414$		0			0
$415$		0			0
$416$		0			0
$417$	3.84394	+	3.84394i	0.188239	+	0.188239i
$418$		0			0
$419$	−15.3986	−	15.3986i	−0.752271	−	0.752271i	0.222631	−	0.974903i	$-0.428535\pi$
$419$	−15.3986	−	15.3986i	−0.752271	−	0.752271i	−0.974903	+	0.222631i	$0.928535\pi$
$420$		0			0
$421$	−3.86468	+	3.86468i	−0.188353	+	0.188353i	−0.794984	−	0.606631i	$-0.792521\pi$
$421$	−3.86468	+	3.86468i	−0.188353	+	0.188353i	0.606631	+	0.794984i	$0.292521\pi$
$422$		0			0
$423$	−11.8223	+	11.8223i	−0.574819	+	0.574819i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		−	5.91523i		−	0.286258i
$428$		0			0
$429$	10.6150	−	10.6150i	0.512497	−	0.512497i
$430$		0			0
$431$			27.2692i			1.31351i	0.754103	+	0.656756i	$0.228072\pi$
$431$			27.2692i			1.31351i	−0.754103	+	0.656756i	$0.771928\pi$
$432$		0			0
$433$	−19.1435	+	19.1435i	−0.919978	+	0.919978i	−0.997027	−	0.0770497i	$-0.975450\pi$
$433$	−19.1435	+	19.1435i	−0.919978	+	0.919978i	0.0770497	+	0.997027i	$0.475450\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	2.88474			0.137996
$438$		0			0
$439$			30.1995i			1.44134i	0.693276	+	0.720672i	$0.256167\pi$
$439$			30.1995i			1.44134i	−0.693276	+	0.720672i	$0.743833\pi$
$440$		0			0
$441$		−	5.97118i		−	0.284342i
$442$		0			0
$443$	27.7051			1.31631			0.658153	−	0.752884i	$-0.271338\pi$
$443$	27.7051			1.31631			0.658153	+	0.752884i	$0.271338\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	4.75220	−	4.75220i	0.224772	−	0.224772i
$448$		0			0
$449$			9.78315i			0.461695i	0.972990	+	0.230848i	$0.0741499\pi$
$449$			9.78315i			0.461695i	−0.972990	+	0.230848i	$0.925850\pi$
$450$		0			0
$451$	14.0454	−	14.0454i	0.661371	−	0.661371i
$452$		0			0
$453$			4.80992i			0.225990i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−0.557108	+	0.557108i	−0.0260604	+	0.0260604i	−0.720017	−	0.693957i	$-0.755866\pi$
$457$	−0.557108	+	0.557108i	−0.0260604	+	0.0260604i	0.693957	+	0.720017i	$0.255866\pi$
$458$		0			0
$459$	0.609992	−	0.609992i	0.0284720	−	0.0284720i
$460$		0			0
$461$	−12.5791	−	12.5791i	−0.585865	−	0.585865i	0.350644	−	0.936509i	$-0.385963\pi$
$461$	−12.5791	−	12.5791i	−0.585865	−	0.585865i	−0.936509	+	0.350644i	$0.885963\pi$
$462$		0			0
$463$	3.29549	+	3.29549i	0.153154	+	0.153154i	0.779525	−	0.626371i	$-0.215460\pi$
$463$	3.29549	+	3.29549i	0.153154	+	0.153154i	−0.626371	+	0.779525i	$0.715460\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	10.1995			0.471979			0.235989	−	0.971756i	$-0.424167\pi$
$467$	10.1995			0.471979			0.235989	+	0.971756i	$0.424167\pi$
$468$		0			0
$469$	17.1311	+	17.1311i	0.791042	+	0.791042i
$470$		0			0
$471$	−4.95314			−0.228229
$472$		0			0
$473$	−18.8459	−	18.8459i	−0.866534	−	0.866534i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		−	2.77149i		−	0.126898i
$478$		0			0
$479$	5.65795			0.258518			0.129259	−	0.991611i	$-0.458740\pi$
$479$	5.65795			0.258518			0.129259	+	0.991611i	$0.458740\pi$
$480$		0			0
$481$	−26.1205			−1.19099
$482$		0			0
$483$			2.63516i			0.119904i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−19.7470	−	19.7470i	−0.894823	−	0.894823i	0.100149	−	0.994972i	$-0.468068\pi$
$487$	−19.7470	−	19.7470i	−0.894823	−	0.894823i	−0.994972	+	0.100149i	$0.968068\pi$
$488$		0			0
$489$	4.70802			0.212904
$490$		0			0
$491$	4.21405	+	4.21405i	0.190177	+	0.190177i	0.795773	−	0.605595i	$-0.207065\pi$
$491$	4.21405	+	4.21405i	0.190177	+	0.190177i	−0.605595	+	0.795773i	$0.707065\pi$
$492$		0			0
$493$	0.959871			0.0432304
$494$		0			0
$495$		0			0
$496$		0			0
$497$	16.3712	+	16.3712i	0.734348	+	0.734348i
$498$		0			0
$499$	−16.8862	−	16.8862i	−0.755928	−	0.755928i	0.219650	−	0.975579i	$-0.429508\pi$
$499$	−16.8862	−	16.8862i	−0.755928	−	0.755928i	−0.975579	+	0.219650i	$0.929508\pi$
$500$		0			0
$501$	4.68450	−	4.68450i	0.209288	−	0.209288i
$502$		0			0
$503$	−20.3714	+	20.3714i	−0.908317	+	0.908317i	−0.996136	−	0.0878190i	$-0.972010\pi$
$503$	−20.3714	+	20.3714i	−0.908317	+	0.908317i	0.0878190	+	0.996136i	$0.472010\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$			6.43550i			0.285811i
$508$		0			0
$509$	20.6309	−	20.6309i	0.914448	−	0.914448i	−0.0821701	−	0.996618i	$-0.526185\pi$
$509$	20.6309	−	20.6309i	0.914448	−	0.914448i	0.996618	+	0.0821701i	$0.0261851\pi$
$510$		0			0
$511$			8.41952i			0.372458i
$512$		0			0
$513$	−2.41307	+	2.41307i	−0.106540	+	0.106540i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−36.0320			−1.58469
$518$		0			0
$519$		−	4.44215i		−	0.194989i
$520$		0			0
$521$			19.0433i			0.834300i	0.908838	+	0.417150i	$0.136971\pi$
$521$			19.0433i			0.834300i	−0.908838	+	0.417150i	$0.863029\pi$
$522$		0			0
$523$	−19.1782			−0.838603			−0.419301	−	0.907847i	$-0.637725\pi$
$523$	−19.1782			−0.838603			−0.419301	+	0.907847i	$0.637725\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	0.174722	−	0.174722i	0.00761101	−	0.00761101i
$528$		0			0
$529$			17.1693i			0.746490i
$530$		0			0
$531$	−21.0644	+	21.0644i	−0.914118	+	0.914118i
$532$		0			0
$533$			17.0553i			0.738749i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	1.50153	−	1.50153i	0.0647957	−	0.0647957i
$538$		0			0
$539$	9.09950	−	9.09950i	0.391943	−	0.391943i
$540$		0			0
$541$	14.5231	+	14.5231i	0.624398	+	0.624398i	0.946653	−	0.322255i	$-0.104441\pi$
$541$	14.5231	+	14.5231i	0.624398	+	0.624398i	−0.322255	+	0.946653i	$0.604441\pi$
$542$		0			0
$543$	−0.768787	−	0.768787i	−0.0329918	−	0.0329918i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−9.97058			−0.426311			−0.213156	−	0.977018i	$-0.568374\pi$
$547$	−9.97058			−0.426311			−0.213156	+	0.977018i	$0.568374\pi$
$548$		0			0
$549$	−5.23967	−	5.23967i	−0.223624	−	0.223624i
$550$		0			0
$551$	−3.79716			−0.161765
$552$		0			0
$553$	−12.9316	−	12.9316i	−0.549906	−	0.549906i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	11.4424i		−	0.484831i	−0.970173	−	0.242416i	$-0.922060\pi$
$557$		−	11.4424i		−	0.484831i	0.970173	−	0.242416i	$-0.0779397\pi$
$558$		0			0
$559$	22.8846			0.967915
$560$		0			0
$561$	0.889743			0.0375650
$562$		0			0
$563$		−	47.0585i		−	1.98328i	−0.129034	−	0.991640i	$-0.541188\pi$
$563$		−	47.0585i		−	1.98328i	0.129034	−	0.991640i	$-0.458812\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	10.6346	+	10.6346i	0.446612	+	0.446612i
$568$		0			0
$569$	−41.4684			−1.73845			−0.869224	−	0.494419i	$-0.835381\pi$
$569$	−41.4684			−1.73845			−0.869224	+	0.494419i	$0.835381\pi$
$570$		0			0
$571$	−16.1745	−	16.1745i	−0.676881	−	0.676881i	0.282412	−	0.959293i	$-0.408865\pi$
$571$	−16.1745	−	16.1745i	−0.676881	−	0.676881i	−0.959293	+	0.282412i	$0.908865\pi$
$572$		0			0
$573$	9.97963			0.416905
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−20.0316	−	20.0316i	−0.833926	−	0.833926i	0.154125	−	0.988051i	$-0.450744\pi$
$577$	−20.0316	−	20.0316i	−0.833926	−	0.833926i	−0.988051	+	0.154125i	$0.950744\pi$
$578$		0			0
$579$	−1.89898	−	1.89898i	−0.0789189	−	0.0789189i
$580$		0			0
$581$	−14.2562	+	14.2562i	−0.591447	+	0.591447i
$582$		0			0
$583$	4.22349	−	4.22349i	0.174919	−	0.174919i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			29.1190i			1.20187i	0.799298	+	0.600935i	$0.205205\pi$
$587$			29.1190i			1.20187i	−0.799298	+	0.600935i	$0.794795\pi$
$588$		0			0
$589$	−0.691185	+	0.691185i	−0.0284798	+	0.0284798i
$590$		0			0
$591$			0.554866i			0.0228242i
$592$		0			0
$593$	10.3431	−	10.3431i	0.424740	−	0.424740i	−0.462092	−	0.886832i	$-0.652901\pi$
$593$	10.3431	−	10.3431i	0.424740	−	0.424740i	0.886832	+	0.462092i	$0.152901\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	12.7016			0.519843
$598$		0			0
$599$			2.59479i			0.106020i	0.998594	+	0.0530101i	$0.0168816\pi$
$599$			2.59479i			0.106020i	−0.998594	+	0.0530101i	$0.983118\pi$
$600$		0			0
$601$		−	14.4092i		−	0.587765i	−0.955842	−	0.293882i	$-0.905053\pi$
$601$		−	14.4092i		−	0.587765i	0.955842	−	0.293882i	$-0.0949474\pi$
$602$		0			0
$603$	30.3493			1.23592
$604$		0			0
$605$		0			0
$606$		0			0
$607$	11.8502	−	11.8502i	0.480985	−	0.480985i	−0.424461	−	0.905446i	$-0.639536\pi$
$607$	11.8502	−	11.8502i	0.480985	−	0.480985i	0.905446	+	0.424461i	$0.139536\pi$
$608$		0			0
$609$		−	3.46864i		−	0.140557i
$610$		0			0
$611$	21.8769	−	21.8769i	0.885045	−	0.885045i
$612$		0			0
$613$			16.8256i			0.679579i	0.940502	+	0.339789i	$0.110356\pi$
$613$			16.8256i			0.679579i	−0.940502	+	0.339789i	$0.889644\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−22.4849	+	22.4849i	−0.905209	+	0.905209i	−0.995881	−	0.0906720i	$-0.971098\pi$
$617$	−22.4849	+	22.4849i	−0.905209	+	0.905209i	0.0906720	+	0.995881i	$0.471098\pi$
$618$		0			0
$619$	−14.1269	+	14.1269i	−0.567809	+	0.567809i	−0.931514	−	0.363705i	$-0.881512\pi$
$619$	−14.1269	+	14.1269i	−0.567809	+	0.567809i	0.363705	+	0.931514i	$0.381512\pi$
$620$		0			0
$621$	4.87738	+	4.87738i	0.195723	+	0.195723i
$622$		0			0
$623$	6.60715	+	6.60715i	0.264710	+	0.264710i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−3.51974			−0.140565
$628$		0			0
$629$	−1.09470	−	1.09470i	−0.0436486	−	0.0436486i
$630$		0			0
$631$	33.9235			1.35047			0.675236	−	0.737601i	$-0.264042\pi$
$631$	33.9235			1.35047			0.675236	+	0.737601i	$0.264042\pi$
$632$		0			0
$633$	−0.204361	−	0.204361i	−0.00812261	−	0.00812261i
$634$		0			0
$635$		0			0
$636$		0			0
$637$			11.0496i			0.437799i
$638$		0			0
$639$	29.0030			1.14734
$640$		0			0
$641$	18.8495			0.744509			0.372254	−	0.928131i	$-0.378585\pi$
$641$	18.8495			0.744509			0.372254	+	0.928131i	$0.378585\pi$
$642$		0			0
$643$			16.4916i			0.650364i	0.945652	+	0.325182i	$0.105426\pi$
$643$			16.4916i			0.650364i	−0.945652	+	0.325182i	$0.894574\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−0.316870	−	0.316870i	−0.0124574	−	0.0124574i	0.700851	−	0.713308i	$-0.252804\pi$
$647$	−0.316870	−	0.316870i	−0.0124574	−	0.0124574i	−0.713308	+	0.700851i	$0.752804\pi$
$648$		0			0
$649$	−64.2002			−2.52008
$650$		0			0
$651$	−0.631386	−	0.631386i	−0.0247460	−	0.0247460i
$652$		0			0
$653$	17.0751			0.668200			0.334100	−	0.942538i	$-0.391568\pi$
$653$	17.0751			0.668200			0.334100	+	0.942538i	$0.391568\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	7.45796	+	7.45796i	0.290963	+	0.290963i
$658$		0			0
$659$	−7.42245	−	7.42245i	−0.289138	−	0.289138i	0.547601	−	0.836739i	$-0.315541\pi$
$659$	−7.42245	−	7.42245i	−0.289138	−	0.289138i	−0.836739	+	0.547601i	$0.815541\pi$
$660$		0			0
$661$	31.7614	−	31.7614i	1.23538	−	1.23538i	0.273507	−	0.961870i	$-0.411816\pi$
$661$	31.7614	−	31.7614i	1.23538	−	1.23538i	0.961870	−	0.273507i	$-0.0881837\pi$
$662$		0			0
$663$	−0.540209	+	0.540209i	−0.0209800	+	0.0209800i
$664$		0			0
$665$		0			0
$666$		0			0
$667$			7.67495i			0.297175i
$668$		0			0
$669$	−8.31446	+	8.31446i	−0.321456	+	0.321456i
$670$		0			0
$671$		−	15.9695i		−	0.616496i
$672$		0			0
$673$	−4.14672	+	4.14672i	−0.159844	+	0.159844i	−0.782498	−	0.622653i	$-0.786055\pi$
$673$	−4.14672	+	4.14672i	−0.159844	+	0.159844i	0.622653	+	0.782498i	$0.286055\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	25.2618			0.970890			0.485445	−	0.874267i	$-0.338658\pi$
$677$	25.2618			0.970890			0.485445	+	0.874267i	$0.338658\pi$
$678$		0			0
$679$			22.2569i			0.854143i
$680$		0			0
$681$			6.84196i			0.262184i
$682$		0			0
$683$	−8.20306			−0.313881			−0.156941	−	0.987608i	$-0.550163\pi$
$683$	−8.20306			−0.313881			−0.156941	+	0.987608i	$0.550163\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−3.92707	+	3.92707i	−0.149827	+	0.149827i
$688$		0			0
$689$			5.12859i			0.195384i
$690$		0			0
$691$	7.89158	−	7.89158i	0.300210	−	0.300210i	−0.540886	−	0.841096i	$-0.681911\pi$
$691$	7.89158	−	7.89158i	0.300210	−	0.300210i	0.841096	+	0.540886i	$0.181911\pi$
$692$		0			0
$693$			35.9153i			1.36431i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−0.714783	+	0.714783i	−0.0270744	+	0.0270744i
$698$		0			0
$699$	0.834083	−	0.834083i	0.0315479	−	0.0315479i
$700$		0			0
$701$	1.50228	+	1.50228i	0.0567405	+	0.0567405i	0.734908	−	0.678167i	$-0.237225\pi$
$701$	1.50228	+	1.50228i	0.0567405	+	0.0567405i	−0.678167	+	0.734908i	$0.737225\pi$
$702$		0			0
$703$	4.33054	+	4.33054i	0.163329	+	0.163329i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	26.0756			0.980675
$708$		0			0
$709$	36.0738	+	36.0738i	1.35478	+	1.35478i	0.880228	+	0.474551i	$0.157390\pi$
$709$	36.0738	+	36.0738i	1.35478	+	1.35478i	0.474551	+	0.880228i	$0.342610\pi$
$710$		0			0
$711$	−22.9094			−0.859170
$712$		0			0
$713$	1.39704	+	1.39704i	0.0523197	+	0.0523197i
$714$		0			0
$715$		0			0
$716$		0			0
$717$			5.84291i			0.218207i
$718$		0			0
$719$	−35.0340			−1.30655			−0.653274	−	0.757121i	$-0.726605\pi$
$719$	−35.0340			−1.30655			−0.653274	+	0.757121i	$0.726605\pi$
$720$		0			0
$721$	−16.0548			−0.597911
$722$		0			0
$723$		−	6.58989i		−	0.245081i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	25.4241	+	25.4241i	0.942928	+	0.942928i	0.998457	−	0.0555295i	$-0.0176847\pi$
$727$	25.4241	+	25.4241i	0.942928	+	0.942928i	−0.0555295	+	0.998457i	$0.517685\pi$
$728$		0			0
$729$	14.5855			0.540203
$730$		0			0
$731$	0.959085	+	0.959085i	0.0354731	+	0.0354731i
$732$		0			0
$733$	7.37554			0.272422			0.136211	−	0.990680i	$-0.456508\pi$
$733$	7.37554			0.272422			0.136211	+	0.990680i	$0.456508\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	46.2494	+	46.2494i	1.70362	+	1.70362i
$738$		0			0
$739$	5.55025	+	5.55025i	0.204169	+	0.204169i	0.801784	−	0.597614i	$-0.203885\pi$
$739$	5.55025	+	5.55025i	0.204169	+	0.204169i	−0.597614	+	0.801784i	$0.703885\pi$
$740$		0			0
$741$	2.13702	−	2.13702i	0.0785053	−	0.0785053i
$742$		0			0
$743$	−6.78835	+	6.78835i	−0.249040	+	0.249040i	−0.820577	−	0.571536i	$-0.806348\pi$
$743$	−6.78835	+	6.78835i	−0.249040	+	0.249040i	0.571536	+	0.820577i	$0.306348\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$			25.2561i			0.924073i
$748$		0			0
$749$	−13.9568	+	13.9568i	−0.509970	+	0.509970i
$750$		0			0
$751$			3.93385i			0.143548i	0.997421	+	0.0717742i	$0.0228661\pi$
$751$			3.93385i			0.143548i	−0.997421	+	0.0717742i	$0.977134\pi$
$752$		0			0
$753$	5.15330	−	5.15330i	0.187797	−	0.187797i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	21.8327			0.793525			0.396762	−	0.917921i	$-0.370134\pi$
$757$	21.8327			0.793525			0.396762	+	0.917921i	$0.370134\pi$
$758$		0			0
$759$			7.11421i			0.258230i
$760$		0			0
$761$			4.27291i			0.154893i	0.996997	+	0.0774464i	$0.0246767\pi$
$761$			4.27291i			0.154893i	−0.996997	+	0.0774464i	$0.975323\pi$
$762$		0			0
$763$	−33.9080			−1.22755
$764$		0			0
$765$		0			0
$766$		0			0
$767$	38.9793	−	38.9793i	1.40746	−	1.40746i
$768$		0			0
$769$			26.1800i			0.944074i	0.881579	+	0.472037i	$0.156481\pi$
$769$			26.1800i			0.944074i	−0.881579	+	0.472037i	$0.843519\pi$
$770$		0			0
$771$	10.1459	−	10.1459i	0.365395	−	0.365395i
$772$		0			0
$773$		−	15.0077i		−	0.539791i	−0.962890	−	0.269895i	$-0.913011\pi$
$773$		−	15.0077i		−	0.539791i	0.962890	−	0.269895i	$-0.0869891\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−3.95587	+	3.95587i	−0.141916	+	0.141916i
$778$		0			0
$779$	2.82762	−	2.82762i	0.101310	−	0.101310i
$780$		0			0
$781$	44.1977	+	44.1977i	1.58152	+	1.58152i
$782$		0			0
$783$	−6.42007	−	6.42007i	−0.229434	−	0.229434i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−42.9223			−1.53001			−0.765007	−	0.644022i	$-0.777264\pi$
$787$	−42.9223			−1.53001			−0.765007	+	0.644022i	$0.777264\pi$
$788$		0			0
$789$	6.98118	+	6.98118i	0.248536	+	0.248536i
$790$		0			0
$791$	13.3568			0.474912
$792$		0			0
$793$	9.69591	+	9.69591i	0.344312	+	0.344312i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	0.280831i		−	0.00994753i	−0.999988	−	0.00497377i	$-0.998417\pi$
$797$		−	0.280831i		−	0.00994753i	0.999988	−	0.00497377i	$-0.00158321\pi$
$798$		0			0
$799$	1.83371			0.0648719
$800$		0			0
$801$	11.7052			0.413581
$802$		0			0
$803$			22.7304i			0.802139i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	3.28398	+	3.28398i	0.115602	+	0.115602i
$808$		0			0
$809$	16.5787			0.582876			0.291438	−	0.956590i	$-0.405866\pi$
$809$	16.5787			0.582876			0.291438	+	0.956590i	$0.405866\pi$
$810$		0			0
$811$	−7.25384	−	7.25384i	−0.254717	−	0.254717i	0.568184	−	0.822901i	$-0.307646\pi$
$811$	−7.25384	−	7.25384i	−0.254717	−	0.254717i	−0.822901	+	0.568184i	$0.807646\pi$
$812$		0			0
$813$	−5.27366			−0.184955
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−3.79405	−	3.79405i	−0.132737	−	0.132737i
$818$		0			0
$819$	−21.8061	−	21.8061i	−0.761965	−	0.761965i
$820$		0			0
$821$	−15.3525	+	15.3525i	−0.535806	+	0.535806i	−0.922294	−	0.386489i	$-0.873688\pi$
$821$	−15.3525	+	15.3525i	−0.535806	+	0.535806i	0.386489	+	0.922294i	$0.373688\pi$
$822$		0			0
$823$	−26.7794	+	26.7794i	−0.933472	+	0.933472i	−0.997921	−	0.0644492i	$-0.979471\pi$
$823$	−26.7794	+	26.7794i	−0.933472	+	0.933472i	0.0644492	+	0.997921i	$0.479471\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	39.4186i		−	1.37072i	−0.728205	−	0.685359i	$-0.759645\pi$
$827$		−	39.4186i		−	1.37072i	0.728205	−	0.685359i	$-0.240355\pi$
$828$		0			0
$829$	20.7102	−	20.7102i	0.719296	−	0.719296i	−0.249165	−	0.968461i	$-0.580156\pi$
$829$	20.7102	−	20.7102i	0.719296	−	0.719296i	0.968461	+	0.249165i	$0.0801561\pi$
$830$		0			0
$831$			4.11345i			0.142694i
$832$		0			0
$833$	−0.463083	+	0.463083i	−0.0160449	+	0.0160449i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−2.33725			−0.0807870
$838$		0			0
$839$			31.8706i			1.10029i	0.835068	+	0.550147i	$0.185428\pi$
$839$			31.8706i			1.10029i	−0.835068	+	0.550147i	$0.814572\pi$
$840$		0			0
$841$			18.8975i			0.651638i
$842$		0			0
$843$	10.4350			0.359399
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−37.6347	+	37.6347i	−1.29314	+	1.29314i
$848$		0			0
$849$			7.18654i			0.246642i
$850$		0			0
$851$	8.75302	−	8.75302i	0.300050	−	0.300050i
$852$		0			0
$853$		−	26.5538i		−	0.909185i	−0.890700	−	0.454592i	$-0.849785\pi$
$853$		−	26.5538i		−	0.909185i	0.890700	−	0.454592i	$-0.150215\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−20.7249	+	20.7249i	−0.707951	+	0.707951i	−0.966104	−	0.258153i	$-0.916886\pi$
$857$	−20.7249	+	20.7249i	−0.707951	+	0.707951i	0.258153	+	0.966104i	$0.416886\pi$
$858$		0			0
$859$	35.9248	−	35.9248i	1.22574	−	1.22574i	0.260176	−	0.965561i	$-0.416219\pi$
$859$	35.9248	−	35.9248i	1.22574	−	1.22574i	0.965561	−	0.260176i	$-0.0837807\pi$
$860$		0			0
$861$	2.58298	+	2.58298i	0.0880278	+	0.0880278i
$862$		0			0
$863$	−9.19232	−	9.19232i	−0.312910	−	0.312910i	0.533126	−	0.846036i	$-0.321017\pi$
$863$	−9.19232	−	9.19232i	−0.312910	−	0.312910i	−0.846036	+	0.533126i	$0.821017\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	8.39500			0.285109
$868$		0			0
$869$	−34.9117	−	34.9117i	−1.18430	−	1.18430i
$870$		0			0
$871$	−56.1608			−1.90293
$872$		0			0
$873$	19.7151	+	19.7151i	0.667253	+	0.667253i
$874$		0			0
$875$		0			0
$876$		0			0
$877$			17.9106i			0.604799i	0.953181	+	0.302399i	$0.0977877\pi$
$877$			17.9106i			0.604799i	−0.953181	+	0.302399i	$0.902212\pi$
$878$		0			0
$879$	5.91641			0.199556
$880$		0			0
$881$	6.01537			0.202663			0.101332	−	0.994853i	$-0.467690\pi$
$881$	6.01537			0.202663			0.101332	+	0.994853i	$0.467690\pi$
$882$		0			0
$883$		−	19.8374i		−	0.667580i	−0.942647	−	0.333790i	$-0.891672\pi$
$883$		−	19.8374i		−	0.667580i	0.942647	−	0.333790i	$-0.108328\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	14.3740	+	14.3740i	0.482632	+	0.482632i	0.905971	−	0.423339i	$-0.139142\pi$
$887$	14.3740	+	14.3740i	0.482632	+	0.482632i	−0.423339	+	0.905971i	$0.639142\pi$
$888$		0			0
$889$	−2.36149			−0.0792019
$890$		0			0
$891$	28.7106	+	28.7106i	0.961842	+	0.961842i
$892$		0			0
$893$	−7.25398			−0.242745
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−4.31941	−	4.31941i	−0.144221	−	0.144221i
$898$		0			0
$899$	−1.83892	−	1.83892i	−0.0613315	−	0.0613315i
$900$		0			0
$901$	−0.214938	+	0.214938i	−0.00716061	+	0.00716061i
$902$		0			0
$903$	3.46580	−	3.46580i	0.115335	−	0.115335i
$904$		0			0
$905$		0			0
$906$		0			0
$907$		−	39.0417i		−	1.29636i	−0.761487	−	0.648180i	$-0.775531\pi$
$907$		−	39.0417i		−	1.29636i	0.761487	−	0.648180i	$-0.224469\pi$
$908$		0			0
$909$	23.0976	−	23.0976i	0.766100	−	0.766100i
$910$		0			0
$911$		−	14.0166i		−	0.464392i	−0.972669	−	0.232196i	$-0.925409\pi$
$911$		−	14.0166i		−	0.464392i	0.972669	−	0.232196i	$-0.0745911\pi$
$912$		0			0
$913$	−38.4879	+	38.4879i	−1.27376	+	1.27376i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−23.6245			−0.780150
$918$		0			0
$919$			8.15149i			0.268893i	0.990921	+	0.134446i	$0.0429256\pi$
$919$			8.15149i			0.268893i	−0.990921	+	0.134446i	$0.957074\pi$
$920$		0			0
$921$		−	12.6362i		−	0.416376i
$922$		0			0
$923$	−53.6695			−1.76655
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−14.2212	+	14.2212i	−0.467086	+	0.467086i
$928$		0			0
$929$		−	13.4779i		−	0.442196i	−0.975252	−	0.221098i	$-0.929036\pi$
$929$		−	13.4779i		−	0.442196i	0.975252	−	0.221098i	$-0.0709641\pi$
$930$		0			0
$931$	1.83192	−	1.83192i	0.0600386	−	0.0600386i
$932$		0			0
$933$		−	10.6633i		−	0.349101i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−15.8564	+	15.8564i	−0.518005	+	0.518005i	−0.916967	−	0.398963i	$-0.869370\pi$
$937$	−15.8564	+	15.8564i	−0.518005	+	0.518005i	0.398963	+	0.916967i	$0.369370\pi$
$938$		0			0
$939$	−9.33225	+	9.33225i	−0.304546	+	0.304546i
$940$		0			0
$941$	15.7073	+	15.7073i	0.512044	+	0.512044i	0.915152	−	0.403108i	$-0.132070\pi$
$941$	15.7073	+	15.7073i	0.512044	+	0.512044i	−0.403108	+	0.915152i	$0.632070\pi$
$942$		0			0
$943$	−5.71527	−	5.71527i	−0.186115	−	0.186115i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−33.6925			−1.09486			−0.547430	−	0.836852i	$-0.684394\pi$
$947$	−33.6925			−1.09486			−0.547430	+	0.836852i	$0.684394\pi$
$948$		0			0
$949$	−13.8008	−	13.8008i	−0.447993	−	0.447993i
$950$		0			0
$951$	8.06945			0.261670
$952$		0			0
$953$	−33.5702	−	33.5702i	−1.08745	−	1.08745i	−0.995791	−	0.0916550i	$-0.970784\pi$
$953$	−33.5702	−	33.5702i	−1.08745	−	1.08745i	−0.0916550	−	0.995791i	$-0.529216\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		−	9.36440i		−	0.302708i
$958$		0			0
$959$	39.7234			1.28274
$960$		0			0
$961$	30.3305			0.978404
$962$		0			0
$963$			24.7257i			0.796774i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	28.6436	+	28.6436i	0.921115	+	0.921115i	0.997108	−	0.0759933i	$-0.0242128\pi$
$967$	28.6436	+	28.6436i	0.921115	+	0.921115i	−0.0759933	+	0.997108i	$0.524213\pi$
$968$		0			0
$969$	0.179123			0.00575427
$970$		0			0
$971$	35.7115	+	35.7115i	1.14604	+	1.14604i	0.987325	+	0.158713i	$0.0507345\pi$
$971$	35.7115	+	35.7115i	1.14604	+	1.14604i	0.158713	+	0.987325i	$0.449266\pi$
$972$		0			0
$973$	24.0669			0.771551
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−7.12822	−	7.12822i	−0.228052	−	0.228052i	0.583826	−	0.811879i	$-0.301555\pi$
$977$	−7.12822	−	7.12822i	−0.228052	−	0.228052i	−0.811879	+	0.583826i	$0.801555\pi$
$978$		0			0
$979$	17.8375	+	17.8375i	0.570090	+	0.570090i
$980$		0			0
$981$	−30.0355	+	30.0355i	−0.958959	+	0.958959i
$982$		0			0
$983$	−23.9941	+	23.9941i	−0.765292	+	0.765292i	−0.977274	−	0.211982i	$-0.932008\pi$
$983$	−23.9941	+	23.9941i	−0.765292	+	0.765292i	0.211982	+	0.977274i	$0.432008\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		−	6.62639i		−	0.210920i
$988$		0			0
$989$	−7.66866	+	7.66866i	−0.243849	+	0.243849i
$990$		0			0
$991$			40.6040i			1.28983i	0.764255	+	0.644914i	$0.223107\pi$
$991$			40.6040i			1.28983i	−0.764255	+	0.644914i	$0.776893\pi$
$992$		0			0
$993$	−4.32718	+	4.32718i	−0.137319	+	0.137319i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	54.9087			1.73898			0.869488	−	0.493953i	$-0.164449\pi$
$997$	54.9087			1.73898			0.869488	+	0.493953i	$0.164449\pi$
$998$		0			0
$999$			14.6437i			0.463308i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1600.2.j.d.1007.6		18
4.3	odd	2		400.2.j.d.307.4		18
5.2	odd	4		320.2.s.b.303.6		18
5.3	odd	4		1600.2.s.d.943.4		18
5.4	even	2		320.2.j.b.47.4		18
16.5	even	4		400.2.s.d.107.1		18
16.11	odd	4		1600.2.s.d.207.4		18
20.3	even	4		400.2.s.d.243.1		18
20.7	even	4		80.2.s.b.3.9	yes	18
20.19	odd	2		80.2.j.b.67.6	yes	18
40.19	odd	2		640.2.j.d.607.4		18
40.27	even	4		640.2.s.d.223.6		18
40.29	even	2		640.2.j.c.607.6		18
40.37	odd	4		640.2.s.c.223.4		18
60.47	odd	4		720.2.z.g.163.1		18
60.59	even	2		720.2.bd.g.307.4		18
80.19	odd	4		640.2.s.c.287.4		18
80.27	even	4		320.2.j.b.143.6		18
80.29	even	4		640.2.s.d.287.6		18
80.37	odd	4		80.2.j.b.43.6	✓	18
80.43	even	4	inner	1600.2.j.d.143.4		18
80.53	odd	4		400.2.j.d.43.4		18
80.59	odd	4		320.2.s.b.207.6		18
80.67	even	4		640.2.j.c.543.4		18
80.69	even	4		80.2.s.b.27.9	yes	18
80.77	odd	4		640.2.j.d.543.6		18
240.149	odd	4		720.2.z.g.667.1		18
240.197	even	4		720.2.bd.g.523.4		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
80.2.j.b.43.6	✓	18	80.37	odd	4
80.2.j.b.67.6	yes	18	20.19	odd	2
80.2.s.b.3.9	yes	18	20.7	even	4
80.2.s.b.27.9	yes	18	80.69	even	4
320.2.j.b.47.4		18	5.4	even	2
320.2.j.b.143.6		18	80.27	even	4
320.2.s.b.207.6		18	80.59	odd	4
320.2.s.b.303.6		18	5.2	odd	4
400.2.j.d.43.4		18	80.53	odd	4
400.2.j.d.307.4		18	4.3	odd	2
400.2.s.d.107.1		18	16.5	even	4
400.2.s.d.243.1		18	20.3	even	4
640.2.j.c.543.4		18	80.67	even	4
640.2.j.c.607.6		18	40.29	even	2
640.2.j.d.543.6		18	80.77	odd	4
640.2.j.d.607.4		18	40.19	odd	2
640.2.s.c.223.4		18	40.37	odd	4
640.2.s.c.287.4		18	80.19	odd	4
640.2.s.d.223.6		18	40.27	even	4
640.2.s.d.287.6		18	80.29	even	4
720.2.z.g.163.1		18	60.47	odd	4
720.2.z.g.667.1		18	240.149	odd	4
720.2.bd.g.307.4		18	60.59	even	2
720.2.bd.g.523.4		18	240.197	even	4
1600.2.j.d.143.4		18	80.43	even	4	inner
1600.2.j.d.1007.6		18	1.1	even	1	trivial
1600.2.s.d.207.4		18	16.11	odd	4
1600.2.s.d.943.4		18	5.3	odd	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:	\( N \)	\(=\)	\( 1600 = 2^{6} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1600.j (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+0.496487i q^{3} +(1.55426 + 1.55426i) q^{7} +2.75350 q^{9} +O(q^{10})\)	\(q+0.496487i q^{3} +(1.55426 + 1.55426i) q^{7} +2.75350 q^{9} +(4.19607 + 4.19607i) q^{11} -5.09530 q^{13} +(-0.213542 - 0.213542i) q^{17} +(0.844754 + 0.844754i) q^{19} +(-0.771668 + 0.771668i) q^{21} +(1.70744 - 1.70744i) q^{23} +2.85654i q^{27} +(-2.24750 + 2.24750i) q^{29} +0.818209i q^{31} +(-2.08329 + 2.08329i) q^{33} +5.12639 q^{37} -2.52975i q^{39} -3.34727i q^{41} -4.49131 q^{43} +(-4.29355 + 4.29355i) q^{47} -2.16858i q^{49} +(0.106021 - 0.106021i) q^{51} -1.00653i q^{53} +(-0.419410 + 0.419410i) q^{57} +(-7.65005 + 7.65005i) q^{59} +(-1.90291 - 1.90291i) q^{61} +(4.27964 + 4.27964i) q^{63} +11.0221 q^{67} +(0.847724 + 0.847724i) q^{69} +10.5331 q^{71} +(2.70854 + 2.70854i) q^{73} +13.0435i q^{77} -8.32010 q^{79} +6.84226 q^{81} +9.17237i q^{83} +(-1.11585 - 1.11585i) q^{87} +4.25101 q^{89} +(-7.91940 - 7.91940i) q^{91} -0.406230 q^{93} +(7.16000 + 7.16000i) q^{97} +(11.5539 + 11.5539i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 2 q^{7} - 10 q^{9}+O(q^{10}) \) 18 * q + 2 * q^7 - 10 * q^9	\( 18 q + 2 q^{7} - 10 q^{9} + 2 q^{11} + 6 q^{17} - 2 q^{19} - 16 q^{21} - 2 q^{23} - 14 q^{29} + 8 q^{33} - 8 q^{37} - 44 q^{43} - 38 q^{47} - 8 q^{51} - 24 q^{57} + 10 q^{59} + 14 q^{61} + 6 q^{63} + 12 q^{67} + 32 q^{69} - 24 q^{71} - 14 q^{73} - 16 q^{79} + 2 q^{81} + 24 q^{87} - 12 q^{89} - 16 q^{93} - 18 q^{97} + 22 q^{99}+O(q^{100}) \) 18 * q + 2 * q^7 - 10 * q^9 + 2 * q^11 + 6 * q^17 - 2 * q^19 - 16 * q^21 - 2 * q^23 - 14 * q^29 + 8 * q^33 - 8 * q^37 - 44 * q^43 - 38 * q^47 - 8 * q^51 - 24 * q^57 + 10 * q^59 + 14 * q^61 + 6 * q^63 + 12 * q^67 + 32 * q^69 - 24 * q^71 - 14 * q^73 - 16 * q^79 + 2 * q^81 + 24 * q^87 - 12 * q^89 - 16 * q^93 - 18 * q^97 + 22 * q^99

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