LMFDB - Embedded newform 2028.2.q.j.361.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2028,2,Mod(361,2028)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2028.361"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2028, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 5])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2028 = 2^{2} \cdot 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2028.q (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,6,0,0,0,0,0,-6,0,0,0,0,0,0,0,26] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$16.1936615299$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{6})$
Coefficient field:	12.0.17213603549184.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 5x^{10} + 19x^{8} - 28x^{6} + 31x^{4} - 6x^{2} + 1 $ x^12 - 5x^10 + 19x^8 - 28x^6 + 31x^4 - 6*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			361.4
Root			$1.07992 + 0.623490i$ of defining polynomial
Character	$\chi$	$=$	2028.361
Dual form			2028.2.q.j.1837.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.500000 + 0.866025i) q^{3} +0.554958i q^{5} +(0.908389 + 0.524459i) q^{7} +(-0.500000 + 0.866025i) q^{9} +(2.52174 - 1.45593i) q^{11} +(-0.480608 + 0.277479i) q^{15} +(1.37651 - 2.38419i) q^{17} +(4.01058 + 2.31551i) q^{19} +1.04892i q^{21} +(-2.88135 - 4.99065i) q^{23} +4.69202 q^{25} -1.00000 q^{27} +(1.40097 + 2.42655i) q^{29} +4.18598i q^{31} +(2.52174 + 1.45593i) q^{33} +(-0.291053 + 0.504118i) q^{35} +(0.404271 - 0.233406i) q^{37} +(3.37730 - 1.94989i) q^{41} +(4.59783 - 7.96368i) q^{43} +(-0.480608 - 0.277479i) q^{45} +11.5211i q^{47} +(-2.94989 - 5.10935i) q^{49} +2.75302 q^{51} +5.62565 q^{53} +(0.807979 + 1.39946i) q^{55} +4.63102i q^{57} +(-2.69327 - 1.55496i) q^{59} +(-5.45257 + 9.44414i) q^{61} +(-0.908389 + 0.524459i) q^{63} +(-6.97057 + 4.02446i) q^{67} +(2.88135 - 4.99065i) q^{69} +(11.8576 + 6.84601i) q^{71} +9.36658i q^{73} +(2.34601 + 4.06341i) q^{75} +3.05429 q^{77} -3.60925 q^{79} +(-0.500000 - 0.866025i) q^{81} -1.65519i q^{83} +(1.32312 + 0.763906i) q^{85} +(-1.40097 + 2.42655i) q^{87} +(15.5629 - 8.98523i) q^{89} +(-3.62517 + 2.09299i) q^{93} +(-1.28501 + 2.22571i) q^{95} +(-1.14113 - 0.658834i) q^{97} +2.91185i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 6 q^{3} - 6 q^{9} + 26 q^{17} + 36 q^{25} - 12 q^{27} + 8 q^{29} + 8 q^{35} - 18 q^{43} + 10 q^{49} + 52 q^{51} + 20 q^{53} + 30 q^{55} - 14 q^{61} + 18 q^{75} - 12 q^{77} + 4 q^{79} - 6 q^{81} - 8 q^{87}+ \cdots + 34 q^{95}+O(q^{100}) $ 12 * q + 6 * q^3 - 6 * q^9 + 26 * q^17 + 36 * q^25 - 12 * q^27 + 8 * q^29 + 8 * q^35 - 18 * q^43 + 10 * q^49 + 52 * q^51 + 20 * q^53 + 30 * q^55 - 14 * q^61 + 18 * q^75 - 12 * q^77 + 4 * q^79 - 6 * q^81 - 8 * q^87 + 34 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2028\mathbb{Z}\right)^\times$.

$n$	$677$	$1015$	$1861$
$\chi(n)$	$1$	$1$	$e\left(\frac{5}{6}\right)$

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
$3$	0.500000	+	0.866025i	0.288675	+	0.500000i
$3$	0.500000	+	0.866025i	0.288675	+	0.500000i
$4$		0			0
$5$			0.554958i			0.248185i	0.992271	+	0.124092i	$0.0396019\pi$
$5$			0.554958i			0.248185i	−0.992271	+	0.124092i	$0.960398\pi$
$6$		0			0
$7$	0.908389	+	0.524459i	0.343339	+	0.198227i	0.661747	−	0.749727i	$-0.269815\pi$
$7$	0.908389	+	0.524459i	0.343339	+	0.198227i	−0.318409	+	0.947954i	$0.603148\pi$
$8$		0			0
$9$	−0.500000	+	0.866025i	−0.166667	+	0.288675i
$10$		0			0
$11$	2.52174	−	1.45593i	0.760333	−	0.438979i	−0.0690822	−	0.997611i	$-0.522007\pi$
$11$	2.52174	−	1.45593i	0.760333	−	0.438979i	0.829415	+	0.558632i	$0.188674\pi$
$12$		0			0
$13$		0			0
$13$		0			0
$14$		0			0
$15$	−0.480608	+	0.277479i	−0.124092	+	0.0716448i
$16$		0			0
$17$	1.37651	−	2.38419i	0.333853	−	0.578250i	−0.649411	−	0.760438i	$-0.724984\pi$
$17$	1.37651	−	2.38419i	0.333853	−	0.578250i	0.983264	+	0.182188i	$0.0583178\pi$
$18$		0			0
$19$	4.01058	+	2.31551i	0.920091	+	0.531215i	0.883664	−	0.468122i	$-0.155069\pi$
$19$	4.01058	+	2.31551i	0.920091	+	0.531215i	0.0364268	+	0.999336i	$0.488402\pi$
$20$		0			0
$21$			1.04892i			0.228893i
$22$		0			0
$23$	−2.88135	−	4.99065i	−0.600804	−	1.04062i	−0.992700	−	0.120613i	$-0.961514\pi$
$23$	−2.88135	−	4.99065i	−0.600804	−	1.04062i	0.391896	−	0.920010i	$-0.371819\pi$
$24$		0			0
$25$	4.69202			0.938404
$26$		0			0
$27$	−1.00000			−0.192450
$28$		0			0
$29$	1.40097	+	2.42655i	0.260153	+	0.450599i	0.966283	−	0.257484i	$-0.0828935\pi$
$29$	1.40097	+	2.42655i	0.260153	+	0.450599i	−0.706129	+	0.708083i	$0.749560\pi$
$30$		0			0
$31$			4.18598i			0.751824i	0.926655	+	0.375912i	$0.122671\pi$
$31$			4.18598i			0.751824i	−0.926655	+	0.375912i	$0.877329\pi$
$32$		0			0
$33$	2.52174	+	1.45593i	0.438979	+	0.253444i
$34$		0			0
$35$	−0.291053	+	0.504118i	−0.0491969	+	0.0852115i
$36$		0			0
$37$	0.404271	−	0.233406i	0.0664618	−	0.0383717i	−0.466401	−	0.884573i	$-0.654450\pi$
$37$	0.404271	−	0.233406i	0.0664618	−	0.0383717i	0.532863	+	0.846202i	$0.321116\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	3.37730	−	1.94989i	0.527446	−	0.304521i	−0.212530	−	0.977155i	$-0.568170\pi$
$41$	3.37730	−	1.94989i	0.527446	−	0.304521i	0.739976	+	0.672634i	$0.234837\pi$
$42$		0			0
$43$	4.59783	−	7.96368i	0.701163	−	1.21445i	−0.266895	−	0.963726i	$-0.585998\pi$
$43$	4.59783	−	7.96368i	0.701163	−	1.21445i	0.968058	−	0.250725i	$-0.0806689\pi$
$44$		0			0
$45$	−0.480608	−	0.277479i	−0.0716448	−	0.0413641i
$46$		0			0
$47$			11.5211i			1.68053i	0.542179	+	0.840263i	$0.317599\pi$
$47$			11.5211i			1.68053i	−0.542179	+	0.840263i	$0.682401\pi$
$48$		0			0
$49$	−2.94989	−	5.10935i	−0.421412	−	0.729908i
$50$		0			0
$51$	2.75302			0.385500
$52$		0			0
$53$	5.62565			0.772742			0.386371	−	0.922343i	$-0.373728\pi$
$53$	5.62565			0.772742			0.386371	+	0.922343i	$0.373728\pi$
$54$		0			0
$55$	0.807979	+	1.39946i	0.108948	+	0.188703i
$56$		0			0
$57$			4.63102i			0.613394i
$58$		0			0
$59$	−2.69327	−	1.55496i	−0.350633	−	0.202438i	0.314331	−	0.949314i	$-0.398220\pi$
$59$	−2.69327	−	1.55496i	−0.350633	−	0.202438i	−0.664964	+	0.746875i	$0.731553\pi$
$60$		0			0
$61$	−5.45257	+	9.44414i	−0.698131	+	1.20920i	0.270983	+	0.962584i	$0.412651\pi$
$61$	−5.45257	+	9.44414i	−0.698131	+	1.20920i	−0.969114	+	0.246614i	$0.920682\pi$
$62$		0			0
$63$	−0.908389	+	0.524459i	−0.114446	+	0.0660756i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−6.97057	+	4.02446i	−0.851590	+	0.491666i	−0.861187	−	0.508288i	$-0.830279\pi$
$67$	−6.97057	+	4.02446i	−0.851590	+	0.491666i	0.00959683	+	0.999954i	$0.496945\pi$
$68$		0			0
$69$	2.88135	−	4.99065i	0.346874	−	0.600804i
$70$		0			0
$71$	11.8576	+	6.84601i	1.40724	+	0.812472i	0.995121	−	0.0986570i	$-0.0314547\pi$
$71$	11.8576	+	6.84601i	1.40724	+	0.812472i	0.412121	+	0.911129i	$0.364788\pi$
$72$		0			0
$73$			9.36658i			1.09628i	0.836388	+	0.548138i	$0.184663\pi$
$73$			9.36658i			1.09628i	−0.836388	+	0.548138i	$0.815337\pi$
$74$		0			0
$75$	2.34601	+	4.06341i	0.270894	+	0.469202i
$76$		0			0
$77$	3.05429			0.348069
$78$		0			0
$79$	−3.60925			−0.406073			−0.203036	−	0.979171i	$-0.565081\pi$
$79$	−3.60925			−0.406073			−0.203036	+	0.979171i	$0.565081\pi$
$80$		0			0
$81$	−0.500000	−	0.866025i	−0.0555556	−	0.0962250i
$82$		0			0
$83$		−	1.65519i		−	0.181680i	−0.995865	−	0.0908401i	$-0.971045\pi$
$83$		−	1.65519i		−	0.181680i	0.995865	−	0.0908401i	$-0.0289552\pi$
$84$		0			0
$85$	1.32312	+	0.763906i	0.143513	+	0.0828572i
$86$		0			0
$87$	−1.40097	+	2.42655i	−0.150200	+	0.260153i
$88$		0			0
$89$	15.5629	−	8.98523i	1.64966	−	0.952432i	0.672456	−	0.740137i	$-0.265239\pi$
$89$	15.5629	−	8.98523i	1.64966	−	0.952432i	0.977206	−	0.212295i	$-0.0680939\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−3.62517	+	2.09299i	−0.375912	+	0.217033i
$94$		0			0
$95$	−1.28501	+	2.22571i	−0.131839	+	0.228353i
$96$		0			0
$97$	−1.14113	−	0.658834i	−0.115865	−	0.0668944i	0.440948	−	0.897533i	$-0.354642\pi$
$97$	−1.14113	−	0.658834i	−0.115865	−	0.0668944i	−0.556812	+	0.830638i	$0.687976\pi$
$98$		0			0
$99$			2.91185i			0.292652i
$100$		0			0
$101$	−7.51238	−	13.0118i	−0.747509	−	1.29472i	−0.949013	−	0.315237i	$-0.897916\pi$
$101$	−7.51238	−	13.0118i	−0.747509	−	1.29472i	0.201504	−	0.979488i	$-0.435417\pi$
$102$		0			0
$103$	9.20775			0.907267			0.453633	−	0.891188i	$-0.350128\pi$
$103$	9.20775			0.907267			0.453633	+	0.891188i	$0.350128\pi$
$104$		0			0
$105$	−0.582105			−0.0568077
$106$		0			0
$107$	−3.61476	−	6.26095i	−0.349452	−	0.605269i	0.636700	−	0.771112i	$-0.280299\pi$
$107$	−3.61476	−	6.26095i	−0.349452	−	0.605269i	−0.986152	+	0.165843i	$0.946966\pi$
$108$		0			0
$109$			15.5036i			1.48498i	0.669857	+	0.742490i	$0.266355\pi$
$109$			15.5036i			1.48498i	−0.669857	+	0.742490i	$0.733645\pi$
$110$		0			0
$111$	0.404271	+	0.233406i	0.0383717	+	0.0221539i
$112$		0			0
$113$	−6.91335	+	11.9743i	−0.650353	+	1.12644i	0.332684	+	0.943038i	$0.392046\pi$
$113$	−6.91335	+	11.9743i	−0.650353	+	1.12644i	−0.983037	+	0.183406i	$0.941288\pi$
$114$		0			0
$115$	2.76960	−	1.59903i	0.258267	−	0.149110i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	2.50081	−	1.44385i	0.229249	−	0.132357i
$120$		0			0
$121$	−1.26055	+	2.18334i	−0.114596	+	0.198486i
$122$		0			0
$123$	3.37730	+	1.94989i	0.304521	+	0.175815i
$124$		0			0
$125$			5.37867i			0.481083i
$126$		0			0
$127$	3.58815	+	6.21485i	0.318396	+	0.551479i	0.980154	−	0.198239i	$-0.0635223\pi$
$127$	3.58815	+	6.21485i	0.318396	+	0.551479i	−0.661757	+	0.749718i	$0.730189\pi$
$128$		0			0
$129$	9.19567			0.809634
$130$		0			0
$131$	13.1304			1.14720			0.573602	−	0.819134i	$-0.305545\pi$
$131$	13.1304			1.14720			0.573602	+	0.819134i	$0.305545\pi$
$132$		0			0
$133$	2.42878	+	4.20677i	0.210602	+	0.364773i
$134$		0			0
$135$		−	0.554958i		−	0.0477632i
$136$		0			0
$137$	−18.8894	−	10.9058i	−1.61383	−	0.931747i	−0.988471	−	0.151411i	$-0.951618\pi$
$137$	−18.8894	−	10.9058i	−1.61383	−	0.931747i	−0.625361	−	0.780335i	$-0.715048\pi$
$138$		0			0
$139$	1.48307	−	2.56876i	0.125793	−	0.217879i	−0.796250	−	0.604968i	$-0.793186\pi$
$139$	1.48307	−	2.56876i	0.125793	−	0.217879i	0.922043	+	0.387089i	$0.126519\pi$
$140$		0			0
$141$	−9.97757	+	5.76055i	−0.840263	+	0.485126i
$142$		0			0
$143$		0			0
$144$		0			0
$145$	−1.34663	+	0.777479i	−0.111832	+	0.0645661i
$146$		0			0
$147$	2.94989	−	5.10935i	0.243303	−	0.421412i
$148$		0			0
$149$	−4.93317	−	2.84817i	−0.404141	−	0.233331i	0.284128	−	0.958786i	$-0.408296\pi$
$149$	−4.93317	−	2.84817i	−0.404141	−	0.233331i	−0.688269	+	0.725455i	$0.741629\pi$
$150$		0			0
$151$			19.1468i			1.55814i	0.626937	+	0.779070i	$0.284309\pi$
$151$			19.1468i			1.55814i	−0.626937	+	0.779070i	$0.715691\pi$
$152$		0			0
$153$	1.37651	+	2.38419i	0.111284	+	0.192750i
$154$		0			0
$155$	−2.32304			−0.186591
$156$		0			0
$157$	−8.38404			−0.669119			−0.334560	−	0.942375i	$-0.608588\pi$
$157$	−8.38404			−0.669119			−0.334560	+	0.942375i	$0.608588\pi$
$158$		0			0
$159$	2.81282	+	4.87195i	0.223071	+	0.386371i
$160$		0			0
$161$		−	6.04461i		−	0.476382i
$162$		0			0
$163$	1.70647	+	0.985230i	0.133661	+	0.0771692i	0.565339	−	0.824858i	$-0.308745\pi$
$163$	1.70647	+	0.985230i	0.133661	+	0.0771692i	−0.431679	+	0.902028i	$0.642079\pi$
$164$		0			0
$165$	−0.807979	+	1.39946i	−0.0629010	+	0.108948i
$166$		0			0
$167$	21.4770	−	12.3998i	1.66194	−	0.959523i	0.690157	−	0.723660i	$-0.257542\pi$
$167$	21.4770	−	12.3998i	1.66194	−	0.959523i	0.971786	−	0.235863i	$-0.0757917\pi$
$168$		0			0
$169$		0			0
$170$		0			0
$171$	−4.01058	+	2.31551i	−0.306697	+	0.177072i
$172$		0			0
$173$	−1.28568	+	2.22686i	−0.0977481	+	0.169305i	−0.910752	−	0.412953i	$-0.864497\pi$
$173$	−1.28568	+	2.22686i	−0.0977481	+	0.169305i	0.813004	+	0.582258i	$0.197831\pi$
$174$		0			0
$175$	4.26218	+	2.46077i	0.322191	+	0.186017i
$176$		0			0
$177$		−	3.10992i		−	0.233756i
$178$		0			0
$179$	−1.50269	−	2.60273i	−0.112316	−	0.194537i	0.804388	−	0.594105i	$-0.202494\pi$
$179$	−1.50269	−	2.60273i	−0.112316	−	0.194537i	−0.916704	+	0.399568i	$0.869160\pi$
$180$		0			0
$181$	14.6843			1.09147			0.545736	−	0.837957i	$-0.316250\pi$
$181$	14.6843			1.09147			0.545736	+	0.837957i	$0.316250\pi$
$182$		0			0
$183$	−10.9051			−0.806132
$184$		0			0
$185$	0.129531	+	0.224354i	0.00952328	+	0.0164948i
$186$		0			0
$187$		−	8.01639i		−	0.586217i
$188$		0			0
$189$	−0.908389	−	0.524459i	−0.0660756	−	0.0381488i
$190$		0			0
$191$	8.54019	−	14.7920i	0.617946	−	1.07031i	−0.371914	−	0.928267i	$-0.621298\pi$
$191$	8.54019	−	14.7920i	0.617946	−	1.07031i	0.989860	−	0.142047i	$-0.0453684\pi$
$192$		0			0
$193$	−12.9695	+	7.48792i	−0.933562	+	0.538992i	−0.887936	−	0.459966i	$-0.847861\pi$
$193$	−12.9695	+	7.48792i	−0.933562	+	0.538992i	−0.0456255	+	0.998959i	$0.514528\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−0.166870	+	0.0963427i	−0.0118890	+	0.00686414i	−0.505933	−	0.862573i	$-0.668852\pi$
$197$	−0.166870	+	0.0963427i	−0.0118890	+	0.00686414i	0.494044	+	0.869437i	$0.335518\pi$
$198$		0			0
$199$	5.93631	−	10.2820i	0.420814	−	0.728871i	−0.575205	−	0.818009i	$-0.695078\pi$
$199$	5.93631	−	10.2820i	0.420814	−	0.728871i	0.996019	+	0.0891378i	$0.0284112\pi$
$200$		0			0
$201$	−6.97057	−	4.02446i	−0.491666	−	0.283863i
$202$		0			0
$203$			2.93900i			0.206277i
$204$		0			0
$205$	1.08211	+	1.87426i	0.0755775	+	0.130904i
$206$		0			0
$207$	5.76271			0.400536
$208$		0			0
$209$	13.4849			0.932767
$210$		0			0
$211$	−3.01693	−	5.22547i	−0.207694	−	0.359736i	0.743294	−	0.668965i	$-0.233262\pi$
$211$	−3.01693	−	5.22547i	−0.207694	−	0.359736i	−0.950988	+	0.309229i	$0.899929\pi$
$212$		0			0
$213$			13.6920i			0.938162i
$214$		0			0
$215$	4.41951	+	2.55161i	0.301408	+	0.174018i
$216$		0			0
$217$	−2.19537	+	3.80250i	−0.149032	+	0.258130i
$218$		0			0
$219$	−8.11170	+	4.68329i	−0.548138	+	0.316468i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−7.38415	+	4.26324i	−0.494479	+	0.285488i	−0.726431	−	0.687240i	$-0.758822\pi$
$223$	−7.38415	+	4.26324i	−0.494479	+	0.285488i	0.231952	+	0.972727i	$0.425489\pi$
$224$		0			0
$225$	−2.34601	+	4.06341i	−0.156401	+	0.270894i
$226$		0			0
$227$	9.52628	+	5.50000i	0.632281	+	0.365048i	0.781635	−	0.623736i	$-0.214386\pi$
$227$	9.52628	+	5.50000i	0.632281	+	0.365048i	−0.149354	+	0.988784i	$0.547719\pi$
$228$		0			0
$229$		−	17.9119i		−	1.18365i	−0.806067	−	0.591824i	$-0.798408\pi$
$229$		−	17.9119i		−	1.18365i	0.806067	−	0.591824i	$-0.201592\pi$
$230$		0			0
$231$	1.52715	+	2.64510i	0.100479	+	0.174035i
$232$		0			0
$233$	−22.8159			−1.49472			−0.747361	−	0.664418i	$-0.768679\pi$
$233$	−22.8159			−1.49472			−0.747361	+	0.664418i	$0.768679\pi$
$234$		0			0
$235$	−6.39373			−0.417081
$236$		0			0
$237$	−1.80463	−	3.12570i	−0.117223	−	0.203036i
$238$		0			0
$239$		−	8.34481i		−	0.539781i	−0.962891	−	0.269891i	$-0.913012\pi$
$239$		−	8.34481i		−	0.539781i	0.962891	−	0.269891i	$-0.0869875\pi$
$240$		0			0
$241$	3.25228	+	1.87771i	0.209498	+	0.120954i	0.601078	−	0.799190i	$-0.294738\pi$
$241$	3.25228	+	1.87771i	0.209498	+	0.120954i	−0.391580	+	0.920144i	$0.628071\pi$
$242$		0			0
$243$	0.500000	−	0.866025i	0.0320750	−	0.0555556i
$244$		0			0
$245$	2.83548	−	1.63706i	0.181152	−	0.104588i
$246$		0			0
$247$		0			0
$248$		0			0
$249$	1.43343	−	0.827593i	0.0908401	−	0.0524466i
$250$		0			0
$251$	−15.1543	+	26.2480i	−0.956530	+	1.65676i	−0.225703	+	0.974196i	$0.572468\pi$
$251$	−15.1543	+	26.2480i	−0.956530	+	1.65676i	−0.730827	+	0.682562i	$0.760866\pi$
$252$		0			0
$253$	−14.5321	−	8.39008i	−0.913622	−	0.527480i
$254$		0			0
$255$			1.52781i			0.0956752i
$256$		0			0
$257$	−10.4206	−	18.0490i	−0.650018	−	1.12586i	−0.983118	−	0.182973i	$-0.941428\pi$
$257$	−10.4206	−	18.0490i	−0.650018	−	1.12586i	0.333100	−	0.942892i	$-0.391905\pi$
$258$		0			0
$259$	0.489647			0.0304252
$260$		0			0
$261$	−2.80194			−0.173436
$262$		0			0
$263$	−4.29321	−	7.43606i	−0.264731	−	0.458527i	0.702762	−	0.711425i	$-0.251950\pi$
$263$	−4.29321	−	7.43606i	−0.264731	−	0.458527i	−0.967493	+	0.252898i	$0.918616\pi$
$264$		0			0
$265$			3.12200i			0.191783i
$266$		0			0
$267$	15.5629	+	8.98523i	0.952432	+	0.549887i
$268$		0			0
$269$	−7.41335	+	12.8403i	−0.452000	+	0.782886i	−0.998510	−	0.0545658i	$-0.982623\pi$
$269$	−7.41335	+	12.8403i	−0.452000	+	0.782886i	0.546510	+	0.837452i	$0.315956\pi$
$270$		0			0
$271$	−6.26561	+	3.61745i	−0.380608	+	0.219744i	−0.678083	−	0.734985i	$-0.737189\pi$
$271$	−6.26561	+	3.61745i	−0.380608	+	0.219744i	0.297475	+	0.954730i	$0.403856\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	11.8321	−	6.83124i	0.713500	−	0.411939i
$276$		0			0
$277$	5.60268	−	9.70412i	0.336632	−	0.583064i	−0.647165	−	0.762350i	$-0.724045\pi$
$277$	5.60268	−	9.70412i	0.336632	−	0.583064i	0.983797	+	0.179286i	$0.0573787\pi$
$278$		0			0
$279$	−3.62517	−	2.09299i	−0.217033	−	0.125304i
$280$		0			0
$281$		−	12.3002i		−	0.733769i	−0.930267	−	0.366884i	$-0.880424\pi$
$281$		−	12.3002i		−	0.733769i	0.930267	−	0.366884i	$-0.119576\pi$
$282$		0			0
$283$	12.1761	+	21.0895i	0.723791	+	1.25364i	0.959470	+	0.281812i	$0.0909355\pi$
$283$	12.1761	+	21.0895i	0.723791	+	1.25364i	−0.235678	+	0.971831i	$0.575731\pi$
$284$		0			0
$285$	−2.57002			−0.152235
$286$		0			0
$287$	4.09054			0.241457
$288$		0			0
$289$	4.71044	+	8.15872i	0.277085	+	0.479925i
$290$		0			0
$291$		−	1.31767i		−	0.0772430i
$292$		0			0
$293$	−5.37141	−	3.10119i	−0.313801	−	0.181173i	0.334825	−	0.942280i	$-0.391323\pi$
$293$	−5.37141	−	3.10119i	−0.313801	−	0.181173i	−0.648626	+	0.761107i	$0.724656\pi$
$294$		0			0
$295$	0.862937	−	1.49465i	0.0502421	−	0.0870219i
$296$		0			0
$297$	−2.52174	+	1.45593i	−0.146326	+	0.0844815i
$298$		0			0
$299$		0			0
$300$		0			0
$301$	8.35325	−	4.82275i	0.481473	−	0.277979i
$302$		0			0
$303$	7.51238	−	13.0118i	0.431575	−	0.747509i
$304$		0			0
$305$	−5.24110	−	3.02595i	−0.300105	−	0.173265i
$306$		0			0
$307$		−	26.4795i		−	1.51126i	−0.654996	−	0.755632i	$-0.727330\pi$
$307$		−	26.4795i		−	1.51126i	0.654996	−	0.755632i	$-0.272670\pi$
$308$		0			0
$309$	4.60388	+	7.97415i	0.261905	+	0.453633i
$310$		0			0
$311$	−15.5308			−0.880671			−0.440335	−	0.897833i	$-0.645140\pi$
$311$	−15.5308			−0.880671			−0.440335	+	0.897833i	$0.645140\pi$
$312$		0			0
$313$	16.9681			0.959092			0.479546	−	0.877517i	$-0.340801\pi$
$313$	16.9681			0.959092			0.479546	+	0.877517i	$0.340801\pi$
$314$		0			0
$315$	−0.291053	−	0.504118i	−0.0163990	−	0.0284038i
$316$		0			0
$317$		−	25.5066i		−	1.43260i	−0.697795	−	0.716298i	$-0.745835\pi$
$317$		−	25.5066i		−	1.43260i	0.697795	−	0.716298i	$-0.254165\pi$
$318$		0			0
$319$	7.06576	+	4.07942i	0.395606	+	0.228403i
$320$		0			0
$321$	3.61476	−	6.26095i	0.201756	−	0.349452i
$322$		0			0
$323$	11.0412	−	6.37465i	0.614350	−	0.354695i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−13.4266	+	7.75182i	−0.742490	+	0.428677i
$328$		0			0
$329$	−6.04234	+	10.4656i	−0.333125	+	0.576990i
$330$		0			0
$331$	9.19162	+	5.30678i	0.505217	+	0.291687i	0.730865	−	0.682522i	$-0.239117\pi$
$331$	9.19162	+	5.30678i	0.505217	+	0.291687i	−0.225648	+	0.974209i	$0.572450\pi$
$332$		0			0
$333$			0.466812i			0.0255811i
$334$		0			0
$335$	−2.23341	−	3.86837i	−0.122024	−	0.211352i
$336$		0			0
$337$	−29.2717			−1.59453			−0.797266	−	0.603628i	$-0.793721\pi$
$337$	−29.2717			−1.59453			−0.797266	+	0.603628i	$0.793721\pi$
$338$		0			0
$339$	−13.8267			−0.750963
$340$		0			0
$341$	6.09448	+	10.5560i	0.330035	+	0.571637i
$342$		0			0
$343$		−	13.5308i		−	0.730594i
$344$		0			0
$345$	2.76960	+	1.59903i	0.149110	+	0.0860889i
$346$		0			0
$347$	17.4073	−	30.1503i	0.934473	−	1.61855i	0.158902	−	0.987294i	$-0.449205\pi$
$347$	17.4073	−	30.1503i	0.934473	−	1.61855i	0.775571	−	0.631261i	$-0.217462\pi$
$348$		0			0
$349$	−26.1502	+	15.0978i	−1.39979	+	0.808169i	−0.994370	−	0.105962i	$-0.966208\pi$
$349$	−26.1502	+	15.0978i	−1.39979	+	0.808169i	−0.405419	+	0.914131i	$0.632874\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	21.6239	−	12.4846i	1.15092	−	0.664486i	0.201812	−	0.979424i	$-0.435317\pi$
$353$	21.6239	−	12.4846i	1.15092	−	0.664486i	0.949112	+	0.314938i	$0.101984\pi$
$354$		0			0
$355$	−3.79925	+	6.58049i	−0.201643	+	0.349256i
$356$		0			0
$357$	2.50081	+	1.44385i	0.132357	+	0.0764164i
$358$		0			0
$359$		−	14.1661i		−	0.747660i	−0.927497	−	0.373830i	$-0.878044\pi$
$359$		−	14.1661i		−	0.747660i	0.927497	−	0.373830i	$-0.121956\pi$
$360$		0			0
$361$	1.22318	+	2.11862i	0.0643782	+	0.111506i
$362$		0			0
$363$	−2.52111			−0.132324
$364$		0			0
$365$	−5.19806			−0.272079
$366$		0			0
$367$	−4.39344	−	7.60965i	−0.229335	−	0.397221i	0.728276	−	0.685284i	$-0.240322\pi$
$367$	−4.39344	−	7.60965i	−0.229335	−	0.397221i	−0.957611	+	0.288063i	$0.906989\pi$
$368$		0			0
$369$			3.89977i			0.203014i
$370$		0			0
$371$	5.11028	+	2.95042i	0.265312	+	0.153178i
$372$		0			0
$373$	9.16301	−	15.8708i	0.474443	−	0.821759i	−0.525129	−	0.851023i	$-0.675983\pi$
$373$	9.16301	−	15.8708i	0.474443	−	0.821759i	0.999572	+	0.0292636i	$0.00931621\pi$
$374$		0			0
$375$	−4.65806	+	2.68933i	−0.240541	+	0.138877i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	4.13601	−	2.38793i	0.212453	−	0.122660i	−0.389998	−	0.920816i	$-0.627524\pi$
$379$	4.13601	−	2.38793i	0.212453	−	0.122660i	0.602451	+	0.798156i	$0.294191\pi$
$380$		0			0
$381$	−3.58815	+	6.21485i	−0.183826	+	0.318396i
$382$		0			0
$383$	5.36095	+	3.09515i	0.273932	+	0.158155i	0.630673	−	0.776049i	$-0.282779\pi$
$383$	5.36095	+	3.09515i	0.273932	+	0.158155i	−0.356741	+	0.934203i	$0.616112\pi$
$384$		0			0
$385$			1.69501i			0.0863855i
$386$		0			0
$387$	4.59783	+	7.96368i	0.233721	+	0.404817i
$388$		0			0
$389$	−24.8780			−1.26136			−0.630682	−	0.776041i	$-0.717225\pi$
$389$	−24.8780			−1.26136			−0.630682	+	0.776041i	$0.717225\pi$
$390$		0			0
$391$	−15.8649			−0.802320
$392$		0			0
$393$	6.56518	+	11.3712i	0.331169	+	0.573602i
$394$		0			0
$395$		−	2.00298i		−	0.100781i
$396$		0			0
$397$	4.00127	+	2.31013i	0.200818	+	0.115942i	0.597037	−	0.802214i	$-0.296345\pi$
$397$	4.00127	+	2.31013i	0.200818	+	0.115942i	−0.396219	+	0.918156i	$0.629678\pi$
$398$		0			0
$399$	−2.42878	+	4.20677i	−0.121591	+	0.210602i
$400$		0			0
$401$	−1.42763	+	0.824240i	−0.0712923	+	0.0411606i	−0.535222	−	0.844711i	$-0.679772\pi$
$401$	−1.42763	+	0.824240i	−0.0712923	+	0.0411606i	0.463930	+	0.885872i	$0.346439\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	0.480608	−	0.277479i	0.0238816	−	0.0137880i
$406$		0			0
$407$	0.679644	−	1.17718i	0.0336887	−	0.0583506i
$408$		0			0
$409$	−30.1419	−	17.4025i	−1.49042	−	0.860496i	−0.490483	−	0.871451i	$-0.663180\pi$
$409$	−30.1419	−	17.4025i	−1.49042	−	0.860496i	−0.999940	+	0.0109543i	$0.996513\pi$
$410$		0			0
$411$		−	21.8116i		−	1.07589i
$412$		0			0
$413$	−1.63102	−	2.82501i	−0.0802574	−	0.139010i
$414$		0			0
$415$	0.918559			0.0450903
$416$		0			0
$417$	2.96615			0.145253
$418$		0			0
$419$	−10.8177	−	18.7367i	−0.528478	−	0.915350i	−0.999449	−	0.0332014i	$-0.989430\pi$
$419$	−10.8177	−	18.7367i	−0.528478	−	0.915350i	0.470971	−	0.882149i	$-0.343904\pi$
$420$		0			0
$421$			35.1008i			1.71071i	0.518043	+	0.855355i	$0.326661\pi$
$421$			35.1008i			1.71071i	−0.518043	+	0.855355i	$0.673339\pi$
$422$		0			0
$423$	−9.97757	−	5.76055i	−0.485126	−	0.280088i
$424$		0			0
$425$	6.45862	−	11.1867i	0.313289	−	0.542632i
$426$		0			0
$427$	−9.90612	+	5.71930i	−0.479391	+	0.276776i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	17.4006	−	10.0462i	0.838156	−	0.483910i	−0.0184807	−	0.999829i	$-0.505883\pi$
$431$	17.4006	−	10.0462i	0.838156	−	0.483910i	0.856637	+	0.515919i	$0.172550\pi$
$432$		0			0
$433$	−9.34697	+	16.1894i	−0.449187	+	0.778014i	−0.998333	−	0.0577118i	$-0.981620\pi$
$433$	−9.34697	+	16.1894i	−0.449187	+	0.778014i	0.549147	+	0.835726i	$0.314953\pi$
$434$		0			0
$435$	−1.34663	−	0.777479i	−0.0645661	−	0.0372773i
$436$		0			0
$437$		−	26.6872i		−	1.27662i
$438$		0			0
$439$	0.675096	+	1.16930i	0.0322206	+	0.0558076i	0.881686	−	0.471837i	$-0.156409\pi$
$439$	0.675096	+	1.16930i	0.0322206	+	0.0558076i	−0.849465	+	0.527644i	$0.823075\pi$
$440$		0			0
$441$	5.89977			0.280942
$442$		0			0
$443$	−0.400436			−0.0190253			−0.00951265	−	0.999955i	$-0.503028\pi$
$443$	−0.400436			−0.0190253			−0.00951265	+	0.999955i	$0.503028\pi$
$444$		0			0
$445$	4.98643	+	8.63674i	0.236379	+	0.409421i
$446$		0			0
$447$		−	5.69633i		−	0.269427i
$448$		0			0
$449$	−30.9954	−	17.8952i	−1.46277	−	0.844528i	−0.463627	−	0.886031i	$-0.653452\pi$
$449$	−30.9954	−	17.8952i	−1.46277	−	0.844528i	−0.999138	+	0.0415028i	$0.986785\pi$
$450$		0			0
$451$	5.67778	−	9.83421i	0.267356	−	0.463075i
$452$		0			0
$453$	−16.5816	+	9.57338i	−0.779070	+	0.449796i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−4.47814	+	2.58546i	−0.209479	+	0.120943i	−0.601069	−	0.799197i	$-0.705258\pi$
$457$	−4.47814	+	2.58546i	−0.209479	+	0.120943i	0.391590	+	0.920140i	$0.371925\pi$
$458$		0			0
$459$	−1.37651	+	2.38419i	−0.0642500	+	0.111284i
$460$		0			0
$461$	−18.0809	−	10.4390i	−0.842111	−	0.486193i	0.0158705	−	0.999874i	$-0.494948\pi$
$461$	−18.0809	−	10.4390i	−0.842111	−	0.486193i	−0.857981	+	0.513681i	$0.828281\pi$
$462$		0			0
$463$		−	30.7198i		−	1.42767i	−0.700315	−	0.713834i	$-0.746957\pi$
$463$		−	30.7198i		−	1.42767i	0.700315	−	0.713834i	$-0.253043\pi$
$464$		0			0
$465$	−1.16152	−	2.01182i	−0.0538643	−	0.0932957i
$466$		0			0
$467$	−15.2741			−0.706802			−0.353401	−	0.935472i	$-0.614975\pi$
$467$	−15.2741			−0.706802			−0.353401	+	0.935472i	$0.614975\pi$
$468$		0			0
$469$	−8.44265			−0.389845
$470$		0			0
$471$	−4.19202	−	7.26079i	−0.193158	−	0.334560i
$472$		0			0
$473$		−	26.7764i		−	1.23118i
$474$		0			0
$475$	18.8177	+	10.8644i	0.863417	+	0.498494i
$476$		0			0
$477$	−2.81282	+	4.87195i	−0.128790	+	0.223071i
$478$		0			0
$479$	2.64883	−	1.52930i	0.121028	−	0.0698756i	−0.438264	−	0.898846i	$-0.644406\pi$
$479$	2.64883	−	1.52930i	0.121028	−	0.0698756i	0.559292	+	0.828971i	$0.311073\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$	5.23478	−	3.02230i	0.238191	−	0.137520i
$484$		0			0
$485$	0.365625	−	0.633281i	0.0166022	−	0.0287558i
$486$		0			0
$487$	13.1221	+	7.57606i	0.594620	+	0.343304i	0.766922	−	0.641740i	$-0.221787\pi$
$487$	13.1221	+	7.57606i	0.594620	+	0.343304i	−0.172302	+	0.985044i	$0.555120\pi$
$488$		0			0
$489$			1.97046i			0.0891073i
$490$		0			0
$491$	11.7397	+	20.3338i	0.529807	+	0.917653i	0.999395	+	0.0347674i	$0.0110690\pi$
$491$	11.7397	+	20.3338i	0.529807	+	0.917653i	−0.469588	+	0.882886i	$0.655598\pi$
$492$		0			0
$493$	7.71379			0.347412
$494$		0			0
$495$	−1.61596			−0.0726319
$496$		0			0
$497$	7.18090	+	12.4377i	0.322107	+	0.557906i
$498$		0			0
$499$		−	38.0689i		−	1.70420i	−0.523381	−	0.852099i	$-0.675330\pi$
$499$		−	38.0689i		−	1.70420i	0.523381	−	0.852099i	$-0.324670\pi$
$500$		0			0
$501$	21.4770	+	12.3998i	0.959523	+	0.553981i
$502$		0			0
$503$	5.56369	−	9.63659i	0.248073	−	0.429674i	−0.714918	−	0.699208i	$-0.753536\pi$
$503$	5.56369	−	9.63659i	0.248073	−	0.429674i	0.962991	+	0.269534i	$0.0868695\pi$
$504$		0			0
$505$	7.22101	−	4.16905i	0.321331	−	0.185521i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−23.1777	+	13.3817i	−1.02733	+	0.593131i	−0.916220	−	0.400676i	$-0.868775\pi$
$509$	−23.1777	+	13.3817i	−1.02733	+	0.593131i	−0.111114	+	0.993808i	$0.535442\pi$
$510$		0			0
$511$	−4.91239	+	8.50850i	−0.217311	+	0.376394i
$512$		0			0
$513$	−4.01058	−	2.31551i	−0.177072	−	0.102232i
$514$		0			0
$515$			5.10992i			0.225170i
$516$		0			0
$517$	16.7739	+	29.0532i	0.737715	+	1.27776i
$518$		0			0
$519$	−2.57135			−0.112870
$520$		0			0
$521$	20.5797			0.901614			0.450807	−	0.892622i	$-0.351136\pi$
$521$	20.5797			0.901614			0.450807	+	0.892622i	$0.351136\pi$
$522$		0			0
$523$	3.38740	+	5.86714i	0.148120	+	0.256552i	0.930533	−	0.366209i	$-0.119344\pi$
$523$	3.38740	+	5.86714i	0.148120	+	0.256552i	−0.782412	+	0.622761i	$0.786011\pi$
$524$		0			0
$525$			4.92154i			0.214794i
$526$		0			0
$527$	9.98015	+	5.76205i	0.434742	+	0.250999i
$528$		0			0
$529$	−5.10441	+	8.84109i	−0.221931	+	0.384395i
$530$		0			0
$531$	2.69327	−	1.55496i	0.116878	−	0.0674794i
$532$		0			0
$533$		0			0
$534$		0			0
$535$	3.47456	−	2.00604i	0.150219	−	0.0867287i
$536$		0			0
$537$	1.50269	−	2.60273i	0.0648458	−	0.112316i
$538$		0			0
$539$	−14.8777	−	8.58964i	−0.640827	−	0.369982i
$540$		0			0
$541$			1.94331i			0.0835495i	0.999127	+	0.0417748i	$0.0133012\pi$
$541$			1.94331i			0.0835495i	−0.999127	+	0.0417748i	$0.986699\pi$
$542$		0			0
$543$	7.34213	+	12.7169i	0.315081	+	0.545736i
$544$		0			0
$545$	−8.60388			−0.368550
$546$		0			0
$547$	−39.1323			−1.67318			−0.836588	−	0.547833i	$-0.815453\pi$
$547$	−39.1323			−1.67318			−0.836588	+	0.547833i	$0.815453\pi$
$548$		0			0
$549$	−5.45257	−	9.44414i	−0.232710	−	0.403066i
$550$		0			0
$551$			12.9758i			0.552789i
$552$		0			0
$553$	−3.27861	−	1.89290i	−0.139420	−	0.0804945i
$554$		0			0
$555$	−0.129531	+	0.224354i	−0.00549827	+	0.00952328i
$556$		0			0
$557$	−21.2166	+	12.2494i	−0.898978	+	0.519025i	−0.876868	−	0.480731i	$-0.840372\pi$
$557$	−21.2166	+	12.2494i	−0.898978	+	0.519025i	−0.0221094	+	0.999756i	$0.507038\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	6.94240	−	4.00820i	0.293108	−	0.169226i
$562$		0			0
$563$	−10.9133	+	18.9025i	−0.459943	+	0.796644i	−0.998957	−	0.0456522i	$-0.985463\pi$
$563$	−10.9133	+	18.9025i	−0.459943	+	0.796644i	0.539015	+	0.842296i	$0.318797\pi$
$564$		0			0
$565$	−6.64522	−	3.83662i	−0.279566	−	0.161408i
$566$		0			0
$567$		−	1.04892i		−	0.0440504i
$568$		0			0
$569$	−5.78136	−	10.0136i	−0.242367	−	0.419793i	0.719021	−	0.694989i	$-0.244591\pi$
$569$	−5.78136	−	10.0136i	−0.242367	−	0.419793i	−0.961388	+	0.275196i	$0.911257\pi$
$570$		0			0
$571$	−24.7614			−1.03623			−0.518116	−	0.855310i	$-0.673366\pi$
$571$	−24.7614			−1.03623			−0.518116	+	0.855310i	$0.673366\pi$
$572$		0			0
$573$	17.0804			0.713543
$574$		0			0
$575$	−13.5194	−	23.4162i	−0.563797	−	0.976525i
$576$		0			0
$577$		−	13.5090i		−	0.562388i	−0.959651	−	0.281194i	$-0.909270\pi$
$577$		−	13.5090i		−	0.562388i	0.959651	−	0.281194i	$-0.0907304\pi$
$578$		0			0
$579$	−12.9695	−	7.48792i	−0.538992	−	0.311187i
$580$		0			0
$581$	0.868076	−	1.50355i	0.0360139	−	0.0623779i
$582$		0			0
$583$	14.1864	−	8.19053i	0.587541	−	0.339217i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	5.80500	−	3.35152i	0.239598	−	0.138332i	−0.375394	−	0.926865i	$-0.622493\pi$
$587$	5.80500	−	3.35152i	0.239598	−	0.138332i	0.614992	+	0.788533i	$0.289159\pi$
$588$		0			0
$589$	−9.69269	+	16.7882i	−0.399380	+	0.691747i
$590$		0			0
$591$	−0.166870	−	0.0963427i	−0.00686414	−	0.00396301i
$592$		0			0
$593$		−	16.0194i		−	0.657837i	−0.944358	−	0.328918i	$-0.893316\pi$
$593$		−	16.0194i		−	0.657837i	0.944358	−	0.328918i	$-0.106684\pi$
$594$		0			0
$595$	0.801274	+	1.38785i	0.0328490	+	0.0568962i
$596$		0			0
$597$	11.8726			0.485914
$598$		0			0
$599$	19.8243			0.809999			0.404999	−	0.914317i	$-0.367272\pi$
$599$	19.8243			0.809999			0.404999	+	0.914317i	$0.367272\pi$
$600$		0			0
$601$	−14.5954	−	25.2800i	−0.595360	−	1.03119i	−0.993496	−	0.113868i	$-0.963676\pi$
$601$	−14.5954	−	25.2800i	−0.595360	−	1.03119i	0.398135	−	0.917327i	$-0.369657\pi$
$602$		0			0
$603$		−	8.04892i		−	0.327777i
$604$		0			0
$605$	−1.21166	−	0.699554i	−0.0492611	−	0.0284409i
$606$		0			0
$607$	3.18114	−	5.50989i	0.129118	−	0.223640i	−0.794217	−	0.607634i	$-0.792119\pi$
$607$	3.18114	−	5.50989i	0.129118	−	0.223640i	0.923335	+	0.383995i	$0.125452\pi$
$608$		0			0
$609$	−2.54525	+	1.46950i	−0.103139	+	0.0595472i
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−26.6485	+	15.3855i	−1.07632	+	0.621416i	−0.929902	−	0.367806i	$-0.880109\pi$
$613$	−26.6485	+	15.3855i	−1.07632	+	0.621416i	−0.146422	+	0.989222i	$0.546776\pi$
$614$		0			0
$615$	−1.08211	+	1.87426i	−0.0436347	+	0.0755775i
$616$		0			0
$617$	10.4132	+	6.01208i	0.419221	+	0.242037i	0.694744	−	0.719257i	$-0.255518\pi$
$617$	10.4132	+	6.01208i	0.419221	+	0.242037i	−0.275523	+	0.961294i	$0.588851\pi$
$618$		0			0
$619$			9.02715i			0.362832i	0.983406	+	0.181416i	$0.0580680\pi$
$619$			9.02715i			0.362832i	−0.983406	+	0.181416i	$0.941932\pi$
$620$		0			0
$621$	2.88135	+	4.99065i	0.115625	+	0.200268i
$622$		0			0
$623$	18.8495			0.755190
$624$		0			0
$625$	20.4752			0.819007
$626$		0			0
$627$	6.74243	+	11.6782i	0.269267	+	0.466384i
$628$		0			0
$629$		−	1.28514i		−	0.0512420i
$630$		0			0
$631$	−13.4689	−	7.77628i	−0.536189	−	0.309569i	0.207344	−	0.978268i	$-0.433518\pi$
$631$	−13.4689	−	7.77628i	−0.536189	−	0.309569i	−0.743533	+	0.668699i	$0.766851\pi$
$632$		0			0
$633$	3.01693	−	5.22547i	0.119912	−	0.207694i
$634$		0			0
$635$	−3.44898	+	1.99127i	−0.136869	+	0.0790212i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	−11.8576	+	6.84601i	−0.469081	+	0.270824i
$640$		0			0
$641$	−9.71044	+	16.8190i	−0.383539	+	0.664310i	−0.991565	−	0.129608i	$-0.958628\pi$
$641$	−9.71044	+	16.8190i	−0.383539	+	0.664310i	0.608026	+	0.793917i	$0.291962\pi$
$642$		0			0
$643$	−23.2326	−	13.4133i	−0.916204	−	0.528971i	−0.0337821	−	0.999429i	$-0.510755\pi$
$643$	−23.2326	−	13.4133i	−0.916204	−	0.528971i	−0.882422	+	0.470458i	$0.844089\pi$
$644$		0			0
$645$			5.10321i			0.200939i
$646$		0			0
$647$	2.54407	+	4.40646i	0.100018	+	0.173236i	0.911692	−	0.410875i	$-0.134777\pi$
$647$	2.54407	+	4.40646i	0.100018	+	0.173236i	−0.811674	+	0.584111i	$0.801443\pi$
$648$		0			0
$649$	−9.05562			−0.355464
$650$		0			0
$651$	−4.39075			−0.172087
$652$		0			0
$653$	9.85786	+	17.0743i	0.385768	+	0.668169i	0.991875	−	0.127213i	$-0.0406033\pi$
$653$	9.85786	+	17.0743i	0.385768	+	0.668169i	−0.606108	+	0.795383i	$0.707270\pi$
$654$		0			0
$655$			7.28680i			0.284719i
$656$		0			0
$657$	−8.11170	−	4.68329i	−0.316468	−	0.182713i
$658$		0			0
$659$	−4.71260	+	8.16245i	−0.183577	+	0.317964i	−0.943096	−	0.332521i	$-0.892101\pi$
$659$	−4.71260	+	8.16245i	−0.183577	+	0.317964i	0.759519	+	0.650485i	$0.225434\pi$
$660$		0			0
$661$	−25.9761	+	14.9973i	−1.01035	+	0.583328i	−0.911295	−	0.411754i	$-0.864916\pi$
$661$	−25.9761	+	14.9973i	−1.01035	+	0.583328i	−0.0990583	+	0.995082i	$0.531583\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−2.33458	+	1.34787i	−0.0905312	+	0.0522682i
$666$		0			0
$667$	8.07338	−	13.9835i	0.312602	−	0.541443i
$668$		0			0
$669$	−7.38415	−	4.26324i	−0.285488	−	0.164826i
$670$		0			0
$671$			31.7542i			1.22586i
$672$		0			0
$673$	−23.8647	−	41.3349i	−0.919918	−	1.59334i	−0.799538	−	0.600616i	$-0.794922\pi$
$673$	−23.8647	−	41.3349i	−0.919918	−	1.59334i	−0.120380	−	0.992728i	$-0.538411\pi$
$674$		0			0
$675$	−4.69202			−0.180596
$676$		0			0
$677$	−42.9124			−1.64926			−0.824630	−	0.565673i	$-0.808616\pi$
$677$	−42.9124			−1.64926			−0.824630	+	0.565673i	$0.808616\pi$
$678$		0			0
$679$	−0.691062	−	1.19695i	−0.0265205	−	0.0459349i
$680$		0			0
$681$			11.0000i			0.421521i
$682$		0			0
$683$	−0.840443	−	0.485230i	−0.0321587	−	0.0185668i	0.483834	−	0.875160i	$-0.339244\pi$
$683$	−0.840443	−	0.485230i	−0.0321587	−	0.0185668i	−0.515993	+	0.856593i	$0.672577\pi$
$684$		0			0
$685$	6.05227	−	10.4828i	0.231245	−	0.400529i
$686$		0			0
$687$	15.5121	−	8.95593i	0.591824	−	0.341690i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−1.01031	+	0.583302i	−0.0384339	+	0.0221898i	−0.519094	−	0.854717i	$-0.673730\pi$
$691$	−1.01031	+	0.583302i	−0.0384339	+	0.0221898i	0.480660	+	0.876907i	$0.340397\pi$
$692$		0			0
$693$	−1.52715	+	2.64510i	−0.0580115	+	0.100479i
$694$		0			0
$695$	1.42555	+	0.823044i	0.0540743	+	0.0312198i
$696$		0			0
$697$		−	10.7362i		−	0.406661i
$698$		0			0
$699$	−11.4080	−	19.7592i	−0.431489	−	0.747361i
$700$		0			0
$701$	17.3274			0.654445			0.327223	−	0.944947i	$-0.393887\pi$
$701$	17.3274			0.654445			0.327223	+	0.944947i	$0.393887\pi$
$702$		0			0
$703$	2.16182			0.0815345
$704$		0			0
$705$	−3.19687	−	5.53713i	−0.120401	−	0.208541i
$706$		0			0
$707$		−	15.7597i		−	0.592705i
$708$		0			0
$709$	−14.2549	−	8.23005i	−0.535353	−	0.309086i	0.207841	−	0.978163i	$-0.433356\pi$
$709$	−14.2549	−	8.23005i	−0.535353	−	0.309086i	−0.743193	+	0.669077i	$0.766690\pi$
$710$		0			0
$711$	1.80463	−	3.12570i	0.0676788	−	0.117223i
$712$		0			0
$713$	20.8908	−	12.0613i	0.782366	−	0.451699i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	7.22682	−	4.17241i	0.269891	−	0.155821i
$718$		0			0
$719$	7.08575	−	12.2729i	0.264254	−	0.457701i	−0.703114	−	0.711077i	$-0.748208\pi$
$719$	7.08575	−	12.2729i	0.264254	−	0.457701i	0.967368	+	0.253376i	$0.0815409\pi$
$720$		0			0
$721$	8.36422	+	4.82908i	0.311500	+	0.179845i
$722$		0			0
$723$			3.75541i			0.139665i
$724$		0			0
$725$	6.57338	+	11.3854i	0.244129	+	0.422844i
$726$		0			0
$727$	37.8810			1.40493			0.702464	−	0.711719i	$-0.252083\pi$
$727$	37.8810			1.40493			0.702464	+	0.711719i	$0.252083\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	−12.6579	−	21.9242i	−0.468171	−	0.810895i
$732$		0			0
$733$		−	0.377338i		−	0.0139373i	−0.999976	−	0.00696865i	$-0.997782\pi$
$733$		−	0.377338i		−	0.0139373i	0.999976	−	0.00696865i	$-0.00221821\pi$
$734$		0			0
$735$	2.83548	+	1.63706i	0.104588	+	0.0603840i
$736$		0			0
$737$	−11.7186	+	20.2973i	−0.431662	+	0.747660i
$738$		0			0
$739$	0.975413	−	0.563155i	0.0358811	−	0.0207160i	−0.481952	−	0.876198i	$-0.660072\pi$
$739$	0.975413	−	0.563155i	0.0358811	−	0.0207160i	0.517833	+	0.855482i	$0.326739\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	19.6628	−	11.3523i	0.721360	−	0.416477i	−0.0938932	−	0.995582i	$-0.529931\pi$
$743$	19.6628	−	11.3523i	0.721360	−	0.416477i	0.815253	+	0.579105i	$0.196598\pi$
$744$		0			0
$745$	1.58061	−	2.73770i	0.0579092	−	0.100302i
$746$		0			0
$747$	1.43343	+	0.827593i	0.0524466	+	0.0302800i
$748$		0			0
$749$		−	7.58317i		−	0.277083i
$750$		0			0
$751$	6.96412	+	12.0622i	0.254124	+	0.440157i	0.964657	−	0.263508i	$-0.0848794\pi$
$751$	6.96412	+	12.0622i	0.254124	+	0.440157i	−0.710533	+	0.703664i	$0.751546\pi$
$752$		0			0
$753$	−30.3086			−1.10451
$754$		0			0
$755$	−10.6256			−0.386707
$756$		0			0
$757$	−0.583302	−	1.01031i	−0.0212005	−	0.0367203i	0.855231	−	0.518248i	$-0.173415\pi$
$757$	−0.583302	−	1.01031i	−0.0212005	−	0.0367203i	−0.876431	+	0.481527i	$0.840082\pi$
$758$		0			0
$759$		−	16.7802i		−	0.609081i
$760$		0			0
$761$	−0.711690	−	0.410895i	−0.0257987	−	0.0148949i	0.487045	−	0.873377i	$-0.338075\pi$
$761$	−0.711690	−	0.410895i	−0.0257987	−	0.0148949i	−0.512844	+	0.858482i	$0.671408\pi$
$762$		0			0
$763$	−8.13102	+	14.0833i	−0.294363	+	0.509851i
$764$		0			0
$765$	−1.32312	+	0.763906i	−0.0478376	+	0.0276191i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	29.1865	−	16.8509i	1.05249	−	0.607657i	0.129147	−	0.991625i	$-0.458776\pi$
$769$	29.1865	−	16.8509i	1.05249	−	0.607657i	0.923346	+	0.383968i	$0.125443\pi$
$770$		0			0
$771$	10.4206	−	18.0490i	0.375288	−	0.650018i
$772$		0			0
$773$	−0.661675	−	0.382019i	−0.0237988	−	0.0137403i	0.488053	−	0.872814i	$-0.337707\pi$
$773$	−0.661675	−	0.382019i	−0.0237988	−	0.0137403i	−0.511852	+	0.859074i	$0.671040\pi$
$774$		0			0
$775$			19.6407i			0.705515i
$776$		0			0
$777$	0.244824	+	0.424047i	0.00878300	+	0.0152126i
$778$		0			0
$779$	18.0599			0.647064
$780$		0			0
$781$	39.8692			1.42663
$782$		0			0
$783$	−1.40097	−	2.42655i	−0.0500665	−	0.0867178i
$784$		0			0
$785$		−	4.65279i		−	0.166065i
$786$		0			0
$787$	−10.5429	−	6.08695i	−0.375814	−	0.216976i	0.300181	−	0.953882i	$-0.402953\pi$
$787$	−10.5429	−	6.08695i	−0.375814	−	0.216976i	−0.675995	+	0.736906i	$0.736286\pi$
$788$		0			0
$789$	4.29321	−	7.43606i	0.152842	−	0.264731i
$790$		0			0
$791$	−12.5600	+	7.25153i	−0.446583	+	0.257835i
$792$		0			0
$793$		0			0
$794$		0			0
$795$	−2.70373	+	1.56100i	−0.0958914	+	0.0553629i
$796$		0			0
$797$	19.1444	−	33.1590i	0.678128	−	1.17455i	−0.297416	−	0.954748i	$-0.596125\pi$
$797$	19.1444	−	33.1590i	0.678128	−	1.17455i	0.975544	−	0.219804i	$-0.0705418\pi$
$798$		0			0
$799$	27.4685	+	15.8589i	0.971764	+	0.561048i
$800$		0			0
$801$			17.9705i			0.634955i
$802$		0			0
$803$	13.6371	+	23.6201i	0.481242	+	0.833535i
$804$		0			0
$805$	3.35450			0.118231
$806$		0			0
$807$	−14.8267			−0.521924
$808$		0			0
$809$	14.9819	+	25.9494i	0.526735	+	0.912331i	0.999515	+	0.0311508i	$0.00991722\pi$
$809$	14.9819	+	25.9494i	0.526735	+	0.912331i	−0.472780	+	0.881181i	$0.656749\pi$
$810$		0			0
$811$			46.3521i			1.62764i	0.581115	+	0.813821i	$0.302617\pi$
$811$			46.3521i			1.62764i	−0.581115	+	0.813821i	$0.697383\pi$
$812$		0			0
$813$	−6.26561	−	3.61745i	−0.219744	−	0.126869i
$814$		0			0
$815$	−0.546761	+	0.947019i	−0.0191522	+	0.0331726i
$816$		0			0
$817$	36.8800	−	21.2927i	1.29027	−	0.744936i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−41.8381	+	24.1552i	−1.46016	+	0.843024i	−0.999018	−	0.0443025i	$-0.985893\pi$
$821$	−41.8381	+	24.1552i	−1.46016	+	0.843024i	−0.461142	+	0.887326i	$0.652560\pi$
$822$		0			0
$823$	−2.65615	+	4.60058i	−0.0925874	+	0.160366i	−0.908599	−	0.417669i	$-0.862847\pi$
$823$	−2.65615	+	4.60058i	−0.0925874	+	0.160366i	0.816012	+	0.578035i	$0.196180\pi$
$824$		0			0
$825$	11.8321	+	6.83124i	0.411939	+	0.237833i
$826$		0			0
$827$			22.0968i			0.768380i	0.923254	+	0.384190i	$0.125519\pi$
$827$			22.0968i			0.768380i	−0.923254	+	0.384190i	$0.874481\pi$
$828$		0			0
$829$	−0.808274	−	1.39997i	−0.0280725	−	0.0486230i	0.851648	−	0.524114i	$-0.175604\pi$
$829$	−0.808274	−	1.39997i	−0.0280725	−	0.0486230i	−0.879720	+	0.475491i	$0.842270\pi$
$830$		0			0
$831$	11.2054			0.388710
$832$		0			0
$833$	−16.2422			−0.562759
$834$		0			0
$835$	6.88135	+	11.9189i	0.238139	+	0.412469i
$836$		0			0
$837$		−	4.18598i		−	0.144689i
$838$		0			0
$839$	35.5783	+	20.5411i	1.22830	+	0.709159i	0.966674	−	0.256010i	$-0.0824081\pi$
$839$	35.5783	+	20.5411i	1.22830	+	0.709159i	0.261626	+	0.965169i	$0.415741\pi$
$840$		0			0
$841$	10.5746	−	18.3157i	0.364640	−	0.631576i
$842$		0			0
$843$	10.6523	−	6.15010i	0.366884	−	0.211821i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−2.29015	+	1.32222i	−0.0786903	+	0.0454319i
$848$		0			0
$849$	−12.1761	+	21.0895i	−0.417881	+	0.723791i
$850$		0			0
$851$	−2.32970	−	1.34505i	−0.0798610	−	0.0461078i
$852$		0			0
$853$			35.0441i			1.19989i	0.800042	+	0.599944i	$0.204811\pi$
$853$			35.0441i			1.19989i	−0.800042	+	0.599944i	$0.795189\pi$
$854$		0			0
$855$	−1.28501	−	2.22571i	−0.0439465	−	0.0761175i
$856$		0			0
$857$	3.76079			0.128466			0.0642331	−	0.997935i	$-0.479540\pi$
$857$	3.76079			0.128466			0.0642331	+	0.997935i	$0.479540\pi$
$858$		0			0
$859$	7.11662			0.242816			0.121408	−	0.992603i	$-0.461259\pi$
$859$	7.11662			0.242816			0.121408	+	0.992603i	$0.461259\pi$
$860$		0			0
$861$	2.04527	+	3.54251i	0.0697026	+	0.120728i
$862$		0			0
$863$			57.5900i			1.96039i	0.198044	+	0.980193i	$0.436541\pi$
$863$			57.5900i			1.96039i	−0.198044	+	0.980193i	$0.563459\pi$
$864$		0			0
$865$	−1.23581	−	0.713496i	−0.0420189	−	0.0242596i
$866$		0			0
$867$	−4.71044	+	8.15872i	−0.159975	+	0.277085i
$868$		0			0
$869$	−9.10159	+	5.25481i	−0.308750	+	0.178257i
$870$		0			0
$871$		0			0
$872$		0			0
$873$	1.14113	−	0.658834i	0.0386215	−	0.0222981i
$874$		0			0
$875$	−2.82089	+	4.88592i	−0.0953634	+	0.165174i
$876$		0			0
$877$	16.6407	+	9.60752i	0.561917	+	0.324423i	0.753915	−	0.656972i	$-0.228163\pi$
$877$	16.6407	+	9.60752i	0.561917	+	0.324423i	−0.191997	+	0.981395i	$0.561497\pi$
$878$		0			0
$879$		−	6.20237i		−	0.209201i
$880$		0			0
$881$	−5.57218	−	9.65130i	−0.187731	−	0.325160i	0.756762	−	0.653690i	$-0.226780\pi$
$881$	−5.57218	−	9.65130i	−0.187731	−	0.325160i	−0.944494	+	0.328530i	$0.893447\pi$
$882$		0			0
$883$	4.28919			0.144343			0.0721714	−	0.997392i	$-0.477007\pi$
$883$	4.28919			0.144343			0.0721714	+	0.997392i	$0.477007\pi$
$884$		0			0
$885$	1.72587			0.0580146
$886$		0			0
$887$	16.5869	+	28.7294i	0.556935	+	0.964640i	0.997750	+	0.0670421i	$0.0213562\pi$
$887$	16.5869	+	28.7294i	0.556935	+	0.964640i	−0.440815	+	0.897598i	$0.645310\pi$
$888$		0			0
$889$			7.52734i			0.252459i
$890$		0			0
$891$	−2.52174	−	1.45593i	−0.0844815	−	0.0487754i
$892$		0			0
$893$	−26.6773	+	46.2064i	−0.892720	+	1.54624i
$894$		0			0
$895$	1.44441	−	0.833929i	0.0482812	−	0.0278752i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−10.1575	+	5.86443i	−0.338771	+	0.195590i
$900$		0			0
$901$	7.74376	−	13.4126i	0.257982	−	0.446838i
$902$		0			0
$903$	8.35325	+	4.82275i	0.277979	+	0.160491i
$904$		0			0
$905$			8.14914i			0.270887i
$906$		0			0
$907$	−20.3141	−	35.1850i	−0.674518	−	1.16830i	−0.976610	−	0.215020i	$-0.931018\pi$
$907$	−20.3141	−	35.1850i	−0.674518	−	1.16830i	0.302092	−	0.953279i	$-0.402315\pi$
$908$		0			0
$909$	15.0248			0.498340
$910$		0			0
$911$	46.2731			1.53309			0.766547	−	0.642188i	$-0.221973\pi$
$911$	46.2731			1.53309			0.766547	+	0.642188i	$0.221973\pi$
$912$		0			0
$913$	−2.40983	−	4.17395i	−0.0797537	−	0.138137i
$914$		0			0
$915$		−	6.05190i		−	0.200070i
$916$		0			0
$917$	11.9275	+	6.88633i	0.393880	+	0.227407i
$918$		0			0
$919$	−25.2337	+	43.7061i	−0.832383	+	1.44173i	0.0637605	+	0.997965i	$0.479691\pi$
$919$	−25.2337	+	43.7061i	−0.832383	+	1.44173i	−0.896143	+	0.443764i	$0.853643\pi$
$920$		0			0
$921$	22.9319	−	13.2397i	0.755632	−	0.436264i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	1.89685	−	1.09515i	0.0623680	−	0.0360082i
$926$		0			0
$927$	−4.60388	+	7.97415i	−0.151211	+	0.261905i
$928$		0			0
$929$	15.5235	+	8.96250i	0.509310	+	0.294050i	0.732550	−	0.680713i	$-0.238330\pi$
$929$	15.5235	+	8.96250i	0.509310	+	0.294050i	−0.223240	+	0.974763i	$0.571663\pi$
$930$		0			0
$931$		−	27.3220i		−	0.895442i
$932$		0			0
$933$	−7.76540	−	13.4501i	−0.254228	−	0.440335i
$934$		0			0
$935$	4.44876			0.145490
$936$		0			0
$937$	5.85325			0.191217			0.0956086	−	0.995419i	$-0.469520\pi$
$937$	5.85325			0.191217			0.0956086	+	0.995419i	$0.469520\pi$
$938$		0			0
$939$	8.48403	+	14.6948i	0.276866	+	0.479546i
$940$		0			0
$941$			56.7958i			1.85149i	0.378147	+	0.925745i	$0.376561\pi$
$941$			56.7958i			1.85149i	−0.378147	+	0.925745i	$0.623439\pi$
$942$		0			0
$943$	−19.4624	−	11.2366i	−0.633783	−	0.365915i
$944$		0			0
$945$	0.291053	−	0.504118i	0.00946794	−	0.0163990i
$946$		0			0
$947$	−5.32583	+	3.07487i	−0.173066	+	0.0999198i	−0.584031	−	0.811731i	$-0.698525\pi$
$947$	−5.32583	+	3.07487i	−0.173066	+	0.0999198i	0.410965	+	0.911651i	$0.365192\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	22.0894	−	12.7533i	0.716298	−	0.413555i
$952$		0			0
$953$	24.4889	−	42.4160i	0.793273	−	1.37399i	−0.130658	−	0.991428i	$-0.541709\pi$
$953$	24.4889	−	42.4160i	0.793273	−	1.37399i	0.923930	−	0.382561i	$-0.124958\pi$
$954$		0			0
$955$	8.20896	+	4.73945i	0.265636	+	0.153365i
$956$		0			0
$957$			8.15883i			0.263738i
$958$		0			0
$959$	−11.4393	−	19.8134i	−0.369394	−	0.639809i
$960$		0			0
$961$	13.4776			0.434760
$962$		0			0
$963$	7.22952			0.232968
$964$		0			0
$965$	−4.15548	−	7.19750i	−0.133770	−	0.231696i
$966$		0			0
$967$		−	54.2616i		−	1.74493i	−0.488673	−	0.872467i	$-0.662519\pi$
$967$		−	54.2616i		−	1.74493i	0.488673	−	0.872467i	$-0.337481\pi$
$968$		0			0
$969$	11.0412	+	6.37465i	0.354695	+	0.204783i
$970$		0			0
$971$	18.1571	−	31.4490i	0.582689	−	1.00925i	−0.412470	−	0.910971i	$-0.635334\pi$
$971$	18.1571	−	31.4490i	0.582689	−	1.00925i	0.995159	−	0.0982761i	$-0.0313328\pi$
$972$		0			0
$973$	2.69442	−	1.55562i	0.0863790	−	0.0498710i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−43.0739	+	24.8687i	−1.37806	+	0.795621i	−0.991925	−	0.126822i	$-0.959522\pi$
$977$	−43.0739	+	24.8687i	−1.37806	+	0.795621i	−0.386131	+	0.922444i	$0.626189\pi$
$978$		0			0
$979$	26.1637	−	45.3168i	0.836195	−	1.44833i
$980$		0			0
$981$	−13.4266	−	7.75182i	−0.428677	−	0.247497i
$982$		0			0
$983$		−	10.9981i		−	0.350784i	−0.984499	−	0.175392i	$-0.943881\pi$
$983$		−	10.9981i		−	0.350784i	0.984499	−	0.175392i	$-0.0561193\pi$
$984$		0			0
$985$	−0.0534662	−	0.0926061i	−0.00170357	−	0.00295068i
$986$		0			0
$987$	−12.0847			−0.384660
$988$		0			0
$989$	−52.9920			−1.68505
$990$		0			0
$991$	−3.36539	−	5.82902i	−0.106905	−	0.185165i	0.807610	−	0.589717i	$-0.200761\pi$
$991$	−3.36539	−	5.82902i	−0.106905	−	0.185165i	−0.914515	+	0.404552i	$0.867427\pi$
$992$		0			0
$993$			10.6136i			0.336811i
$994$		0			0
$995$	5.70608	+	3.29440i	0.180895	+	0.104440i
$996$		0			0
$997$	−21.6749	+	37.5420i	−0.686450	+	1.18897i	0.286529	+	0.958072i	$0.407498\pi$
$997$	−21.6749	+	37.5420i	−0.686450	+	1.18897i	−0.972979	+	0.230894i	$0.925835\pi$
$998$		0			0
$999$	−0.404271	+	0.233406i	−0.0127906	+	0.00738464i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2028.2.q.j.361.4		12
13.2	odd	12		2028.2.a.j.1.2	yes	3
13.3	even	3		2028.2.b.f.337.4		6
13.4	even	6	inner	2028.2.q.j.1837.3		12
13.5	odd	4		2028.2.i.m.2005.2		6
13.6	odd	12		2028.2.i.m.529.2		6
13.7	odd	12		2028.2.i.l.529.2		6
13.8	odd	4		2028.2.i.l.2005.2		6
13.9	even	3	inner	2028.2.q.j.1837.4		12
13.10	even	6		2028.2.b.f.337.3		6
13.11	odd	12		2028.2.a.i.1.2	✓	3
13.12	even	2	inner	2028.2.q.j.361.3		12
39.2	even	12		6084.2.a.y.1.2		3
39.11	even	12		6084.2.a.bb.1.2		3
39.23	odd	6		6084.2.b.r.4393.4		6
39.29	odd	6		6084.2.b.r.4393.3		6
52.11	even	12		8112.2.a.ch.1.2		3
52.15	even	12		8112.2.a.co.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2028.2.a.i.1.2	✓	3	13.11	odd	12
2028.2.a.j.1.2	yes	3	13.2	odd	12
2028.2.b.f.337.3		6	13.10	even	6
2028.2.b.f.337.4		6	13.3	even	3
2028.2.i.l.529.2		6	13.7	odd	12
2028.2.i.l.2005.2		6	13.8	odd	4
2028.2.i.m.529.2		6	13.6	odd	12
2028.2.i.m.2005.2		6	13.5	odd	4
2028.2.q.j.361.3		12	13.12	even	2	inner
2028.2.q.j.361.4		12	1.1	even	1	trivial
2028.2.q.j.1837.3		12	13.4	even	6	inner
2028.2.q.j.1837.4		12	13.9	even	3	inner
6084.2.a.y.1.2		3	39.2	even	12
6084.2.a.bb.1.2		3	39.11	even	12
6084.2.b.r.4393.3		6	39.29	odd	6
6084.2.b.r.4393.4		6	39.23	odd	6
8112.2.a.ch.1.2		3	52.11	even	12
8112.2.a.co.1.2		3	52.15	even	12

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2028 = 2^{2} \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2028.q (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 + 0.866025i) q^{3} +0.554958i q^{5} +(0.908389 + 0.524459i) q^{7} +(-0.500000 + 0.866025i) q^{9} +(2.52174 - 1.45593i) q^{11} +(-0.480608 + 0.277479i) q^{15} +(1.37651 - 2.38419i) q^{17} +(4.01058 + 2.31551i) q^{19} +1.04892i q^{21} +(-2.88135 - 4.99065i) q^{23} +4.69202 q^{25} -1.00000 q^{27} +(1.40097 + 2.42655i) q^{29} +4.18598i q^{31} +(2.52174 + 1.45593i) q^{33} +(-0.291053 + 0.504118i) q^{35} +(0.404271 - 0.233406i) q^{37} +(3.37730 - 1.94989i) q^{41} +(4.59783 - 7.96368i) q^{43} +(-0.480608 - 0.277479i) q^{45} +11.5211i q^{47} +(-2.94989 - 5.10935i) q^{49} +2.75302 q^{51} +5.62565 q^{53} +(0.807979 + 1.39946i) q^{55} +4.63102i q^{57} +(-2.69327 - 1.55496i) q^{59} +(-5.45257 + 9.44414i) q^{61} +(-0.908389 + 0.524459i) q^{63} +(-6.97057 + 4.02446i) q^{67} +(2.88135 - 4.99065i) q^{69} +(11.8576 + 6.84601i) q^{71} +9.36658i q^{73} +(2.34601 + 4.06341i) q^{75} +3.05429 q^{77} -3.60925 q^{79} +(-0.500000 - 0.866025i) q^{81} -1.65519i q^{83} +(1.32312 + 0.763906i) q^{85} +(-1.40097 + 2.42655i) q^{87} +(15.5629 - 8.98523i) q^{89} +(-3.62517 + 2.09299i) q^{93} +(-1.28501 + 2.22571i) q^{95} +(-1.14113 - 0.658834i) q^{97} +2.91185i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 6 q^{3} - 6 q^{9} + 26 q^{17} + 36 q^{25} - 12 q^{27} + 8 q^{29} + 8 q^{35} - 18 q^{43} + 10 q^{49} + 52 q^{51} + 20 q^{53} + 30 q^{55} - 14 q^{61} + 18 q^{75} - 12 q^{77} + 4 q^{79} - 6 q^{81} - 8 q^{87}+ \cdots + 34 q^{95}+O(q^{100}) \) 12 * q + 6 * q^3 - 6 * q^9 + 26 * q^17 + 36 * q^25 - 12 * q^27 + 8 * q^29 + 8 * q^35 - 18 * q^43 + 10 * q^49 + 52 * q^51 + 20 * q^53 + 30 * q^55 - 14 * q^61 + 18 * q^75 - 12 * q^77 + 4 * q^79 - 6 * q^81 - 8 * q^87 + 34 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more