LMFDB - Embedded newform 1620.2.i.m.541.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1620,2,Mod(541,1620)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1620, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4, 0]))

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

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

chi := DirichletCharacter("1620.541");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1620 = 2^{2} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1620.i (of order $3$, 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:	$12.9357651274$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			541.2
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1620.541
Dual form			1620.2.i.m.1081.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 - 0.866025i) q^{5} +(1.36603 - 2.36603i) q^{7} +O(q^{10})$	$q+(-0.500000 - 0.866025i) q^{5} +(1.36603 - 2.36603i) q^{7} +(0.866025 - 1.50000i) q^{11} +(-2.73205 - 4.73205i) q^{13} +4.73205 q^{17} -4.46410 q^{19} +(1.73205 + 3.00000i) q^{23} +(-0.500000 + 0.866025i) q^{25} +(-3.86603 + 6.69615i) q^{29} +(-2.96410 - 5.13397i) q^{31} -2.73205 q^{35} -6.19615 q^{37} +(-5.59808 - 9.69615i) q^{41} +(-1.63397 + 2.83013i) q^{43} +(-0.633975 + 1.09808i) q^{47} +(-0.232051 - 0.401924i) q^{49} +7.26795 q^{53} -1.73205 q^{55} +(-3.86603 - 6.69615i) q^{59} +(2.00000 - 3.46410i) q^{61} +(-2.73205 + 4.73205i) q^{65} +(-3.19615 - 5.53590i) q^{67} +11.1962 q^{71} -0.196152 q^{73} +(-2.36603 - 4.09808i) q^{77} +(7.19615 - 12.4641i) q^{79} +(-7.56218 + 13.0981i) q^{83} +(-2.36603 - 4.09808i) q^{85} -5.19615 q^{89} -14.9282 q^{91} +(2.23205 + 3.86603i) q^{95} +(-0.366025 + 0.633975i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{5} + 2 q^{7}+O(q^{10}) $ 4 * q - 2 * q^5 + 2 * q^7	$ 4 q - 2 q^{5} + 2 q^{7} - 4 q^{13} + 12 q^{17} - 4 q^{19} - 2 q^{25} - 12 q^{29} + 2 q^{31} - 4 q^{35} - 4 q^{37} - 12 q^{41} - 10 q^{43} - 6 q^{47} + 6 q^{49} + 36 q^{53} - 12 q^{59} + 8 q^{61} - 4 q^{65} + 8 q^{67} + 24 q^{71} + 20 q^{73} - 6 q^{77} + 8 q^{79} - 6 q^{83} - 6 q^{85} - 32 q^{91} + 2 q^{95} + 2 q^{97}+O(q^{100}) $ 4 * q - 2 * q^5 + 2 * q^7 - 4 * q^13 + 12 * q^17 - 4 * q^19 - 2 * q^25 - 12 * q^29 + 2 * q^31 - 4 * q^35 - 4 * q^37 - 12 * q^41 - 10 * q^43 - 6 * q^47 + 6 * q^49 + 36 * q^53 - 12 * q^59 + 8 * q^61 - 4 * q^65 + 8 * q^67 + 24 * q^71 + 20 * q^73 - 6 * q^77 + 8 * q^79 - 6 * q^83 - 6 * q^85 - 32 * q^91 + 2 * q^95 + 2 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times$.

$n$	$811$	$1297$	$1541$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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			0
$3$		0			0
$4$		0			0
$5$	−0.500000	−	0.866025i	−0.223607	−	0.387298i
$5$	−0.500000	−	0.866025i	−0.223607	−	0.387298i
$6$		0			0
$7$	1.36603	−	2.36603i	0.516309	−	0.894274i	−0.483512	−	0.875338i	$-0.660639\pi$
$7$	1.36603	−	2.36603i	0.516309	−	0.894274i	0.999821	−	0.0189356i	$-0.00602775\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	0.866025	−	1.50000i	0.261116	−	0.452267i	−0.705422	−	0.708787i	$-0.749243\pi$
$11$	0.866025	−	1.50000i	0.261116	−	0.452267i	0.966539	+	0.256520i	$0.0825760\pi$
$12$		0			0
$13$	−2.73205	−	4.73205i	−0.757735	−	1.31243i	−0.944003	−	0.329936i	$-0.892973\pi$
$13$	−2.73205	−	4.73205i	−0.757735	−	1.31243i	0.186269	−	0.982499i	$-0.440360\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	4.73205			1.14769			0.573845	−	0.818964i	$-0.305451\pi$
$17$	4.73205			1.14769			0.573845	+	0.818964i	$0.305451\pi$
$18$		0			0
$19$	−4.46410			−1.02414			−0.512068	−	0.858945i	$-0.671120\pi$
$19$	−4.46410			−1.02414			−0.512068	+	0.858945i	$0.671120\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	1.73205	+	3.00000i	0.361158	+	0.625543i	0.988152	−	0.153481i	$-0.0490483\pi$
$23$	1.73205	+	3.00000i	0.361158	+	0.625543i	−0.626994	+	0.779024i	$0.715715\pi$
$24$		0			0
$25$	−0.500000	+	0.866025i	−0.100000	+	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−3.86603	+	6.69615i	−0.717903	+	1.24344i	0.243926	+	0.969794i	$0.421564\pi$
$29$	−3.86603	+	6.69615i	−0.717903	+	1.24344i	−0.961829	+	0.273651i	$0.911769\pi$
$30$		0			0
$31$	−2.96410	−	5.13397i	−0.532368	−	0.922089i	−0.999286	−	0.0377881i	$-0.987969\pi$
$31$	−2.96410	−	5.13397i	−0.532368	−	0.922089i	0.466917	−	0.884301i	$-0.345365\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−2.73205			−0.461801
$36$		0			0
$37$	−6.19615			−1.01864			−0.509321	−	0.860577i	$-0.670103\pi$
$37$	−6.19615			−1.01864			−0.509321	+	0.860577i	$0.670103\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−5.59808	−	9.69615i	−0.874273	−	1.51428i	−0.857536	−	0.514425i	$-0.828006\pi$
$41$	−5.59808	−	9.69615i	−0.874273	−	1.51428i	−0.0167371	−	0.999860i	$-0.505328\pi$
$42$		0			0
$43$	−1.63397	+	2.83013i	−0.249179	+	0.431590i	−0.963298	−	0.268434i	$-0.913494\pi$
$43$	−1.63397	+	2.83013i	−0.249179	+	0.431590i	0.714119	+	0.700024i	$0.246827\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−0.633975	+	1.09808i	−0.0924747	+	0.160171i	−0.908552	−	0.417772i	$-0.862811\pi$
$47$	−0.633975	+	1.09808i	−0.0924747	+	0.160171i	0.816077	+	0.577943i	$0.196144\pi$
$48$		0			0
$49$	−0.232051	−	0.401924i	−0.0331501	−	0.0574177i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	7.26795			0.998330			0.499165	−	0.866507i	$-0.333640\pi$
$53$	7.26795			0.998330			0.499165	+	0.866507i	$0.333640\pi$
$54$		0			0
$55$	−1.73205			−0.233550
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−3.86603	−	6.69615i	−0.503314	−	0.871765i	−0.999993	−	0.00383049i	$-0.998781\pi$
$59$	−3.86603	−	6.69615i	−0.503314	−	0.871765i	0.496679	−	0.867934i	$-0.334553\pi$
$60$		0			0
$61$	2.00000	−	3.46410i	0.256074	−	0.443533i	−0.709113	−	0.705095i	$-0.750904\pi$
$61$	2.00000	−	3.46410i	0.256074	−	0.443533i	0.965187	+	0.261562i	$0.0842377\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−2.73205	+	4.73205i	−0.338869	+	0.586939i
$66$		0			0
$67$	−3.19615	−	5.53590i	−0.390472	−	0.676318i	0.602040	−	0.798466i	$-0.294355\pi$
$67$	−3.19615	−	5.53590i	−0.390472	−	0.676318i	−0.992512	+	0.122149i	$0.961022\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	11.1962			1.32874			0.664369	−	0.747404i	$-0.268700\pi$
$71$	11.1962			1.32874			0.664369	+	0.747404i	$0.268700\pi$
$72$		0			0
$73$	−0.196152			−0.0229579			−0.0114790	−	0.999934i	$-0.503654\pi$
$73$	−0.196152			−0.0229579			−0.0114790	+	0.999934i	$0.503654\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−2.36603	−	4.09808i	−0.269634	−	0.467019i
$78$		0			0
$79$	7.19615	−	12.4641i	0.809630	−	1.40232i	−0.103490	−	0.994631i	$-0.533001\pi$
$79$	7.19615	−	12.4641i	0.809630	−	1.40232i	0.913120	−	0.407690i	$-0.133666\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−7.56218	+	13.0981i	−0.830057	+	1.43770i	0.0679356	+	0.997690i	$0.478359\pi$
$83$	−7.56218	+	13.0981i	−0.830057	+	1.43770i	−0.897992	+	0.440011i	$0.854975\pi$
$84$		0			0
$85$	−2.36603	−	4.09808i	−0.256631	−	0.444499i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−5.19615			−0.550791			−0.275396	−	0.961331i	$-0.588809\pi$
$89$	−5.19615			−0.550791			−0.275396	+	0.961331i	$0.588809\pi$
$90$		0			0
$91$	−14.9282			−1.56490
$92$		0			0
$93$		0			0
$94$		0			0
$95$	2.23205	+	3.86603i	0.229004	+	0.396646i
$96$		0			0
$97$	−0.366025	+	0.633975i	−0.0371642	+	0.0643704i	−0.884009	−	0.467469i	$-0.845166\pi$
$97$	−0.366025	+	0.633975i	−0.0371642	+	0.0643704i	0.846845	+	0.531840i	$0.178499\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	3.06218	−	5.30385i	0.304698	−	0.527753i	−0.672496	−	0.740101i	$-0.734778\pi$
$101$	3.06218	−	5.30385i	0.304698	−	0.527753i	0.977194	+	0.212348i	$0.0681111\pi$
$102$		0			0
$103$	−9.19615	−	15.9282i	−0.906124	−	1.56945i	−0.819402	−	0.573220i	$-0.805694\pi$
$103$	−9.19615	−	15.9282i	−0.906124	−	1.56945i	−0.0867223	−	0.996233i	$-0.527639\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−3.46410			−0.334887			−0.167444	−	0.985882i	$-0.553551\pi$
$107$	−3.46410			−0.334887			−0.167444	+	0.985882i	$0.553551\pi$
$108$		0			0
$109$	−7.92820			−0.759384			−0.379692	−	0.925113i	$-0.623970\pi$
$109$	−7.92820			−0.759384			−0.379692	+	0.925113i	$0.623970\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−0.169873	−	0.294229i	−0.0159803	−	0.0276787i	0.857925	−	0.513776i	$-0.171754\pi$
$113$	−0.169873	−	0.294229i	−0.0159803	−	0.0276787i	−0.873905	+	0.486097i	$0.838420\pi$
$114$		0			0
$115$	1.73205	−	3.00000i	0.161515	−	0.279751i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	6.46410	−	11.1962i	0.592563	−	1.02635i
$120$		0			0
$121$	4.00000	+	6.92820i	0.363636	+	0.629837i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$	4.19615			0.372348			0.186174	−	0.982517i	$-0.440391\pi$
$127$	4.19615			0.372348			0.186174	+	0.982517i	$0.440391\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	5.13397	+	8.89230i	0.448557	+	0.776924i	0.998292	−	0.0584149i	$-0.0186046\pi$
$131$	5.13397	+	8.89230i	0.448557	+	0.776924i	−0.549735	+	0.835339i	$0.685271\pi$
$132$		0			0
$133$	−6.09808	+	10.5622i	−0.528770	+	0.915857i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	2.19615	−	3.80385i	0.187630	−	0.324985i	−0.756830	−	0.653612i	$-0.773253\pi$
$137$	2.19615	−	3.80385i	0.187630	−	0.324985i	0.944460	+	0.328628i	$0.106586\pi$
$138$		0			0
$139$	−7.69615	−	13.3301i	−0.652779	−	1.13065i	−0.982446	−	0.186549i	$-0.940270\pi$
$139$	−7.69615	−	13.3301i	−0.652779	−	1.13065i	0.329666	−	0.944097i	$-0.393064\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−9.46410			−0.791428
$144$		0			0
$145$	7.73205			0.642112
$146$		0			0
$147$		0			0
$148$		0			0
$149$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$149$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$150$		0			0
$151$	8.69615	−	15.0622i	0.707683	−	1.22574i	−0.258032	−	0.966136i	$-0.583074\pi$
$151$	8.69615	−	15.0622i	0.707683	−	1.22574i	0.965715	−	0.259606i	$-0.0835928\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−2.96410	+	5.13397i	−0.238082	+	0.412371i
$156$		0			0
$157$	−1.63397	−	2.83013i	−0.130405	−	0.225869i	0.793428	−	0.608665i	$-0.208295\pi$
$157$	−1.63397	−	2.83013i	−0.130405	−	0.225869i	−0.923833	+	0.382796i	$0.874961\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	9.46410			0.745876
$162$		0			0
$163$	18.7321			1.46721			0.733604	−	0.679577i	$-0.237837\pi$
$163$	18.7321			1.46721			0.733604	+	0.679577i	$0.237837\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−10.5622	−	18.2942i	−0.817326	−	1.41565i	−0.907646	−	0.419737i	$-0.862122\pi$
$167$	−10.5622	−	18.2942i	−0.817326	−	1.41565i	0.0903199	−	0.995913i	$-0.471211\pi$
$168$		0			0
$169$	−8.42820	+	14.5981i	−0.648323	+	1.12293i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−12.1244	+	21.0000i	−0.921798	+	1.59660i	−0.125166	+	0.992136i	$0.539946\pi$
$173$	−12.1244	+	21.0000i	−0.921798	+	1.59660i	−0.796632	+	0.604465i	$0.793387\pi$
$174$		0			0
$175$	1.36603	+	2.36603i	0.103262	+	0.178855i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	12.1244			0.906217			0.453108	−	0.891455i	$-0.350315\pi$
$179$	12.1244			0.906217			0.453108	+	0.891455i	$0.350315\pi$
$180$		0			0
$181$	−16.4641			−1.22377			−0.611884	−	0.790948i	$-0.709588\pi$
$181$	−16.4641			−1.22377			−0.611884	+	0.790948i	$0.709588\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	3.09808	+	5.36603i	0.227775	+	0.394518i
$186$		0			0
$187$	4.09808	−	7.09808i	0.299681	−	0.519063i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	9.52628	−	16.5000i	0.689297	−	1.19390i	−0.282768	−	0.959188i	$-0.591253\pi$
$191$	9.52628	−	16.5000i	0.689297	−	1.19390i	0.972065	−	0.234710i	$-0.0754140\pi$
$192$		0			0
$193$	5.29423	+	9.16987i	0.381087	+	0.660062i	0.991218	−	0.132239i	$-0.0422165\pi$
$193$	5.29423	+	9.16987i	0.381087	+	0.660062i	−0.610131	+	0.792301i	$0.708883\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	13.8564			0.987228			0.493614	−	0.869681i	$-0.335676\pi$
$197$	13.8564			0.987228			0.493614	+	0.869681i	$0.335676\pi$
$198$		0			0
$199$	15.8564			1.12403			0.562015	−	0.827127i	$-0.310026\pi$
$199$	15.8564			1.12403			0.562015	+	0.827127i	$0.310026\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	10.5622	+	18.2942i	0.741320	+	1.28400i
$204$		0			0
$205$	−5.59808	+	9.69615i	−0.390987	+	0.677209i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−3.86603	+	6.69615i	−0.267419	+	0.463183i
$210$		0			0
$211$	9.96410	+	17.2583i	0.685957	+	1.18811i	0.973135	+	0.230235i	$0.0739496\pi$
$211$	9.96410	+	17.2583i	0.685957	+	1.18811i	−0.287178	+	0.957877i	$0.592717\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	3.26795			0.222872
$216$		0			0
$217$	−16.1962			−1.09947
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−12.9282	−	22.3923i	−0.869645	−	1.50627i
$222$		0			0
$223$	2.92820	−	5.07180i	0.196087	−	0.339633i	−0.751169	−	0.660110i	$-0.770510\pi$
$223$	2.92820	−	5.07180i	0.196087	−	0.339633i	0.947256	+	0.320477i	$0.103843\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	13.0981	−	22.6865i	0.869350	−	1.50576i	0.00668763	−	0.999978i	$-0.497871\pi$
$227$	13.0981	−	22.6865i	0.869350	−	1.50576i	0.862662	−	0.505780i	$-0.168795\pi$
$228$		0			0
$229$	8.92820	+	15.4641i	0.589992	+	1.02190i	0.994233	+	0.107244i	$0.0342026\pi$
$229$	8.92820	+	15.4641i	0.589992	+	1.02190i	−0.404240	+	0.914653i	$0.632464\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−24.5885			−1.61084			−0.805422	−	0.592702i	$-0.798061\pi$
$233$	−24.5885			−1.61084			−0.805422	+	0.592702i	$0.798061\pi$
$234$		0			0
$235$	1.26795			0.0827119
$236$		0			0
$237$		0			0
$238$		0			0
$239$	4.26795	+	7.39230i	0.276071	+	0.478168i	0.970405	−	0.241484i	$-0.0776343\pi$
$239$	4.26795	+	7.39230i	0.276071	+	0.478168i	−0.694334	+	0.719653i	$0.744301\pi$
$240$		0			0
$241$	−5.16025	+	8.93782i	−0.332401	+	0.575736i	−0.982982	−	0.183701i	$-0.941192\pi$
$241$	−5.16025	+	8.93782i	−0.332401	+	0.575736i	0.650581	+	0.759437i	$0.274525\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−0.232051	+	0.401924i	−0.0148252	+	0.0256780i
$246$		0			0
$247$	12.1962	+	21.1244i	0.776023	+	1.34411i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−20.5359			−1.29621			−0.648107	−	0.761549i	$-0.724439\pi$
$251$	−20.5359			−1.29621			−0.648107	+	0.761549i	$0.724439\pi$
$252$		0			0
$253$	6.00000			0.377217
$254$		0			0
$255$		0			0
$256$		0			0
$257$	7.73205	+	13.3923i	0.482312	+	0.835389i	0.999794	−	0.0203052i	$-0.00646380\pi$
$257$	7.73205	+	13.3923i	0.482312	+	0.835389i	−0.517482	+	0.855694i	$0.673130\pi$
$258$		0			0
$259$	−8.46410	+	14.6603i	−0.525934	+	0.910944i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−12.1244	+	21.0000i	−0.747620	+	1.29492i	0.201341	+	0.979521i	$0.435470\pi$
$263$	−12.1244	+	21.0000i	−0.747620	+	1.29492i	−0.948961	+	0.315394i	$0.897863\pi$
$264$		0			0
$265$	−3.63397	−	6.29423i	−0.223233	−	0.386651i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	16.2679			0.991874			0.495937	−	0.868358i	$-0.334825\pi$
$269$	16.2679			0.991874			0.495937	+	0.868358i	$0.334825\pi$
$270$		0			0
$271$	16.7846			1.01959			0.509796	−	0.860295i	$-0.329721\pi$
$271$	16.7846			1.01959			0.509796	+	0.860295i	$0.329721\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	0.866025	+	1.50000i	0.0522233	+	0.0904534i
$276$		0			0
$277$	−2.56218	+	4.43782i	−0.153946	+	0.266643i	−0.932675	−	0.360718i	$-0.882532\pi$
$277$	−2.56218	+	4.43782i	−0.153946	+	0.266643i	0.778729	+	0.627361i	$0.215865\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	14.6603	−	25.3923i	0.874557	−	1.51478i	0.0173240	−	0.999850i	$-0.494485\pi$
$281$	14.6603	−	25.3923i	0.874557	−	1.51478i	0.857233	−	0.514928i	$-0.172181\pi$
$282$		0			0
$283$	−8.26795	−	14.3205i	−0.491479	−	0.851266i	0.508473	−	0.861078i	$-0.330210\pi$
$283$	−8.26795	−	14.3205i	−0.491479	−	0.851266i	−0.999952	+	0.00981191i	$0.996877\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−30.5885			−1.80558
$288$		0			0
$289$	5.39230			0.317194
$290$		0			0
$291$		0			0
$292$		0			0
$293$	12.2942	+	21.2942i	0.718237	+	1.24402i	0.961698	+	0.274112i	$0.0883837\pi$
$293$	12.2942	+	21.2942i	0.718237	+	1.24402i	−0.243461	+	0.969911i	$0.578283\pi$
$294$		0			0
$295$	−3.86603	+	6.69615i	−0.225089	+	0.389865i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	9.46410	−	16.3923i	0.547323	−	0.947991i
$300$		0			0
$301$	4.46410	+	7.73205i	0.257307	+	0.445668i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−4.00000			−0.229039
$306$		0			0
$307$	−1.80385			−0.102951			−0.0514755	−	0.998674i	$-0.516392\pi$
$307$	−1.80385			−0.102951			−0.0514755	+	0.998674i	$0.516392\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−8.25833	−	14.3038i	−0.468287	−	0.811097i	0.531056	−	0.847337i	$-0.321795\pi$
$311$	−8.25833	−	14.3038i	−0.468287	−	0.811097i	−0.999343	+	0.0362398i	$0.988462\pi$
$312$		0			0
$313$	−7.46410	+	12.9282i	−0.421896	+	0.730745i	−0.996125	−	0.0879495i	$-0.971969\pi$
$313$	−7.46410	+	12.9282i	−0.421896	+	0.730745i	0.574229	+	0.818695i	$0.305302\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	1.56218	−	2.70577i	0.0877406	−	0.151971i	−0.818815	−	0.574057i	$-0.805369\pi$
$317$	1.56218	−	2.70577i	0.0877406	−	0.151971i	0.906556	+	0.422086i	$0.138702\pi$
$318$		0			0
$319$	6.69615	+	11.5981i	0.374913	+	0.649368i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−21.1244			−1.17539
$324$		0			0
$325$	5.46410			0.303094
$326$		0			0
$327$		0			0
$328$		0			0
$329$	1.73205	+	3.00000i	0.0954911	+	0.165395i
$330$		0			0
$331$	14.6962	−	25.4545i	0.807774	−	1.39910i	−0.106629	−	0.994299i	$-0.534006\pi$
$331$	14.6962	−	25.4545i	0.807774	−	1.39910i	0.914403	−	0.404806i	$-0.132661\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−3.19615	+	5.53590i	−0.174624	+	0.302458i
$336$		0			0
$337$	17.1244	+	29.6603i	0.932823	+	1.61570i	0.778471	+	0.627681i	$0.215996\pi$
$337$	17.1244	+	29.6603i	0.932823	+	1.61570i	0.154353	+	0.988016i	$0.450671\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−10.2679			−0.556041
$342$		0			0
$343$	17.8564			0.964155
$344$		0			0
$345$		0			0
$346$		0			0
$347$	10.9019	+	18.8827i	0.585246	+	1.01368i	0.994845	+	0.101410i	$0.0323354\pi$
$347$	10.9019	+	18.8827i	0.585246	+	1.01368i	−0.409599	+	0.912266i	$0.634331\pi$
$348$		0			0
$349$	12.5000	−	21.6506i	0.669110	−	1.15893i	−0.309044	−	0.951048i	$-0.600009\pi$
$349$	12.5000	−	21.6506i	0.669110	−	1.15893i	0.978153	−	0.207884i	$-0.0666577\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−14.4904	+	25.0981i	−0.771245	+	1.33584i	0.165636	+	0.986187i	$0.447032\pi$
$353$	−14.4904	+	25.0981i	−0.771245	+	1.33584i	−0.936881	+	0.349649i	$0.886301\pi$
$354$		0			0
$355$	−5.59808	−	9.69615i	−0.297115	−	0.514618i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−9.33975			−0.492933			−0.246466	−	0.969151i	$-0.579270\pi$
$359$	−9.33975			−0.492933			−0.246466	+	0.969151i	$0.579270\pi$
$360$		0			0
$361$	0.928203			0.0488528
$362$		0			0
$363$		0			0
$364$		0			0
$365$	0.0980762	+	0.169873i	0.00513354	+	0.00889156i
$366$		0			0
$367$	13.1962	−	22.8564i	0.688834	−	1.19309i	−0.283382	−	0.959007i	$-0.591456\pi$
$367$	13.1962	−	22.8564i	0.688834	−	1.19309i	0.972216	−	0.234088i	$-0.0752102\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	9.92820	−	17.1962i	0.515447	−	0.892780i
$372$		0			0
$373$	7.02628	+	12.1699i	0.363807	+	0.630132i	0.988584	−	0.150671i	$-0.0481434\pi$
$373$	7.02628	+	12.1699i	0.363807	+	0.630132i	−0.624777	+	0.780803i	$0.714810\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	42.2487			2.17592
$378$		0			0
$379$	4.53590			0.232993			0.116497	−	0.993191i	$-0.462834\pi$
$379$	4.53590			0.232993			0.116497	+	0.993191i	$0.462834\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	9.12436	+	15.8038i	0.466233	+	0.807539i	0.999256	−	0.0385616i	$-0.0122776\pi$
$383$	9.12436	+	15.8038i	0.466233	+	0.807539i	−0.533023	+	0.846100i	$0.678944\pi$
$384$		0			0
$385$	−2.36603	+	4.09808i	−0.120584	+	0.208857i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	13.7321	−	23.7846i	0.696243	−	1.20593i	−0.273517	−	0.961867i	$-0.588187\pi$
$389$	13.7321	−	23.7846i	0.696243	−	1.20593i	0.969760	−	0.244061i	$-0.0784796\pi$
$390$		0			0
$391$	8.19615	+	14.1962i	0.414497	+	0.717930i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−14.3923			−0.724155
$396$		0			0
$397$	37.3205			1.87306			0.936531	−	0.350584i	$-0.114017\pi$
$397$	37.3205			1.87306			0.936531	+	0.350584i	$0.114017\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−7.39230	−	12.8038i	−0.369154	−	0.639394i	0.620279	−	0.784381i	$-0.287019\pi$
$401$	−7.39230	−	12.8038i	−0.369154	−	0.639394i	−0.989434	+	0.144987i	$0.953686\pi$
$402$		0			0
$403$	−16.1962	+	28.0526i	−0.806788	+	1.39740i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−5.36603	+	9.29423i	−0.265984	+	0.460698i
$408$		0			0
$409$	8.92820	+	15.4641i	0.441471	+	0.764651i	0.997799	−	0.0663124i	$-0.0211234\pi$
$409$	8.92820	+	15.4641i	0.441471	+	0.764651i	−0.556328	+	0.830963i	$0.687790\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−21.1244			−1.03946
$414$		0			0
$415$	15.1244			0.742425
$416$		0			0
$417$		0			0
$418$		0			0
$419$	10.7321	+	18.5885i	0.524295	+	0.908106i	0.999600	+	0.0282844i	$0.00900441\pi$
$419$	10.7321	+	18.5885i	0.524295	+	0.908106i	−0.475305	+	0.879821i	$0.657662\pi$
$420$		0			0
$421$	−6.89230	+	11.9378i	−0.335910	+	0.581814i	−0.983659	−	0.180040i	$-0.942377\pi$
$421$	−6.89230	+	11.9378i	−0.335910	+	0.581814i	0.647749	+	0.761854i	$0.275711\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−2.36603	+	4.09808i	−0.114769	+	0.198786i
$426$		0			0
$427$	−5.46410	−	9.46410i	−0.264426	−	0.458000i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	33.5885			1.61790			0.808950	−	0.587878i	$-0.200037\pi$
$431$	33.5885			1.61790			0.808950	+	0.587878i	$0.200037\pi$
$432$		0			0
$433$	11.4641			0.550930			0.275465	−	0.961311i	$-0.411168\pi$
$433$	11.4641			0.550930			0.275465	+	0.961311i	$0.411168\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−7.73205	−	13.3923i	−0.369874	−	0.640641i
$438$		0			0
$439$	−3.30385	+	5.72243i	−0.157684	+	0.273117i	−0.934033	−	0.357186i	$-0.883736\pi$
$439$	−3.30385	+	5.72243i	−0.157684	+	0.273117i	0.776349	+	0.630303i	$0.217069\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	18.6340	−	32.2750i	0.885327	−	1.53343i	0.0399883	−	0.999200i	$-0.487268\pi$
$443$	18.6340	−	32.2750i	0.885327	−	1.53343i	0.845339	−	0.534231i	$-0.179399\pi$
$444$		0			0
$445$	2.59808	+	4.50000i	0.123161	+	0.213320i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−24.1244			−1.13850			−0.569249	−	0.822165i	$-0.692766\pi$
$449$	−24.1244			−1.13850			−0.569249	+	0.822165i	$0.692766\pi$
$450$		0			0
$451$	−19.3923			−0.913148
$452$		0			0
$453$		0			0
$454$		0			0
$455$	7.46410	+	12.9282i	0.349922	+	0.606084i
$456$		0			0
$457$	−11.0981	+	19.2224i	−0.519146	+	0.899187i	0.480606	+	0.876936i	$0.340417\pi$
$457$	−11.0981	+	19.2224i	−0.519146	+	0.899187i	−0.999752	+	0.0222510i	$0.992917\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	4.79423	−	8.30385i	0.223289	−	0.386749i	−0.732515	−	0.680750i	$-0.761654\pi$
$461$	4.79423	−	8.30385i	0.223289	−	0.386749i	0.955805	+	0.294002i	$0.0949871\pi$
$462$		0			0
$463$	1.19615	+	2.07180i	0.0555899	+	0.0962846i	0.892481	−	0.451085i	$-0.148963\pi$
$463$	1.19615	+	2.07180i	0.0555899	+	0.0962846i	−0.836891	+	0.547369i	$0.815629\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−26.1962			−1.21221			−0.606107	−	0.795383i	$-0.707270\pi$
$467$	−26.1962			−1.21221			−0.606107	+	0.795383i	$0.707270\pi$
$468$		0			0
$469$	−17.4641			−0.806417
$470$		0			0
$471$		0			0
$472$		0			0
$473$	2.83013	+	4.90192i	0.130129	+	0.225391i
$474$		0			0
$475$	2.23205	−	3.86603i	0.102414	−	0.177385i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−1.66987	+	2.89230i	−0.0762984	+	0.132153i	−0.901650	−	0.432466i	$-0.857644\pi$
$479$	−1.66987	+	2.89230i	−0.0762984	+	0.132153i	0.825352	+	0.564619i	$0.190977\pi$
$480$		0			0
$481$	16.9282	+	29.3205i	0.771860	+	1.33690i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	0.732051			0.0332407
$486$		0			0
$487$	−30.5359			−1.38371			−0.691857	−	0.722035i	$-0.743207\pi$
$487$	−30.5359			−1.38371			−0.691857	+	0.722035i	$0.743207\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−2.13397	−	3.69615i	−0.0963049	−	0.166805i	0.813848	−	0.581078i	$-0.197369\pi$
$491$	−2.13397	−	3.69615i	−0.0963049	−	0.166805i	−0.910152	+	0.414273i	$0.864036\pi$
$492$		0			0
$493$	−18.2942	+	31.6865i	−0.823931	+	1.42709i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	15.2942	−	26.4904i	0.686040	−	1.18826i
$498$		0			0
$499$	−13.6962	−	23.7224i	−0.613124	−	1.06196i	−0.990711	−	0.135988i	$-0.956579\pi$
$499$	−13.6962	−	23.7224i	−0.613124	−	1.06196i	0.377587	−	0.925974i	$-0.376754\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	35.3205			1.57486			0.787432	−	0.616402i	$-0.211410\pi$
$503$	35.3205			1.57486			0.787432	+	0.616402i	$0.211410\pi$
$504$		0			0
$505$	−6.12436			−0.272530
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−13.3923	−	23.1962i	−0.593603	−	1.02815i	−0.993742	−	0.111697i	$-0.964371\pi$
$509$	−13.3923	−	23.1962i	−0.593603	−	1.02815i	0.400139	−	0.916455i	$-0.368962\pi$
$510$		0			0
$511$	−0.267949	+	0.464102i	−0.0118534	+	0.0205306i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−9.19615	+	15.9282i	−0.405231	+	0.701880i
$516$		0			0
$517$	1.09808	+	1.90192i	0.0482933	+	0.0836465i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	10.3923			0.455295			0.227648	−	0.973744i	$-0.426897\pi$
$521$	10.3923			0.455295			0.227648	+	0.973744i	$0.426897\pi$
$522$		0			0
$523$	3.60770			0.157753			0.0788767	−	0.996884i	$-0.474867\pi$
$523$	3.60770			0.157753			0.0788767	+	0.996884i	$0.474867\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−14.0263	−	24.2942i	−0.610994	−	1.05827i
$528$		0			0
$529$	5.50000	−	9.52628i	0.239130	−	0.414186i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−30.5885	+	52.9808i	−1.32493	+	2.29485i
$534$		0			0
$535$	1.73205	+	3.00000i	0.0748831	+	0.129701i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−0.803848			−0.0346242
$540$		0			0
$541$	2.46410			0.105940			0.0529700	−	0.998596i	$-0.483131\pi$
$541$	2.46410			0.105940			0.0529700	+	0.998596i	$0.483131\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	3.96410	+	6.86603i	0.169803	+	0.294108i
$546$		0			0
$547$	−2.39230	+	4.14359i	−0.102288	+	0.177167i	−0.912627	−	0.408794i	$-0.865950\pi$
$547$	−2.39230	+	4.14359i	−0.102288	+	0.177167i	0.810339	+	0.585961i	$0.199283\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	17.2583	−	29.8923i	0.735230	−	1.27346i
$552$		0			0
$553$	−19.6603	−	34.0526i	−0.836039	−	1.44806i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	4.39230			0.186108			0.0930540	−	0.995661i	$-0.470337\pi$
$557$	4.39230			0.186108			0.0930540	+	0.995661i	$0.470337\pi$
$558$		0			0
$559$	17.8564			0.755246
$560$		0			0
$561$		0			0
$562$		0			0
$563$	11.3660	+	19.6865i	0.479021	+	0.829688i	0.999711	−	0.0240575i	$-0.00765849\pi$
$563$	11.3660	+	19.6865i	0.479021	+	0.829688i	−0.520690	+	0.853746i	$0.674325\pi$
$564$		0			0
$565$	−0.169873	+	0.294229i	−0.00714661	+	0.0123783i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−21.5263	+	37.2846i	−0.902429	+	1.56305i	−0.0780964	+	0.996946i	$0.524884\pi$
$569$	−21.5263	+	37.2846i	−0.902429	+	1.56305i	−0.824332	+	0.566106i	$0.808449\pi$
$570$		0			0
$571$	−0.428203	−	0.741670i	−0.0179197	−	0.0310379i	0.856926	−	0.515439i	$-0.172371\pi$
$571$	−0.428203	−	0.741670i	−0.0179197	−	0.0310379i	−0.874846	+	0.484401i	$0.839038\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−3.46410			−0.144463
$576$		0			0
$577$	11.8038			0.491401			0.245700	−	0.969346i	$-0.420982\pi$
$577$	11.8038			0.491401			0.245700	+	0.969346i	$0.420982\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	20.6603	+	35.7846i	0.857132	+	1.48460i
$582$		0			0
$583$	6.29423	−	10.9019i	0.260680	−	0.451512i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	1.90192	−	3.29423i	0.0785008	−	0.135967i	−0.824102	−	0.566441i	$-0.808320\pi$
$587$	1.90192	−	3.29423i	0.0785008	−	0.135967i	0.902603	+	0.430473i	$0.141653\pi$
$588$		0			0
$589$	13.2321	+	22.9186i	0.545217	+	0.944344i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	23.0718			0.947445			0.473723	−	0.880674i	$-0.342910\pi$
$593$	23.0718			0.947445			0.473723	+	0.880674i	$0.342910\pi$
$594$		0			0
$595$	−12.9282			−0.530005
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−7.20577	−	12.4808i	−0.294420	−	0.509950i	0.680430	−	0.732813i	$-0.261793\pi$
$599$	−7.20577	−	12.4808i	−0.294420	−	0.509950i	−0.974850	+	0.222863i	$0.928460\pi$
$600$		0			0
$601$	12.1603	−	21.0622i	0.496027	−	0.859144i	−0.503962	−	0.863726i	$-0.668125\pi$
$601$	12.1603	−	21.0622i	0.496027	−	0.859144i	0.999990	+	0.00458144i	$0.00145832\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	4.00000	−	6.92820i	0.162623	−	0.281672i
$606$		0			0
$607$	−1.29423	−	2.24167i	−0.0525311	−	0.0909866i	0.838564	−	0.544803i	$-0.183396\pi$
$607$	−1.29423	−	2.24167i	−0.0525311	−	0.0909866i	−0.891095	+	0.453816i	$0.850062\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	6.92820			0.280285
$612$		0			0
$613$	24.3923			0.985196			0.492598	−	0.870257i	$-0.336047\pi$
$613$	24.3923			0.985196			0.492598	+	0.870257i	$0.336047\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−6.00000	−	10.3923i	−0.241551	−	0.418378i	0.719605	−	0.694383i	$-0.244323\pi$
$617$	−6.00000	−	10.3923i	−0.241551	−	0.418378i	−0.961156	+	0.276005i	$0.910989\pi$
$618$		0			0
$619$	5.00000	−	8.66025i	0.200967	−	0.348085i	−0.747873	−	0.663842i	$-0.768925\pi$
$619$	5.00000	−	8.66025i	0.200967	−	0.348085i	0.948840	+	0.315757i	$0.102258\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−7.09808	+	12.2942i	−0.284378	+	0.492558i
$624$		0			0
$625$	−0.500000	−	0.866025i	−0.0200000	−	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−29.3205			−1.16909
$630$		0			0
$631$	−0.0717968			−0.00285818			−0.00142909	−	0.999999i	$-0.500455\pi$
$631$	−0.0717968			−0.00285818			−0.00142909	+	0.999999i	$0.500455\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−2.09808	−	3.63397i	−0.0832596	−	0.144210i
$636$		0			0
$637$	−1.26795	+	2.19615i	−0.0502380	+	0.0870147i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−6.40192	+	11.0885i	−0.252861	+	0.437968i	−0.964312	−	0.264767i	$-0.914705\pi$
$641$	−6.40192	+	11.0885i	−0.252861	+	0.437968i	0.711451	+	0.702735i	$0.248038\pi$
$642$		0			0
$643$	−7.29423	−	12.6340i	−0.287656	−	0.498235i	0.685594	−	0.727985i	$-0.259543\pi$
$643$	−7.29423	−	12.6340i	−0.287656	−	0.498235i	−0.973250	+	0.229749i	$0.926209\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−0.248711			−0.00977785			−0.00488893	−	0.999988i	$-0.501556\pi$
$647$	−0.248711			−0.00977785			−0.00488893	+	0.999988i	$0.501556\pi$
$648$		0			0
$649$	−13.3923			−0.525694
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−1.26795	−	2.19615i	−0.0496187	−	0.0859421i	0.840149	−	0.542355i	$-0.182467\pi$
$653$	−1.26795	−	2.19615i	−0.0496187	−	0.0859421i	−0.889768	+	0.456413i	$0.849134\pi$
$654$		0			0
$655$	5.13397	−	8.89230i	0.200601	−	0.347451i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−1.26795	+	2.19615i	−0.0493923	+	0.0855500i	−0.889665	−	0.456615i	$-0.849062\pi$
$659$	−1.26795	+	2.19615i	−0.0493923	+	0.0855500i	0.840272	+	0.542165i	$0.182395\pi$
$660$		0			0
$661$	−7.69615	−	13.3301i	−0.299346	−	0.518482i	0.676641	−	0.736313i	$-0.263435\pi$
$661$	−7.69615	−	13.3301i	−0.299346	−	0.518482i	−0.975986	+	0.217831i	$0.930102\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	12.1962			0.472947
$666$		0			0
$667$	−26.7846			−1.03710
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−3.46410	−	6.00000i	−0.133730	−	0.231627i
$672$		0			0
$673$	19.1962	−	33.2487i	0.739957	−	1.28164i	−0.212557	−	0.977149i	$-0.568179\pi$
$673$	19.1962	−	33.2487i	0.739957	−	1.28164i	0.952514	−	0.304495i	$-0.0984877\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	20.3205	−	35.1962i	0.780981	−	1.35270i	−0.150390	−	0.988627i	$-0.548053\pi$
$677$	20.3205	−	35.1962i	0.780981	−	1.35270i	0.931371	−	0.364071i	$-0.118614\pi$
$678$		0			0
$679$	1.00000	+	1.73205i	0.0383765	+	0.0664700i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−21.4641			−0.821301			−0.410651	−	0.911793i	$-0.634698\pi$
$683$	−21.4641			−0.821301			−0.410651	+	0.911793i	$0.634698\pi$
$684$		0			0
$685$	−4.39230			−0.167821
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−19.8564	−	34.3923i	−0.756469	−	1.31024i
$690$		0			0
$691$	5.00000	−	8.66025i	0.190209	−	0.329452i	−0.755110	−	0.655598i	$-0.772417\pi$
$691$	5.00000	−	8.66025i	0.190209	−	0.329452i	0.945319	+	0.326146i	$0.105750\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−7.69615	+	13.3301i	−0.291932	+	0.505641i
$696$		0			0
$697$	−26.4904	−	45.8827i	−1.00339	−	1.73793i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	17.8756			0.675154			0.337577	−	0.941298i	$-0.390393\pi$
$701$	17.8756			0.675154			0.337577	+	0.941298i	$0.390393\pi$
$702$		0			0
$703$	27.6603			1.04323
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−8.36603	−	14.4904i	−0.314637	−	0.544967i
$708$		0			0
$709$	−5.26795	+	9.12436i	−0.197842	+	0.342672i	−0.947828	−	0.318781i	$-0.896727\pi$
$709$	−5.26795	+	9.12436i	−0.197842	+	0.342672i	0.749986	+	0.661453i	$0.230060\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	10.2679	−	17.7846i	0.384538	−	0.666039i
$714$		0			0
$715$	4.73205	+	8.19615i	0.176969	+	0.306519i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	17.1962			0.641308			0.320654	−	0.947196i	$-0.396097\pi$
$719$	17.1962			0.641308			0.320654	+	0.947196i	$0.396097\pi$
$720$		0			0
$721$	−50.2487			−1.87136
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−3.86603	−	6.69615i	−0.143581	−	0.248689i
$726$		0			0
$727$	14.5885	−	25.2679i	0.541056	−	0.937136i	−0.457788	−	0.889061i	$-0.651358\pi$
$727$	14.5885	−	25.2679i	0.541056	−	0.937136i	0.998844	−	0.0480749i	$-0.0153086\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−7.73205	+	13.3923i	−0.285980	+	0.495332i
$732$		0			0
$733$	−21.7846	−	37.7321i	−0.804633	−	1.39367i	−0.916539	−	0.399946i	$-0.869029\pi$
$733$	−21.7846	−	37.7321i	−0.804633	−	1.39367i	0.111906	−	0.993719i	$-0.464305\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−11.0718			−0.407835
$738$		0			0
$739$	−26.1769			−0.962933			−0.481467	−	0.876464i	$-0.659896\pi$
$739$	−26.1769			−0.962933			−0.481467	+	0.876464i	$0.659896\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−2.70577	−	4.68653i	−0.0992651	−	0.171932i	0.812116	−	0.583497i	$-0.198316\pi$
$743$	−2.70577	−	4.68653i	−0.0992651	−	0.171932i	−0.911381	+	0.411564i	$0.864983\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−4.73205	+	8.19615i	−0.172905	+	0.299481i
$750$		0			0
$751$	−20.3923	−	35.3205i	−0.744126	−	1.28886i	−0.950602	−	0.310412i	$-0.899533\pi$
$751$	−20.3923	−	35.3205i	−0.744126	−	1.28886i	0.206476	−	0.978452i	$-0.433800\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−17.3923			−0.632971
$756$		0			0
$757$	−20.3923			−0.741171			−0.370585	−	0.928798i	$-0.620843\pi$
$757$	−20.3923			−0.741171			−0.370585	+	0.928798i	$0.620843\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−15.0622	−	26.0885i	−0.546004	−	0.945706i	−0.998543	−	0.0539615i	$-0.982815\pi$
$761$	−15.0622	−	26.0885i	−0.546004	−	0.945706i	0.452539	−	0.891744i	$-0.350518\pi$
$762$		0			0
$763$	−10.8301	+	18.7583i	−0.392077	+	0.679097i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−21.1244	+	36.5885i	−0.762756	+	1.32113i
$768$		0			0
$769$	−17.6244	−	30.5263i	−0.635551	−	1.10081i	−0.986398	−	0.164373i	$-0.947440\pi$
$769$	−17.6244	−	30.5263i	−0.635551	−	1.10081i	0.350848	−	0.936433i	$-0.385893\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−17.6603			−0.635195			−0.317598	−	0.948226i	$-0.602876\pi$
$773$	−17.6603			−0.635195			−0.317598	+	0.948226i	$0.602876\pi$
$774$		0			0
$775$	5.92820			0.212947
$776$		0			0
$777$		0			0
$778$		0			0
$779$	24.9904	+	43.2846i	0.895373	+	1.55083i
$780$		0			0
$781$	9.69615	−	16.7942i	0.346956	−	0.600945i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−1.63397	+	2.83013i	−0.0583191	+	0.101012i
$786$		0			0
$787$	18.0981	+	31.3468i	0.645127	+	1.11739i	0.984272	+	0.176658i	$0.0565288\pi$
$787$	18.0981	+	31.3468i	0.645127	+	1.11739i	−0.339145	+	0.940734i	$0.610138\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−0.928203			−0.0330031
$792$		0			0
$793$	−21.8564			−0.776144
$794$		0			0
$795$		0			0
$796$		0			0
$797$	11.6603	+	20.1962i	0.413027	+	0.715384i	0.995219	−	0.0976671i	$-0.0311380\pi$
$797$	11.6603	+	20.1962i	0.413027	+	0.715384i	&mi

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:	\( N \)	\(=\)	\( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1620.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 - 0.866025i) q^{5} +(1.36603 - 2.36603i) q^{7} +O(q^{10})\)	\(q+(-0.500000 - 0.866025i) q^{5} +(1.36603 - 2.36603i) q^{7} +(0.866025 - 1.50000i) q^{11} +(-2.73205 - 4.73205i) q^{13} +4.73205 q^{17} -4.46410 q^{19} +(1.73205 + 3.00000i) q^{23} +(-0.500000 + 0.866025i) q^{25} +(-3.86603 + 6.69615i) q^{29} +(-2.96410 - 5.13397i) q^{31} -2.73205 q^{35} -6.19615 q^{37} +(-5.59808 - 9.69615i) q^{41} +(-1.63397 + 2.83013i) q^{43} +(-0.633975 + 1.09808i) q^{47} +(-0.232051 - 0.401924i) q^{49} +7.26795 q^{53} -1.73205 q^{55} +(-3.86603 - 6.69615i) q^{59} +(2.00000 - 3.46410i) q^{61} +(-2.73205 + 4.73205i) q^{65} +(-3.19615 - 5.53590i) q^{67} +11.1962 q^{71} -0.196152 q^{73} +(-2.36603 - 4.09808i) q^{77} +(7.19615 - 12.4641i) q^{79} +(-7.56218 + 13.0981i) q^{83} +(-2.36603 - 4.09808i) q^{85} -5.19615 q^{89} -14.9282 q^{91} +(2.23205 + 3.86603i) q^{95} +(-0.366025 + 0.633975i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{5} + 2 q^{7}+O(q^{10}) \) 4 * q - 2 * q^5 + 2 * q^7	\( 4 q - 2 q^{5} + 2 q^{7} - 4 q^{13} + 12 q^{17} - 4 q^{19} - 2 q^{25} - 12 q^{29} + 2 q^{31} - 4 q^{35} - 4 q^{37} - 12 q^{41} - 10 q^{43} - 6 q^{47} + 6 q^{49} + 36 q^{53} - 12 q^{59} + 8 q^{61} - 4 q^{65} + 8 q^{67} + 24 q^{71} + 20 q^{73} - 6 q^{77} + 8 q^{79} - 6 q^{83} - 6 q^{85} - 32 q^{91} + 2 q^{95} + 2 q^{97}+O(q^{100}) \) 4 * q - 2 * q^5 + 2 * q^7 - 4 * q^13 + 12 * q^17 - 4 * q^19 - 2 * q^25 - 12 * q^29 + 2 * q^31 - 4 * q^35 - 4 * q^37 - 12 * q^41 - 10 * q^43 - 6 * q^47 + 6 * q^49 + 36 * q^53 - 12 * q^59 + 8 * q^61 - 4 * q^65 + 8 * q^67 + 24 * q^71 + 20 * q^73 - 6 * q^77 + 8 * q^79 - 6 * q^83 - 6 * q^85 - 32 * q^91 + 2 * q^95 + 2 * q^97

Introduction

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