LMFDB - Embedded newform 1620.2.i.n.1081.2

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			1081.2
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1620.1081
Dual form			1620.2.i.n.541.2

$q$-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 $\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times$.

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

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	0.999821	+	0.0189356i	$0.00602775\pi$
$7$	1.36603	+	2.36603i	0.516309	+	0.894274i	−0.483512	+	0.875338i	$0.660639\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−0.866025	−	1.50000i	−0.261116	−	0.452267i	0.705422	−	0.708787i	$-0.250757\pi$
$11$	−0.866025	−	1.50000i	−0.261116	−	0.452267i	−0.966539	+	0.256520i	$0.917424\pi$
$12$		0			0
$13$	−2.73205	+	4.73205i	−0.757735	+	1.31243i	0.186269	+	0.982499i	$0.440360\pi$
$13$	−2.73205	+	4.73205i	−0.757735	+	1.31243i	−0.944003	+	0.329936i	$0.892973\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−4.73205			−1.14769			−0.573845	−	0.818964i	$-0.694549\pi$
$17$	−4.73205			−1.14769			−0.573845	+	0.818964i	$0.694549\pi$
$18$		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	$-0.950952\pi$
$23$	−1.73205	+	3.00000i	−0.361158	+	0.625543i	0.626994	+	0.779024i	$0.284285\pi$
$24$		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	0.961829	+	0.273651i	$0.0882312\pi$
$29$	3.86603	+	6.69615i	0.717903	+	1.24344i	−0.243926	+	0.969794i	$0.578436\pi$
$30$		0			0
$31$	−2.96410	+	5.13397i	−0.532368	+	0.922089i	0.466917	+	0.884301i	$0.345365\pi$
$31$	−2.96410	+	5.13397i	−0.532368	+	0.922089i	−0.999286	+	0.0377881i	$0.987969\pi$
$32$		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	0.0167371	−	0.999860i	$-0.494672\pi$
$41$	5.59808	−	9.69615i	0.874273	−	1.51428i	0.857536	−	0.514425i	$-0.171994\pi$
$42$		0			0
$43$	−1.63397	−	2.83013i	−0.249179	−	0.431590i	0.714119	−	0.700024i	$-0.246827\pi$
$43$	−1.63397	−	2.83013i	−0.249179	−	0.431590i	−0.963298	+	0.268434i	$0.913494\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	0.633975	+	1.09808i	0.0924747	+	0.160171i	0.908552	−	0.417772i	$-0.137189\pi$
$47$	0.633975	+	1.09808i	0.0924747	+	0.160171i	−0.816077	+	0.577943i	$0.803856\pi$
$48$		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	$-0.666360\pi$
$53$	−7.26795			−0.998330			−0.499165	+	0.866507i	$0.666360\pi$
$54$		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	−0.496679	−	0.867934i	$-0.665447\pi$
$59$	3.86603	−	6.69615i	0.503314	−	0.871765i	0.999993	−	0.00383049i	$-0.00121929\pi$
$60$		0			0
$61$	2.00000	+	3.46410i	0.256074	+	0.443533i	0.965187	−	0.261562i	$-0.0842377\pi$
$61$	2.00000	+	3.46410i	0.256074	+	0.443533i	−0.709113	+	0.705095i	$0.750904\pi$
$62$		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	−0.992512	−	0.122149i	$-0.961022\pi$
$67$	−3.19615	+	5.53590i	−0.390472	+	0.676318i	0.602040	+	0.798466i	$0.294355\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−11.1962			−1.32874			−0.664369	−	0.747404i	$-0.731300\pi$
$71$	−11.1962			−1.32874			−0.664369	+	0.747404i	$0.731300\pi$
$72$		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	0.913120	+	0.407690i	$0.133666\pi$
$79$	7.19615	+	12.4641i	0.809630	+	1.40232i	−0.103490	+	0.994631i	$0.533001\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	7.56218	+	13.0981i	0.830057	+	1.43770i	0.897992	+	0.440011i	$0.145025\pi$
$83$	7.56218	+	13.0981i	0.830057	+	1.43770i	−0.0679356	+	0.997690i	$0.521641\pi$
$84$		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	$-0.411191\pi$
$89$	5.19615			0.550791			0.275396	+	0.961331i	$0.411191\pi$
$90$		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	0.846845	−	0.531840i	$-0.178499\pi$
$97$	−0.366025	−	0.633975i	−0.0371642	−	0.0643704i	−0.884009	+	0.467469i	$0.845166\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−3.06218	−	5.30385i	−0.304698	−	0.527753i	0.672496	−	0.740101i	$-0.265222\pi$
$101$	−3.06218	−	5.30385i	−0.304698	−	0.527753i	−0.977194	+	0.212348i	$0.931889\pi$
$102$		0			0
$103$	−9.19615	+	15.9282i	−0.906124	+	1.56945i	−0.0867223	+	0.996233i	$0.527639\pi$
$103$	−9.19615	+	15.9282i	−0.906124	+	1.56945i	−0.819402	+	0.573220i	$0.805694\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	3.46410			0.334887			0.167444	−	0.985882i	$-0.446449\pi$
$107$	3.46410			0.334887			0.167444	+	0.985882i	$0.446449\pi$
$108$		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	$-0.828246\pi$
$113$	0.169873	−	0.294229i	0.0159803	−	0.0276787i	0.873905	+	0.486097i	$0.161580\pi$
$114$		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	$-0.981395\pi$
$131$	−5.13397	+	8.89230i	−0.448557	+	0.776924i	0.549735	+	0.835339i	$0.314729\pi$
$132$		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	$-0.226747\pi$
$137$	−2.19615	−	3.80385i	−0.187630	−	0.324985i	−0.944460	+	0.328628i	$0.893414\pi$
$138$		0			0
$139$	−7.69615	+	13.3301i	−0.652779	+	1.13065i	0.329666	+	0.944097i	$0.393064\pi$
$139$	−7.69615	+	13.3301i	−0.652779	+	1.13065i	−0.982446	+	0.186549i	$0.940270\pi$
$140$		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	$-0.833333\pi$
$149$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$150$		0			0
$151$	8.69615	+	15.0622i	0.707683	+	1.22574i	0.965715	+	0.259606i	$0.0835928\pi$
$151$	8.69615	+	15.0622i	0.707683	+	1.22574i	−0.258032	+	0.966136i	$0.583074\pi$
$152$		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	−0.923833	−	0.382796i	$-0.874961\pi$
$157$	−1.63397	+	2.83013i	−0.130405	+	0.225869i	0.793428	+	0.608665i	$0.208295\pi$
$158$		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	−0.0903199	−	0.995913i	$-0.528789\pi$
$167$	10.5622	−	18.2942i	0.817326	−	1.41565i	0.907646	−	0.419737i	$-0.137878\pi$
$168$		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	0.796632	+	0.604465i	$0.206613\pi$
$173$	12.1244	+	21.0000i	0.921798	+	1.59660i	0.125166	+	0.992136i	$0.460054\pi$
$174$		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	$-0.649685\pi$
$179$	−12.1244			−0.906217			−0.453108	+	0.891455i	$0.649685\pi$
$180$		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	−0.972065	−	0.234710i	$-0.924586\pi$
$191$	−9.52628	−	16.5000i	−0.689297	−	1.19390i	0.282768	−	0.959188i	$-0.408747\pi$
$192$		0			0
$193$	5.29423	−	9.16987i	0.381087	−	0.660062i	−0.610131	−	0.792301i	$-0.708883\pi$
$193$	5.29423	−	9.16987i	0.381087	−	0.660062i	0.991218	+	0.132239i	$0.0422165\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−13.8564			−0.987228			−0.493614	−	0.869681i	$-0.664324\pi$
$197$	−13.8564			−0.987228			−0.493614	+	0.869681i	$0.664324\pi$
$198$		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	−0.287178	−	0.957877i	$-0.592717\pi$
$211$	9.96410	−	17.2583i	0.685957	−	1.18811i	0.973135	−	0.230235i	$-0.0739496\pi$
$212$		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	0.947256	−	0.320477i	$-0.103843\pi$
$223$	2.92820	+	5.07180i	0.196087	+	0.339633i	−0.751169	+	0.660110i	$0.770510\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−13.0981	−	22.6865i	−0.869350	−	1.50576i	−0.862662	−	0.505780i	$-0.831205\pi$
$227$	−13.0981	−	22.6865i	−0.869350	−	1.50576i	−0.00668763	−	0.999978i	$-0.502129\pi$
$228$		0			0
$229$	8.92820	−	15.4641i	0.589992	−	1.02190i	−0.404240	−	0.914653i	$-0.632464\pi$
$229$	8.92820	−	15.4641i	0.589992	−	1.02190i	0.994233	−	0.107244i	$-0.0342026\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	24.5885			1.61084			0.805422	−	0.592702i	$-0.201939\pi$
$233$	24.5885			1.61084			0.805422	+	0.592702i	$0.201939\pi$
$234$		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	$-0.922366\pi$
$239$	−4.26795	+	7.39230i	−0.276071	+	0.478168i	0.694334	+	0.719653i	$0.255699\pi$
$240$		0			0
$241$	−5.16025	−	8.93782i	−0.332401	−	0.575736i	0.650581	−	0.759437i	$-0.274525\pi$
$241$	−5.16025	−	8.93782i	−0.332401	−	0.575736i	−0.982982	+	0.183701i	$0.941192\pi$
$242$		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	$-0.275561\pi$
$251$	20.5359			1.29621			0.648107	+	0.761549i	$0.275561\pi$
$252$		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	$-0.993536\pi$
$257$	−7.73205	+	13.3923i	−0.482312	+	0.835389i	0.517482	+	0.855694i	$0.326870\pi$
$258$		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	0.948961	+	0.315394i	$0.102137\pi$
$263$	12.1244	+	21.0000i	0.747620	+	1.29492i	−0.201341	+	0.979521i	$0.564530\pi$
$264$		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	$-0.665175\pi$
$269$	−16.2679			−0.991874			−0.495937	+	0.868358i	$0.665175\pi$
$270$		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	0.778729	−	0.627361i	$-0.215865\pi$
$277$	−2.56218	−	4.43782i	−0.153946	−	0.266643i	−0.932675	+	0.360718i	$0.882532\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−14.6603	−	25.3923i	−0.874557	−	1.51478i	−0.857233	−	0.514928i	$-0.827819\pi$
$281$	−14.6603	−	25.3923i	−0.874557	−	1.51478i	−0.0173240	−	0.999850i	$-0.505515\pi$
$282$		0			0
$283$	−8.26795	+	14.3205i	−0.491479	+	0.851266i	−0.999952	−	0.00981191i	$-0.996877\pi$
$283$	−8.26795	+	14.3205i	−0.491479	+	0.851266i	0.508473	+	0.861078i	$0.330210\pi$
$284$		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	0.243461	+	0.969911i	$0.421717\pi$
$293$	−12.2942	+	21.2942i	−0.718237	+	1.24402i	−0.961698	+	0.274112i	$0.911616\pi$
$294$		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	$-0.678205\pi$
$311$	8.25833	−	14.3038i	0.468287	−	0.811097i	0.999343	+	0.0362398i	$0.0115380\pi$
$312$		0			0
$313$	−7.46410	−	12.9282i	−0.421896	−	0.730745i	0.574229	−	0.818695i	$-0.305302\pi$
$313$	−7.46410	−	12.9282i	−0.421896	−	0.730745i	−0.996125	+	0.0879495i	$0.971969\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−1.56218	−	2.70577i	−0.0877406	−	0.151971i	0.818815	−	0.574057i	$-0.194631\pi$
$317$	−1.56218	−	2.70577i	−0.0877406	−	0.151971i	−0.906556	+	0.422086i	$0.861298\pi$
$318$		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	0.914403	+	0.404806i	$0.132661\pi$
$331$	14.6962	+	25.4545i	0.807774	+	1.39910i	−0.106629	+	0.994299i	$0.534006\pi$
$332$		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	0.154353	−	0.988016i	$-0.450671\pi$
$337$	17.1244	−	29.6603i	0.932823	−	1.61570i	0.778471	−	0.627681i	$-0.215996\pi$
$338$		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	0.409599	+	0.912266i	$0.365669\pi$
$347$	−10.9019	+	18.8827i	−0.585246	+	1.01368i	−0.994845	+	0.101410i	$0.967665\pi$
$348$		0			0
$349$	12.5000	+	21.6506i	0.669110	+	1.15893i	0.978153	+	0.207884i	$0.0666577\pi$
$349$	12.5000	+	21.6506i	0.669110	+	1.15893i	−0.309044	+	0.951048i	$0.600009\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	14.4904	+	25.0981i	0.771245	+	1.33584i	0.936881	+	0.349649i	$0.113699\pi$
$353$	14.4904	+	25.0981i	0.771245	+	1.33584i	−0.165636	+	0.986187i	$0.552968\pi$
$354$		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	$-0.420730\pi$
$359$	9.33975			0.492933			0.246466	+	0.969151i	$0.420730\pi$
$360$		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	0.972216	+	0.234088i	$0.0752102\pi$
$367$	13.1962	+	22.8564i	0.688834	+	1.19309i	−0.283382	+	0.959007i	$0.591456\pi$
$368$		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	−0.624777	−	0.780803i	$-0.714810\pi$
$373$	7.02628	−	12.1699i	0.363807	−	0.630132i	0.988584	+	0.150671i	$0.0481434\pi$
$374$		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	$-0.987722\pi$
$383$	−9.12436	+	15.8038i	−0.466233	+	0.807539i	0.533023	+	0.846100i	$0.321056\pi$
$384$		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	−0.969760	−	0.244061i	$-0.921520\pi$
$389$	−13.7321	−	23.7846i	−0.696243	−	1.20593i	0.273517	−	0.961867i	$-0.411813\pi$
$390$		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	$-0.712981\pi$
$401$	7.39230	−	12.8038i	0.369154	−	0.639394i	0.989434	+	0.144987i	$0.0463141\pi$
$402$		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	−0.556328	−	0.830963i	$-0.687790\pi$
$409$	8.92820	−	15.4641i	0.441471	−	0.764651i	0.997799	+	0.0663124i	$0.0211234\pi$
$410$		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	0.475305	+	0.879821i	$0.342338\pi$
$419$	−10.7321	+	18.5885i	−0.524295	+	0.908106i	−0.999600	+	0.0282844i	$0.990996\pi$
$420$		0			0
$421$	−6.89230	−	11.9378i	−0.335910	−	0.581814i	0.647749	−	0.761854i	$-0.275711\pi$
$421$	−6.89230	−	11.9378i	−0.335910	−	0.581814i	−0.983659	+	0.180040i	$0.942377\pi$
$422$		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	$-0.799963\pi$
$431$	−33.5885			−1.61790			−0.808950	+	0.587878i	$0.799963\pi$
$432$		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	0.776349	−	0.630303i	$-0.217069\pi$
$439$	−3.30385	−	5.72243i	−0.157684	−	0.273117i	−0.934033	+	0.357186i	$0.883736\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−18.6340	−	32.2750i	−0.885327	−	1.53343i	−0.845339	−	0.534231i	$-0.820601\pi$
$443$	−18.6340	−	32.2750i	−0.885327	−	1.53343i	−0.0399883	−	0.999200i	$-0.512732\pi$
$444$		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	$-0.307234\pi$
$449$	24.1244			1.13850			0.569249	+	0.822165i	$0.307234\pi$
$450$		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	−0.999752	−	0.0222510i	$-0.992917\pi$
$457$	−11.0981	−	19.2224i	−0.519146	−	0.899187i	0.480606	−	0.876936i	$-0.340417\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−4.79423	−	8.30385i	−0.223289	−	0.386749i	0.732515	−	0.680750i	$-0.238346\pi$
$461$	−4.79423	−	8.30385i	−0.223289	−	0.386749i	−0.955805	+	0.294002i	$0.905013\pi$
$462$		0			0
$463$	1.19615	−	2.07180i	0.0555899	−	0.0962846i	−0.836891	−	0.547369i	$-0.815629\pi$
$463$	1.19615	−	2.07180i	0.0555899	−	0.0962846i	0.892481	+	0.451085i	$0.148963\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	26.1962			1.21221			0.606107	−	0.795383i	$-0.292730\pi$
$467$	26.1962			1.21221			0.606107	+	0.795383i	$0.292730\pi$
$468$		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	$-0.142356\pi$
$479$	1.66987	+	2.89230i	0.0762984	+	0.132153i	−0.825352	+	0.564619i	$0.809023\pi$
$480$		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	$-0.802631\pi$
$491$	2.13397	−	3.69615i	0.0963049	−	0.166805i	0.910152	+	0.414273i	$0.135964\pi$
$492$		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	0.377587	+	0.925974i	$0.376754\pi$
$499$	−13.6962	+	23.7224i	−0.613124	+	1.06196i	−0.990711	+	0.135988i	$0.956579\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−35.3205			−1.57486			−0.787432	−	0.616402i	$-0.788590\pi$
$503$	−35.3205			−1.57486			−0.787432	+	0.616402i	$0.788590\pi$
$504$		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	−0.400139	−	0.916455i	$-0.631038\pi$
$509$	13.3923	−	23.1962i	0.593603	−	1.02815i	0.993742	−	0.111697i	$-0.0356286\pi$
$510$		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	$-0.573103\pi$
$521$	−10.3923			−0.455295			−0.227648	+	0.973744i	$0.573103\pi$
$522$		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	0.810339	−	0.585961i	$-0.199283\pi$
$547$	−2.39230	−	4.14359i	−0.102288	−	0.177167i	−0.912627	+	0.408794i	$0.865950\pi$
$548$		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	$-0.529663\pi$
$557$	−4.39230			−0.186108			−0.0930540	+	0.995661i	$0.529663\pi$
$558$		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	$-0.992342\pi$
$563$	−11.3660	+	19.6865i	−0.479021	+	0.829688i	0.520690	+	0.853746i	$0.325675\pi$
$564$		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	0.824332	+	0.566106i	$0.191551\pi$
$569$	21.5263	+	37.2846i	0.902429	+	1.56305i	0.0780964	+	0.996946i	$0.475116\pi$
$570$		0			0
$571$	−0.428203	+	0.741670i	−0.0179197	+	0.0310379i	−0.874846	−	0.484401i	$-0.839038\pi$
$571$	−0.428203	+	0.741670i	−0.0179197	+	0.0310379i	0.856926	+	0.515439i	$0.172371\pi$
$572$		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	$-0.191680\pi$
$587$	−1.90192	−	3.29423i	−0.0785008	−	0.135967i	−0.902603	+	0.430473i	$0.858347\pi$
$588$		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	$-0.657090\pi$
$593$	−23.0718			−0.947445			−0.473723	+	0.880674i	$0.657090\pi$
$594$		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	$-0.738207\pi$
$599$	7.20577	−	12.4808i	0.294420	−	0.509950i	0.974850	+	0.222863i	$0.0715403\pi$
$600$		0			0
$601$	12.1603	+	21.0622i	0.496027	+	0.859144i	0.999990	−	0.00458144i	$-0.00145832\pi$
$601$	12.1603	+	21.0622i	0.496027	+	0.859144i	−0.503962	+	0.863726i	$0.668125\pi$
$602$		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	−0.891095	−	0.453816i	$-0.850062\pi$
$607$	−1.29423	+	2.24167i	−0.0525311	+	0.0909866i	0.838564	+	0.544803i	$0.183396\pi$
$608$		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	$-0.755677\pi$
$617$	6.00000	−	10.3923i	0.241551	−	0.418378i	0.961156	+	0.276005i	$0.0890106\pi$
$618$		0			0
$619$	5.00000	+	8.66025i	0.200967	+	0.348085i	0.948840	−	0.315757i	$-0.102258\pi$
$619$	5.00000	+	8.66025i	0.200967	+	0.348085i	−0.747873	+	0.663842i	$0.768925\pi$
$620$		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	$-0.0852952\pi$
$641$	6.40192	+	11.0885i	0.252861	+	0.437968i	−0.711451	+	0.702735i	$0.751962\pi$
$642$		0			0
$643$	−7.29423	+	12.6340i	−0.287656	+	0.498235i	−0.973250	−	0.229749i	$-0.926209\pi$
$643$	−7.29423	+	12.6340i	−0.287656	+	0.498235i	0.685594	+	0.727985i	$0.259543\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	0.248711			0.00977785			0.00488893	−	0.999988i	$-0.498444\pi$
$647$	0.248711			0.00977785			0.00488893	+	0.999988i	$0.498444\pi$
$648$		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	$-0.817533\pi$
$653$	1.26795	−	2.19615i	0.0496187	−	0.0859421i	0.889768	+	0.456413i	$0.150866\pi$
$654$		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	$-0.150938\pi$
$659$	1.26795	+	2.19615i	0.0493923	+	0.0855500i	−0.840272	+	0.542165i	$0.817605\pi$
$660$		0			0
$661$	−7.69615	+	13.3301i	−0.299346	+	0.518482i	−0.975986	−	0.217831i	$-0.930102\pi$
$661$	−7.69615	+	13.3301i	−0.299346	+	0.518482i	0.676641	+	0.736313i	$0.263435\pi$
$662$		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	0.952514	+	0.304495i	$0.0984877\pi$
$673$	19.1962	+	33.2487i	0.739957	+	1.28164i	−0.212557	+	0.977149i	$0.568179\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−20.3205	−	35.1962i	−0.780981	−	1.35270i	−0.931371	−	0.364071i	$-0.881386\pi$
$677$	−20.3205	−	35.1962i	−0.780981	−	1.35270i	0.150390	−	0.988627i	$-0.451947\pi$
$678$		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	$-0.365302\pi$
$683$	21.4641			0.821301			0.410651	+	0.911793i	$0.365302\pi$
$684$		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	0.945319	−	0.326146i	$-0.105750\pi$
$691$	5.00000	+	8.66025i	0.190209	+	0.329452i	−0.755110	+	0.655598i	$0.772417\pi$
$692$		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	$-0.609607\pi$
$701$	−17.8756			−0.675154			−0.337577	+	0.941298i	$0.609607\pi$
$702$		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	0.749986	−	0.661453i	$-0.230060\pi$
$709$	−5.26795	−	9.12436i	−0.197842	−	0.342672i	−0.947828	+	0.318781i	$0.896727\pi$
$710$		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	$-0.603903\pi$
$719$	−17.1962			−0.641308			−0.320654	+	0.947196i	$0.603903\pi$
$720$		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	0.998844	+	0.0480749i	$0.0153086\pi$
$727$	14.5885	+	25.2679i	0.541056	+	0.937136i	−0.457788	+	0.889061i	$0.651358\pi$
$728$		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	0.111906	+	0.993719i	$0.464305\pi$
$733$	−21.7846	+	37.7321i	−0.804633	+	1.39367i	−0.916539	+	0.399946i	$0.869029\pi$
$734$		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	$-0.801684\pi$
$743$	2.70577	−	4.68653i	0.0992651	−	0.171932i	0.911381	+	0.411564i	$0.135017\pi$
$744$		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	0.206476	+	0.978452i	$0.433800\pi$
$751$	−20.3923	+	35.3205i	−0.744126	+	1.28886i	−0.950602	+	0.310412i	$0.899533\pi$
$752$		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	−0.452539	−	0.891744i	$-0.649482\pi$
$761$	15.0622	−	26.0885i	0.546004	−	0.945706i	0.998543	−	0.0539615i	$-0.0171848\pi$
$762$		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	0.350848	+	0.936433i	$0.385893\pi$
$769$	−17.6244	+	30.5263i	−0.635551	+	1.10081i	−0.986398	+	0.164373i	$0.947440\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	17.6603			0.635195			0.317598	−	0.948226i	$-0.397124\pi$
$773$	17.6603			0.635195			0.317598	+	0.948226i	$0.397124\pi$
$774$		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	−0.339145	−	0.940734i	$-0.610138\pi$
$787$	18.0981	−	31.3468i	0.645127	−	1.11739i	0.984272	−	0.176658i	$-0.0565288\pi$
$788$		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	$-0.968862\pi$
$797$	−11.6603	+	20.1962i	−0.413027	+	0.715384i	0.582192	+	0.813052i	$0.302195\pi$

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

L-functions

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