LMFDB - Embedded newform 1620.4.i.w.1081.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1620,4,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, 4, names="a")

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

chi := DirichletCharacter("1620.541");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1620 = 2^{2} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
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:	$95.5830942093$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 306x^{10} + 30777x^{8} + 1381040x^{6} + 28918584x^{4} + 243888288x^{2} + 366645904 $ x^12 + 306x^10 + 30777x^8 + 1381040x^6 + 28918584x^4 + 243888288*x^2 + 366645904
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{13} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1081.1
Root			$-1.37492i$ of defining polynomial
Character	$\chi$	$=$	1620.1081
Dual form			1620.4.i.w.541.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.50000 + 4.33013i) q^{5} +(-11.2133 - 19.4221i) q^{7} +O(q^{10})$	$q+(-2.50000 + 4.33013i) q^{5} +(-11.2133 - 19.4221i) q^{7} +(28.2573 + 48.9430i) q^{11} +(-21.7839 + 37.7309i) q^{13} -34.9879 q^{17} +77.1406 q^{19} +(61.2376 - 106.067i) q^{23} +(-12.5000 - 21.6506i) q^{25} +(136.560 + 236.529i) q^{29} +(148.813 - 257.751i) q^{31} +112.133 q^{35} -267.925 q^{37} +(-90.5138 + 156.774i) q^{41} +(-184.930 - 320.308i) q^{43} +(-56.2803 - 97.4803i) q^{47} +(-79.9776 + 138.525i) q^{49} +23.1415 q^{53} -282.573 q^{55} +(139.658 - 241.894i) q^{59} +(196.434 + 340.234i) q^{61} +(-108.920 - 188.654i) q^{65} +(-197.290 + 341.716i) q^{67} -973.017 q^{71} -760.770 q^{73} +(633.716 - 1097.63i) q^{77} +(415.876 + 720.319i) q^{79} +(-259.945 - 450.238i) q^{83} +(87.4698 - 151.502i) q^{85} +1189.22 q^{89} +977.081 q^{91} +(-192.852 + 334.029i) q^{95} +(419.859 + 727.217i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 30 q^{5} - 12 q^{7}+O(q^{10}) $ 12 * q - 30 * q^5 - 12 * q^7	$ 12 q - 30 q^{5} - 12 q^{7} + 84 q^{13} - 24 q^{17} - 228 q^{19} + 30 q^{23} - 150 q^{25} + 168 q^{29} + 324 q^{31} + 120 q^{35} - 984 q^{37} + 312 q^{41} + 156 q^{43} + 462 q^{47} + 588 q^{49} - 2028 q^{53} + 1008 q^{59} - 36 q^{61} + 420 q^{65} - 144 q^{67} - 2424 q^{71} - 1800 q^{73} + 672 q^{77} + 936 q^{79} + 288 q^{83} + 60 q^{85} + 240 q^{89} + 4572 q^{91} + 570 q^{95} + 1188 q^{97}+O(q^{100}) $ 12 * q - 30 * q^5 - 12 * q^7 + 84 * q^13 - 24 * q^17 - 228 * q^19 + 30 * q^23 - 150 * q^25 + 168 * q^29 + 324 * q^31 + 120 * q^35 - 984 * q^37 + 312 * q^41 + 156 * q^43 + 462 * q^47 + 588 * q^49 - 2028 * q^53 + 1008 * q^59 - 36 * q^61 + 420 * q^65 - 144 * q^67 - 2424 * q^71 - 1800 * q^73 + 672 * q^77 + 936 * q^79 + 288 * q^83 + 60 * q^85 + 240 * q^89 + 4572 * q^91 + 570 * q^95 + 1188 * 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
$5$	−2.50000	+	4.33013i	−0.223607	+	0.387298i
$5$	−2.50000	+	4.33013i	−0.223607	+	0.387298i
$6$		0			0
$7$	−11.2133	−	19.4221i	−0.605463	−	1.04869i	−0.991978	−	0.126410i	$-0.959655\pi$
$7$	−11.2133	−	19.4221i	−0.605463	−	1.04869i	0.386515	−	0.922283i	$-0.373679\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	28.2573	+	48.9430i	0.774535	+	1.34153i	0.935056	+	0.354501i	$0.115349\pi$
$11$	28.2573	+	48.9430i	0.774535	+	1.34153i	−0.160521	+	0.987032i	$0.551317\pi$
$12$		0			0
$13$	−21.7839	+	37.7309i	−0.464752	+	0.804974i	−0.999190	−	0.0402336i	$-0.987190\pi$
$13$	−21.7839	+	37.7309i	−0.464752	+	0.804974i	0.534438	+	0.845207i	$0.320523\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−34.9879			−0.499166			−0.249583	−	0.968353i	$-0.580293\pi$
$17$	−34.9879			−0.499166			−0.249583	+	0.968353i	$0.580293\pi$
$18$		0			0
$19$	77.1406			0.931436			0.465718	−	0.884933i	$-0.345796\pi$
$19$	77.1406			0.931436			0.465718	+	0.884933i	$0.345796\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	61.2376	−	106.067i	0.555171	−	0.961584i	−0.442720	−	0.896660i	$-0.645986\pi$
$23$	61.2376	−	106.067i	0.555171	−	0.961584i	0.997890	−	0.0649235i	$-0.0206803\pi$
$24$		0			0
$25$	−12.5000	−	21.6506i	−0.100000	−	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	136.560	+	236.529i	0.874432	+	1.51456i	0.857366	+	0.514707i	$0.172099\pi$
$29$	136.560	+	236.529i	0.874432	+	1.51456i	0.0170663	+	0.999854i	$0.494567\pi$
$30$		0			0
$31$	148.813	−	257.751i	0.862180	−	1.49334i	−0.00764108	−	0.999971i	$-0.502432\pi$
$31$	148.813	−	257.751i	0.862180	−	1.49334i	0.869821	−	0.493368i	$-0.164234\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	112.133			0.541543
$36$		0			0
$37$	−267.925			−1.19045			−0.595224	−	0.803560i	$-0.702937\pi$
$37$	−267.925			−1.19045			−0.595224	+	0.803560i	$0.702937\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−90.5138	+	156.774i	−0.344777	+	0.597172i	−0.985313	−	0.170756i	$-0.945379\pi$
$41$	−90.5138	+	156.774i	−0.344777	+	0.597172i	0.640536	+	0.767928i	$0.278712\pi$
$42$		0			0
$43$	−184.930	−	320.308i	−0.655850	−	1.13597i	−0.981680	−	0.190537i	$-0.938977\pi$
$43$	−184.930	−	320.308i	−0.655850	−	1.13597i	0.325830	−	0.945428i	$-0.394356\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−56.2803	−	97.4803i	−0.174666	−	0.302531i	0.765379	−	0.643579i	$-0.222551\pi$
$47$	−56.2803	−	97.4803i	−0.174666	−	0.302531i	−0.940046	+	0.341048i	$0.889218\pi$
$48$		0			0
$49$	−79.9776	+	138.525i	−0.233171	+	0.403864i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	23.1415			0.0599759			0.0299880	−	0.999550i	$-0.490453\pi$
$53$	23.1415			0.0599759			0.0299880	+	0.999550i	$0.490453\pi$
$54$		0			0
$55$	−282.573			−0.692765
$56$		0			0
$57$		0			0
$58$		0			0
$59$	139.658	−	241.894i	0.308167	−	0.533761i	−0.669794	−	0.742547i	$-0.733618\pi$
$59$	139.658	−	241.894i	0.308167	−	0.533761i	0.977961	+	0.208785i	$0.0669510\pi$
$60$		0			0
$61$	196.434	+	340.234i	0.412309	+	0.714140i	0.995142	−	0.0984523i	$-0.0313892\pi$
$61$	196.434	+	340.234i	0.412309	+	0.714140i	−0.582833	+	0.812592i	$0.698056\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−108.920	−	188.654i	−0.207843	−	0.359995i
$66$		0			0
$67$	−197.290	+	341.716i	−0.359744	+	0.623094i	−0.987918	−	0.154978i	$-0.950469\pi$
$67$	−197.290	+	341.716i	−0.359744	+	0.623094i	0.628174	+	0.778073i	$0.283803\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−973.017			−1.62642			−0.813210	−	0.581970i	$-0.802282\pi$
$71$	−973.017			−1.62642			−0.813210	+	0.581970i	$0.802282\pi$
$72$		0			0
$73$	−760.770			−1.21975			−0.609873	−	0.792500i	$-0.708779\pi$
$73$	−760.770			−1.21975			−0.609873	+	0.792500i	$0.708779\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	633.716	−	1097.63i	0.937904	−	1.62450i
$78$		0			0
$79$	415.876	+	720.319i	0.592275	+	1.02585i	0.993925	+	0.110057i	$0.0351035\pi$
$79$	415.876	+	720.319i	0.592275	+	1.02585i	−0.401650	+	0.915793i	$0.631563\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−259.945	−	450.238i	−0.343767	−	0.595422i	0.641362	−	0.767238i	$-0.278370\pi$
$83$	−259.945	−	450.238i	−0.343767	−	0.595422i	−0.985129	+	0.171816i	$0.945036\pi$
$84$		0			0
$85$	87.4698	−	151.502i	0.111617	−	0.193326i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	1189.22			1.41637			0.708183	−	0.706028i	$-0.249515\pi$
$89$	1189.22			1.41637			0.708183	+	0.706028i	$0.249515\pi$
$90$		0			0
$91$	977.081			1.12556
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−192.852	+	334.029i	−0.208275	+	0.360743i
$96$		0			0
$97$	419.859	+	727.217i	0.439486	+	0.761213i	0.997650	−	0.0685181i	$-0.0218271\pi$
$97$	419.859	+	727.217i	0.439486	+	0.761213i	−0.558163	+	0.829731i	$0.688494\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	520.609	+	901.721i	0.512896	+	0.888362i	0.999888	+	0.0149558i	$0.00476075\pi$
$101$	520.609	+	901.721i	0.512896	+	0.888362i	−0.486992	+	0.873406i	$0.661906\pi$
$102$		0			0
$103$	−934.068	+	1617.85i	−0.893558	+	1.54769i	−0.0579790	+	0.998318i	$0.518466\pi$
$103$	−934.068	+	1617.85i	−0.893558	+	1.54769i	−0.835579	+	0.549370i	$0.814868\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−371.156			−0.335336			−0.167668	−	0.985844i	$-0.553624\pi$
$107$	−371.156			−0.335336			−0.167668	+	0.985844i	$0.553624\pi$
$108$		0			0
$109$	−1204.94			−1.05883			−0.529414	−	0.848364i	$-0.677588\pi$
$109$	−1204.94			−1.05883			−0.529414	+	0.848364i	$0.677588\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−1101.75	+	1908.29i	−0.917205	+	1.58865i	−0.113564	+	0.993531i	$0.536227\pi$
$113$	−1101.75	+	1908.29i	−0.917205	+	1.58865i	−0.803641	+	0.595115i	$0.797107\pi$
$114$		0			0
$115$	306.188	+	530.333i	0.248280	+	0.430033i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	392.331	+	679.537i	0.302226	+	0.523471i
$120$		0			0
$121$	−931.444	+	1613.31i	−0.699808	+	1.21210i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	125.000			0.0894427
$126$		0			0
$127$	2753.36			1.92379			0.961896	−	0.273417i	$-0.0881539\pi$
$127$	2753.36			1.92379			0.961896	+	0.273417i	$0.0881539\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	465.796	−	806.783i	0.310663	−	0.538083i	−0.667843	−	0.744302i	$-0.732782\pi$
$131$	465.796	−	806.783i	0.310663	−	0.538083i	0.978506	+	0.206218i	$0.0661157\pi$
$132$		0			0
$133$	−865.004	−	1498.23i	−0.563950	−	0.976790i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	647.229	+	1121.03i	0.403624	+	0.699098i	0.994160	−	0.107914i	$-0.0344170\pi$
$137$	647.229	+	1121.03i	0.403624	+	0.699098i	−0.590536	+	0.807011i	$0.701084\pi$
$138$		0			0
$139$	−488.332	+	845.817i	−0.297984	+	0.516124i	−0.975675	−	0.219223i	$-0.929648\pi$
$139$	−488.332	+	845.817i	−0.297984	+	0.516124i	0.677690	+	0.735347i	$0.262981\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−2462.22			−1.43987
$144$		0			0
$145$	−1365.60			−0.782116
$146$		0			0
$147$		0			0
$148$		0			0
$149$	869.466	−	1505.96i	0.478050	−	0.828007i	−0.521633	−	0.853170i	$-0.674677\pi$
$149$	869.466	−	1505.96i	0.478050	−	0.828007i	0.999683	+	0.0251630i	$0.00801048\pi$
$150$		0			0
$151$	746.577	+	1293.11i	0.402355	+	0.696899i	0.994010	−	0.109292i	$-0.0348585\pi$
$151$	746.577	+	1293.11i	0.402355	+	0.696899i	−0.591655	+	0.806191i	$0.701525\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	744.064	+	1288.76i	0.385578	+	0.667841i
$156$		0			0
$157$	−830.078	+	1437.74i	−0.421958	+	0.730853i	−0.996131	−	0.0878810i	$-0.971990\pi$
$157$	−830.078	+	1437.74i	−0.421958	+	0.730853i	0.574173	+	0.818734i	$0.305324\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−2746.71			−1.34454
$162$		0			0
$163$	−3344.58			−1.60716			−0.803582	−	0.595195i	$-0.797075\pi$
$163$	−3344.58			−1.60716			−0.803582	+	0.595195i	$0.797075\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−220.558	+	382.018i	−0.102200	+	0.177015i	−0.912591	−	0.408875i	$-0.865921\pi$
$167$	−220.558	+	382.018i	−0.102200	+	0.177015i	0.810391	+	0.585889i	$0.199255\pi$
$168$		0			0
$169$	149.421	+	258.805i	0.0680115	+	0.117799i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1676.70	−	2904.12i	−0.736860	−	1.27628i	−0.953902	−	0.300117i	$-0.902974\pi$
$173$	−1676.70	−	2904.12i	−0.736860	−	1.27628i	0.217042	−	0.976162i	$-0.430359\pi$
$174$		0			0
$175$	−280.333	+	485.551i	−0.121093	+	0.209739i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	823.449			0.343840			0.171920	−	0.985111i	$-0.445003\pi$
$179$	823.449			0.343840			0.171920	+	0.985111i	$0.445003\pi$
$180$		0			0
$181$	−1754.26			−0.720404			−0.360202	−	0.932874i	$-0.617292\pi$
$181$	−1754.26			−0.720404			−0.360202	+	0.932874i	$0.617292\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	669.813	−	1160.15i	0.266192	−	0.461059i
$186$		0			0
$187$	−988.662	−	1712.41i	−0.386621	−	0.669647i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	557.115	+	964.952i	0.211055	+	0.365557i	0.952045	−	0.305958i	$-0.0989769\pi$
$191$	557.115	+	964.952i	0.211055	+	0.365557i	−0.740990	+	0.671516i	$0.765644\pi$
$192$		0			0
$193$	−1541.66	+	2670.23i	−0.574980	+	0.995894i	0.421064	+	0.907031i	$0.361657\pi$
$193$	−1541.66	+	2670.23i	−0.574980	+	0.995894i	−0.996044	+	0.0888631i	$0.971677\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−2968.69			−1.07366			−0.536828	−	0.843691i	$-0.680378\pi$
$197$	−2968.69			−1.07366			−0.536828	+	0.843691i	$0.680378\pi$
$198$		0			0
$199$	−3366.56			−1.19924			−0.599621	−	0.800284i	$-0.704682\pi$
$199$	−3366.56			−1.19924			−0.599621	+	0.800284i	$0.704682\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	3062.58	−	5304.55i	1.05887	−	1.83402i
$204$		0			0
$205$	−452.569	−	783.872i	−0.154189	−	0.267063i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	2179.78	+	3775.49i	0.721429	+	1.24955i
$210$		0			0
$211$	−1492.31	+	2584.75i	−0.486894	+	0.843326i	−0.999886	−	0.0150674i	$-0.995204\pi$
$211$	−1492.31	+	2584.75i	−0.486894	+	0.843326i	0.512992	+	0.858393i	$0.328537\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	1849.30			0.586610
$216$		0			0
$217$	−6674.75			−2.08807
$218$		0			0
$219$		0			0
$220$		0			0
$221$	762.174	−	1320.12i	0.231988	−	0.401815i
$222$		0			0
$223$	1776.48	+	3076.95i	0.533461	+	0.923982i	0.999236	+	0.0390790i	$0.0124424\pi$
$223$	1776.48	+	3076.95i	0.533461	+	0.923982i	−0.465775	+	0.884903i	$0.654224\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	879.065	+	1522.59i	0.257029	+	0.445187i	0.965445	−	0.260608i	$-0.0839230\pi$
$227$	879.065	+	1522.59i	0.257029	+	0.445187i	−0.708416	+	0.705796i	$0.750590\pi$
$228$		0			0
$229$	2122.35	−	3676.01i	0.612439	−	1.06078i	−0.378389	−	0.925647i	$-0.623522\pi$
$229$	2122.35	−	3676.01i	0.612439	−	1.06078i	0.990828	−	0.135129i	$-0.0431449\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−4604.60			−1.29467			−0.647334	−	0.762207i	$-0.724116\pi$
$233$	−4604.60			−1.29467			−0.647334	+	0.762207i	$0.724116\pi$
$234$		0			0
$235$	562.803			0.156226
$236$		0			0
$237$		0			0
$238$		0			0
$239$	1992.67	−	3451.40i	0.539309	−	0.934111i	−0.459632	−	0.888110i	$-0.652019\pi$
$239$	1992.67	−	3451.40i	0.539309	−	0.934111i	0.998941	−	0.0460018i	$-0.0146480\pi$
$240$		0			0
$241$	−1498.26	−	2595.06i	−0.400462	−	0.693620i	0.593320	−	0.804967i	$-0.297817\pi$
$241$	−1498.26	−	2595.06i	−0.400462	−	0.693620i	−0.993782	+	0.111347i	$0.964484\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−399.888	−	692.626i	−0.104277	−	0.180613i
$246$		0			0
$247$	−1680.43	+	2910.58i	−0.432886	+	0.749781i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−1586.05			−0.398847			−0.199423	−	0.979913i	$-0.563907\pi$
$251$	−1586.05			−0.398847			−0.199423	+	0.979913i	$0.563907\pi$
$252$		0			0
$253$	6921.63			1.72000
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−1306.40	+	2262.75i	−0.317085	+	0.549208i	−0.979879	−	0.199594i	$-0.936038\pi$
$257$	−1306.40	+	2262.75i	−0.317085	+	0.549208i	0.662793	+	0.748803i	$0.269371\pi$
$258$		0			0
$259$	3004.33	+	5203.66i	0.720773	+	1.24841i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	1093.88	+	1894.66i	0.256470	+	0.444219i	0.965294	−	0.261167i	$-0.0841072\pi$
$263$	1093.88	+	1894.66i	0.256470	+	0.444219i	−0.708824	+	0.705386i	$0.750774\pi$
$264$		0			0
$265$	−57.8536	+	100.205i	−0.0134110	+	0.0232286i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−3368.46			−0.763489			−0.381745	−	0.924268i	$-0.624677\pi$
$269$	−3368.46			−0.763489			−0.381745	+	0.924268i	$0.624677\pi$
$270$		0			0
$271$	−1111.68			−0.249187			−0.124594	−	0.992208i	$-0.539763\pi$
$271$	−1111.68			−0.249187			−0.124594	+	0.992208i	$0.539763\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	706.431	−	1223.57i	0.154907	−	0.268307i
$276$		0			0
$277$	1868.09	+	3235.63i	0.405209	+	0.701843i	0.994346	−	0.106191i	$-0.0338654\pi$
$277$	1868.09	+	3235.63i	0.405209	+	0.701843i	−0.589137	+	0.808033i	$0.700532\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	204.302	+	353.861i	0.0433723	+	0.0751230i	0.886897	−	0.461968i	$-0.152857\pi$
$281$	204.302	+	353.861i	0.0433723	+	0.0751230i	−0.843524	+	0.537091i	$0.819523\pi$
$282$		0			0
$283$	−663.450	+	1149.13i	−0.139357	+	0.241373i	−0.927253	−	0.374434i	$-0.877837\pi$
$283$	−663.450	+	1149.13i	−0.139357	+	0.241373i	0.787896	+	0.615808i	$0.211170\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	4059.84			0.835000
$288$		0			0
$289$	−3688.85			−0.750834
$290$		0			0
$291$		0			0
$292$		0			0
$293$	672.729	−	1165.20i	0.134134	−	0.232327i	−0.791132	−	0.611645i	$-0.790508\pi$
$293$	672.729	−	1165.20i	0.134134	−	0.232327i	0.925266	+	0.379318i	$0.123841\pi$
$294$		0			0
$295$	698.288	+	1209.47i	0.137817	+	0.238705i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	2667.99	+	4621.10i	0.516033	+	0.893795i
$300$		0			0
$301$	−4147.36	+	7183.44i	−0.794186	+	1.37557i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−1964.34			−0.368780
$306$		0			0
$307$	−5947.29			−1.10563			−0.552817	−	0.833303i	$-0.686447\pi$
$307$	−5947.29			−1.10563			−0.552817	+	0.833303i	$0.686447\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−2422.82	+	4196.44i	−0.441753	+	0.765139i	−0.997820	−	0.0659987i	$-0.978977\pi$
$311$	−2422.82	+	4196.44i	−0.441753	+	0.765139i	0.556066	+	0.831138i	$0.312310\pi$
$312$		0			0
$313$	2612.44	+	4524.88i	0.471770	+	0.817129i	0.999478	−	0.0322964i	$-0.0102821\pi$
$313$	2612.44	+	4524.88i	0.471770	+	0.817129i	−0.527709	+	0.849425i	$0.676949\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	3381.55	+	5857.02i	0.599139	+	1.03774i	0.992948	+	0.118547i	$0.0378236\pi$
$317$	3381.55	+	5857.02i	0.599139	+	1.03774i	−0.393810	+	0.919192i	$0.628843\pi$
$318$		0			0
$319$	−7717.62	+	13367.3i	−1.35456	+	2.34616i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−2698.99			−0.464941
$324$		0			0
$325$	1089.20			0.185901
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−1262.18	+	2186.16i	−0.211508	+	0.366343i
$330$		0			0
$331$	−1921.16	−	3327.55i	−0.319023	−	0.552563i	0.661262	−	0.750155i	$-0.270021\pi$
$331$	−1921.16	−	3327.55i	−0.319023	−	0.552563i	−0.980284	+	0.197592i	$0.936688\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−986.451	−	1708.58i	−0.160882	−	0.278656i
$336$		0			0
$337$	−4208.82	+	7289.89i	−0.680323	+	1.17835i	0.294559	+	0.955633i	$0.404827\pi$
$337$	−4208.82	+	7289.89i	−0.680323	+	1.17835i	−0.974882	+	0.222721i	$0.928506\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	16820.2			2.67115
$342$		0			0
$343$	−4105.09			−0.646221
$344$		0			0
$345$		0			0
$346$		0			0
$347$	471.641	−	816.906i	0.0729655	−	0.126380i	−0.827234	−	0.561857i	$-0.810087\pi$
$347$	471.641	−	816.906i	0.0729655	−	0.126380i	0.900200	+	0.435477i	$0.143420\pi$
$348$		0			0
$349$	−1094.43	−	1895.61i	−0.167861	−	0.290744i	0.769807	−	0.638277i	$-0.220353\pi$
$349$	−1094.43	−	1895.61i	−0.167861	−	0.290744i	−0.937668	+	0.347533i	$0.887019\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	4026.22	+	6973.61i	0.607065	+	1.05147i	0.991722	+	0.128407i	$0.0409865\pi$
$353$	4026.22	+	6973.61i	0.607065	+	1.05147i	−0.384657	+	0.923060i	$0.625680\pi$
$354$		0			0
$355$	2432.54	−	4213.29i	0.363679	−	0.629910i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	5099.37			0.749679			0.374839	−	0.927090i	$-0.377698\pi$
$359$	5099.37			0.749679			0.374839	+	0.927090i	$0.377698\pi$
$360$		0			0
$361$	−908.321			−0.132428
$362$		0			0
$363$		0			0
$364$		0			0
$365$	1901.92	−	3294.23i	0.272743	−	0.472405i
$366$		0			0
$367$	923.372	+	1599.33i	0.131334	+	0.227477i	0.924191	−	0.381930i	$-0.124741\pi$
$367$	923.372	+	1599.33i	0.131334	+	0.227477i	−0.792857	+	0.609408i	$0.791407\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−259.493	−	449.455i	−0.0363132	−	0.0628963i
$372$		0			0
$373$	2584.59	−	4476.64i	0.358780	−	0.621426i	−0.628977	−	0.777424i	$-0.716526\pi$
$373$	2584.59	−	4476.64i	0.358780	−	0.621426i	0.987757	+	0.155998i	$0.0498594\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−11899.2			−1.62558
$378$		0			0
$379$	1189.17			0.161170			0.0805851	−	0.996748i	$-0.474321\pi$
$379$	1189.17			0.161170			0.0805851	+	0.996748i	$0.474321\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	889.184	−	1540.11i	0.118630	−	0.205473i	−0.800595	−	0.599206i	$-0.795483\pi$
$383$	889.184	−	1540.11i	0.118630	−	0.205473i	0.919225	+	0.393733i	$0.128817\pi$
$384$		0			0
$385$	3168.58	+	5488.14i	0.419443	+	0.726497i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−6151.64	−	10655.0i	−0.801801	−	1.38876i	−0.918430	−	0.395585i	$-0.870542\pi$
$389$	−6151.64	−	10655.0i	−0.801801	−	1.38876i	0.116628	−	0.993176i	$-0.462791\pi$
$390$		0			0
$391$	−2142.58	+	3711.05i	−0.277122	+	0.479990i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−4158.76			−0.529747
$396$		0			0
$397$	4361.53			0.551383			0.275692	−	0.961246i	$-0.411093\pi$
$397$	4361.53			0.551383			0.275692	+	0.961246i	$0.411093\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	3444.34	−	5965.77i	0.428933	−	0.742933i	−0.567846	−	0.823135i	$-0.692223\pi$
$401$	3444.34	−	5965.77i	0.428933	−	0.742933i	0.996779	+	0.0802015i	$0.0255564\pi$
$402$		0			0
$403$	6483.45	+	11229.7i	0.801399	+	1.38806i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−7570.82	−	13113.1i	−0.922044	−	1.59703i
$408$		0			0
$409$	2311.31	−	4003.31i	0.279431	−	0.483988i	−0.691813	−	0.722077i	$-0.743188\pi$
$409$	2311.31	−	4003.31i	0.279431	−	0.483988i	0.971243	+	0.238089i	$0.0765210\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−6264.11			−0.746335
$414$		0			0
$415$	2599.45			0.307475
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−5052.83	+	8751.76i	−0.589134	+	1.02041i	0.405213	+	0.914222i	$0.367198\pi$
$419$	−5052.83	+	8751.76i	−0.589134	+	1.02041i	−0.994346	+	0.106187i	$0.966136\pi$
$420$		0			0
$421$	−4803.90	−	8320.60i	−0.556123	−	0.963233i	−0.997815	−	0.0660664i	$-0.978955\pi$
$421$	−4803.90	−	8320.60i	−0.556123	−	0.963233i	0.441692	−	0.897167i	$-0.354378\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	437.349	+	757.511i	0.0499166	+	0.0864580i
$426$		0			0
$427$	4405.36	−	7630.32i	0.499275	−	0.864770i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−6722.50			−0.751303			−0.375651	−	0.926761i	$-0.622581\pi$
$431$	−6722.50			−0.751303			−0.375651	+	0.926761i	$0.622581\pi$
$432$		0			0
$433$	−10997.0			−1.22052			−0.610258	−	0.792202i	$-0.708934\pi$
$433$	−10997.0			−1.22052			−0.610258	+	0.792202i	$0.708934\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	4723.91	−	8182.05i	0.517106	−	0.895653i
$438$		0			0
$439$	−6571.87	−	11382.8i	−0.714484	−	1.23752i	−0.963158	−	0.268935i	$-0.913328\pi$
$439$	−6571.87	−	11382.8i	−0.714484	−	1.23752i	0.248674	−	0.968587i	$-0.420005\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−6922.23	−	11989.7i	−0.742404	−	1.28588i	−0.951398	−	0.307965i	$-0.900352\pi$
$443$	−6922.23	−	11989.7i	−0.742404	−	1.28588i	0.208993	−	0.977917i	$-0.432981\pi$
$444$		0			0
$445$	−2973.04	+	5149.46i	−0.316709	+	0.548557i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−13502.6			−1.41921			−0.709605	−	0.704600i	$-0.751126\pi$
$449$	−13502.6			−1.41921			−0.709605	+	0.704600i	$0.751126\pi$
$450$		0			0
$451$	−10230.7			−1.06817
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−2442.70	+	4230.89i	−0.251683	+	0.435928i
$456$		0			0
$457$	−2332.59	−	4040.17i	−0.238762	−	0.413547i	0.721598	−	0.692313i	$-0.243408\pi$
$457$	−2332.59	−	4040.17i	−0.238762	−	0.413547i	−0.960359	+	0.278765i	$0.910075\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	7610.11	+	13181.1i	0.768847	+	1.33168i	0.938189	+	0.346124i	$0.112502\pi$
$461$	7610.11	+	13181.1i	0.768847	+	1.33168i	−0.169342	+	0.985557i	$0.554164\pi$
$462$		0			0
$463$	5344.55	−	9257.04i	0.536463	−	0.929181i	−0.462628	−	0.886553i	$-0.653093\pi$
$463$	5344.55	−	9257.04i	0.536463	−	0.929181i	0.999091	−	0.0426288i	$-0.0135733\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	10642.9			1.05459			0.527295	−	0.849682i	$-0.323206\pi$
$467$	10642.9			1.05459			0.527295	+	0.849682i	$0.323206\pi$
$468$		0			0
$469$	8849.12			0.871246
$470$		0			0
$471$		0			0
$472$		0			0
$473$	10451.2	−	18102.1i	1.01596	−	1.75969i
$474$		0			0
$475$	−964.258	−	1670.14i	−0.0931436	−	0.161329i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	2596.24	+	4496.82i	0.247652	+	0.428946i	0.962874	−	0.269952i	$-0.0870077\pi$
$479$	2596.24	+	4496.82i	0.247652	+	0.428946i	−0.715222	+	0.698897i	$0.753674\pi$
$480$		0			0
$481$	5836.46	−	10109.0i	0.553263	−	0.958280i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−4198.59			−0.393089
$486$		0			0
$487$	13302.4			1.23776			0.618882	−	0.785484i	$-0.287586\pi$
$487$	13302.4			1.23776			0.618882	+	0.785484i	$0.287586\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−8518.11	+	14753.8i	−0.782926	+	1.35607i	0.147304	+	0.989091i	$0.452941\pi$
$491$	−8518.11	+	14753.8i	−0.782926	+	1.35607i	−0.930230	+	0.366977i	$0.880393\pi$
$492$		0			0
$493$	−4777.95	−	8275.65i	−0.436487	−	0.756017i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	10910.8	+	18898.0i	0.984737	+	1.70561i
$498$		0			0
$499$	−7425.33	+	12861.0i	−0.666139	+	1.15379i	0.312836	+	0.949807i	$0.398721\pi$
$499$	−7425.33	+	12861.0i	−0.666139	+	1.15379i	−0.978975	+	0.203980i	$0.934612\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	12662.6			1.12246			0.561231	−	0.827659i	$-0.310328\pi$
$503$	12662.6			1.12246			0.561231	+	0.827659i	$0.310328\pi$
$504$		0			0
$505$	−5206.09			−0.458748
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−398.665	+	690.509i	−0.0347162	+	0.0601302i	−0.882861	−	0.469634i	$-0.844386\pi$
$509$	−398.665	+	690.509i	−0.0347162	+	0.0601302i	0.848145	+	0.529764i	$0.177719\pi$
$510$		0			0
$511$	8530.76	+	14775.7i	0.738510	+	1.27914i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−4670.34	−	8089.27i	−0.399611	−	0.692147i
$516$		0			0
$517$	3180.65	−	5509.05i	0.270570	−	0.468642i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−8374.09			−0.704176			−0.352088	−	0.935967i	$-0.614528\pi$
$521$	−8374.09			−0.704176			−0.352088	+	0.935967i	$0.614528\pi$
$522$		0			0
$523$	1576.14			0.131778			0.0658889	−	0.997827i	$-0.479012\pi$
$523$	1576.14			0.131778			0.0658889	+	0.997827i	$0.479012\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−5206.65	+	9018.18i	−0.430370	+	0.745424i
$528$		0			0
$529$	−1416.59	−	2453.60i	−0.116429	−	0.201660i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−3943.49	−	6830.33i	−0.320472	−	0.555074i
$534$		0			0
$535$	927.889	−	1607.15i	0.0749834	−	0.129875i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−9039.79			−0.722395
$540$		0			0
$541$	17191.1			1.36618			0.683089	−	0.730335i	$-0.260636\pi$
$541$	17191.1			1.36618			0.683089	+	0.730335i	$0.260636\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	3012.35	−	5217.54i	0.236761	−	0.410082i
$546$		0			0
$547$	−8006.25	−	13867.2i	−0.625818	−	1.08395i	−0.988382	−	0.151990i	$-0.951432\pi$
$547$	−8006.25	−	13867.2i	−0.625818	−	1.08395i	0.362564	−	0.931959i	$-0.381901\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	10534.3	+	18246.0i	0.814477	+	1.41072i
$552$		0			0
$553$	9326.72	−	16154.3i	0.717201	−	1.24223i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	18982.1			1.44398			0.721992	−	0.691902i	$-0.243227\pi$
$557$	18982.1			1.44398			0.721992	+	0.691902i	$0.243227\pi$
$558$		0			0
$559$	16114.0			1.21923
$560$		0			0
$561$		0			0
$562$		0			0
$563$	762.766	−	1321.15i	0.0570991	−	0.0988985i	−0.836063	−	0.548633i	$-0.815148\pi$
$563$	762.766	−	1321.15i	0.0570991	−	0.0988985i	0.893162	+	0.449735i	$0.148482\pi$
$564$		0			0
$565$	−5508.76	−	9541.45i	−0.410186	−	0.710464i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−6405.84	−	11095.2i	−0.471963	−	0.817464i	0.527523	−	0.849541i	$-0.323121\pi$
$569$	−6405.84	−	11095.2i	−0.471963	−	0.817464i	−0.999485	+	0.0320774i	$0.989788\pi$
$570$		0			0
$571$	−4186.43	+	7251.10i	−0.306824	+	0.531435i	−0.977666	−	0.210166i	$-0.932600\pi$
$571$	−4186.43	+	7251.10i	−0.306824	+	0.531435i	0.670842	+	0.741600i	$0.265933\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−3061.88			−0.222068
$576$		0			0
$577$	−7014.66			−0.506107			−0.253054	−	0.967452i	$-0.581435\pi$
$577$	−7014.66			−0.506107			−0.253054	+	0.967452i	$0.581435\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−5829.70	+	10097.3i	−0.416276	+	0.721012i
$582$		0			0
$583$	653.914	+	1132.61i	0.0464534	+	0.0804597i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−1925.96	−	3335.87i	−0.135423	−	0.234559i	0.790336	−	0.612673i	$-0.209906\pi$
$587$	−1925.96	−	3335.87i	−0.135423	−	0.234559i	−0.925759	+	0.378115i	$0.876573\pi$
$588$		0			0
$589$	11479.5	−	19883.1i	0.803065	−	1.39095i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	22196.9			1.53713			0.768563	−	0.639774i	$-0.220972\pi$
$593$	22196.9			1.53713			0.768563	+	0.639774i	$0.220972\pi$
$594$		0			0
$595$	−3923.31			−0.270319
$596$		0			0
$597$		0			0
$598$		0			0
$599$	3407.68	−	5902.28i	0.232444	−	0.402605i	−0.726083	−	0.687607i	$-0.758661\pi$
$599$	3407.68	−	5902.28i	0.232444	−	0.402605i	0.958527	+	0.285002i	$0.0919943\pi$
$600$		0			0
$601$	−4952.25	−	8577.54i	−0.336117	−	0.582172i	0.647582	−	0.761996i	$-0.275780\pi$
$601$	−4952.25	−	8577.54i	−0.336117	−	0.582172i	−0.983699	+	0.179824i	$0.942447\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−4657.22	−	8066.55i	−0.312964	−	0.542069i
$606$		0			0
$607$	−6013.87	+	10416.3i	−0.402135	+	0.696517i	−0.993983	−	0.109532i	$-0.965065\pi$
$607$	−6013.87	+	10416.3i	−0.402135	+	0.696517i	0.591849	+	0.806049i	$0.298398\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	4904.02			0.324706
$612$		0			0
$613$	−21430.6			−1.41203			−0.706014	−	0.708198i	$-0.749508\pi$
$613$	−21430.6			−1.41203			−0.706014	+	0.708198i	$0.749508\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−14095.6	+	24414.2i	−0.919718	+	1.59300i	−0.119875	+	0.992789i	$0.538249\pi$
$617$	−14095.6	+	24414.2i	−0.919718	+	1.59300i	−0.799843	+	0.600210i	$0.795084\pi$
$618$		0			0
$619$	1634.60	+	2831.21i	0.106139	+	0.183838i	0.914203	−	0.405257i	$-0.132818\pi$
$619$	1634.60	+	2831.21i	0.106139	+	0.183838i	−0.808064	+	0.589095i	$0.799484\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−13335.1	−	23097.0i	−0.857558	−	1.48533i
$624$		0			0
$625$	−312.500	+	541.266i	−0.0200000	+	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	9374.14			0.594231
$630$		0			0
$631$	16952.0			1.06949			0.534745	−	0.845013i	$-0.320408\pi$
$631$	16952.0			1.06949			0.534745	+	0.845013i	$0.320408\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−6883.41	+	11922.4i	−0.430173	+	0.745081i
$636$		0			0
$637$	−3484.45	−	6035.25i	−0.216733	−	0.375393i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	5134.39	+	8893.03i	0.316375	+	0.547978i	0.979729	−	0.200328i	$-0.0642009\pi$
$641$	5134.39	+	8893.03i	0.316375	+	0.547978i	−0.663354	+	0.748306i	$0.730868\pi$
$642$		0			0
$643$	−2148.91	+	3722.01i	−0.131796	+	0.228277i	−0.924369	−	0.381500i	$-0.875408\pi$
$643$	−2148.91	+	3722.01i	−0.131796	+	0.228277i	0.792573	+	0.609777i	$0.208741\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	11265.0			0.684503			0.342252	−	0.939608i	$-0.388810\pi$
$647$	11265.0			0.684503			0.342252	+	0.939608i	$0.388810\pi$
$648$		0			0
$649$	15785.4			0.954745
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−2845.82	+	4929.11i	−0.170545	+	0.295392i	−0.938610	−	0.344979i	$-0.887886\pi$
$653$	−2845.82	+	4929.11i	−0.170545	+	0.295392i	0.768066	+	0.640371i	$0.221219\pi$
$654$		0			0
$655$	2328.98	+	4033.91i	0.138933	+	0.240638i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−11002.7	−	19057.1i	−0.650383	−	1.12650i	−0.983030	−	0.183445i	$-0.941275\pi$
$659$	−11002.7	−	19057.1i	−0.650383	−	1.12650i	0.332647	−	0.943051i	$-0.392058\pi$
$660$		0			0
$661$	9710.96	−	16819.9i	0.571426	−	0.989738i	−0.424994	−	0.905196i	$-0.639724\pi$
$661$	9710.96	−	16819.9i	0.571426	−	0.989738i	0.996420	−	0.0845423i	$-0.0269428\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	8650.04			0.504412
$666$		0			0
$667$	33450.4			1.94184
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−11101.4	+	19228.2i	−0.638695	+	1.10625i
$672$		0			0
$673$	7650.79	+	13251.6i	0.438211	+	0.759004i	0.997552	−	0.0699340i	$-0.0222789\pi$
$673$	7650.79	+	13251.6i	0.438211	+	0.759004i	−0.559340	+	0.828938i	$0.688946\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	2216.11	+	3838.41i	0.125808	+	0.217906i	0.922048	−	0.387074i	$-0.126514\pi$
$677$	2216.11	+	3838.41i	0.125808	+	0.217906i	−0.796241	+	0.604980i	$0.793181\pi$
$678$		0			0
$679$	9416.03	−	16309.0i	0.532186	−	0.921772i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	17375.9			0.973455			0.486727	−	0.873554i	$-0.338191\pi$
$683$	17375.9			0.973455			0.486727	+	0.873554i	$0.338191\pi$
$684$		0			0
$685$	−6472.29			−0.361012
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−504.112	+	873.147i	−0.0278739	+	0.0482790i
$690$		0			0
$691$	10453.8	+	18106.5i	0.575515	+	0.996821i	0.995985	+	0.0895150i	$0.0285317\pi$
$691$	10453.8	+	18106.5i	0.575515	+	0.996821i	−0.420470	+	0.907306i	$0.638135\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−2441.66	−	4229.08i	−0.133263	−	0.230818i
$696$		0			0
$697$	3166.89	−	5485.21i	0.172101	−	0.298088i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	28112.3			1.51467			0.757337	−	0.653024i	$-0.226500\pi$
$701$	28112.3			1.51467			0.757337	+	0.653024i	$0.226500\pi$
$702$		0			0
$703$	−20667.9			−1.10883
$704$		0			0
$705$		0			0
$706$		0			0
$707$	11675.5	−	20222.6i	0.621079	−	1.07574i
$708$		0			0
$709$	−3504.67	−	6070.27i	−0.185643	−	0.321542i	0.758150	−	0.652080i	$-0.226103\pi$
$709$	−3504.67	−	6070.27i	−0.185643	−	0.321542i	−0.943793	+	0.330537i	$0.892770\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−18225.9	−	31568.1i	−0.957313	−	1.65812i
$714$		0			0
$715$	6155.54	−	10661.7i	0.321964	−	0.557658i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−15546.9			−0.806401			−0.403200	−	0.915112i	$-0.632102\pi$
$719$	−15546.9			−0.806401			−0.403200	+	0.915112i	$0.632102\pi$
$720$		0			0
$721$	41896.1			2.16407
$722$		0			0
$723$		0			0
$724$		0			0
$725$	3414.00	−	5913.22i	0.174886	−	0.302912i
$726$		0			0
$727$	14835.7	+	25696.2i	0.756846	+	1.31090i	0.944452	+	0.328651i	$0.106594\pi$
$727$	14835.7	+	25696.2i	0.756846	+	1.31090i	−0.187606	+	0.982244i	$0.560073\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	6470.32	+	11206.9i	0.327378	+	0.567035i
$732$		0			0
$733$	17542.4	−	30384.4i	0.883962	−	1.53107i	0.0370635	−	0.999313i	$-0.488200\pi$
$733$	17542.4	−	30384.4i	0.883962	−	1.53107i	0.846899	−	0.531754i	$-0.178467\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−22299.5			−1.11454
$738$		0			0
$739$	−13071.3			−0.650657			−0.325328	−	0.945601i	$-0.605475\pi$
$739$	−13071.3			−0.650657			−0.325328	+	0.945601i	$0.605475\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−3966.90	+	6870.88i	−0.195870	+	0.339257i	−0.947185	−	0.320686i	$-0.896086\pi$
$743$	−3966.90	+	6870.88i	−0.195870	+	0.339257i	0.751315	+	0.659944i	$0.229420\pi$
$744$		0			0
$745$	4347.33	+	7529.80i	0.213790	+	0.370296i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	4161.89	+	7208.61i	0.203034	+	0.351665i
$750$		0			0
$751$	14630.4	−	25340.6i	0.710880	−	1.23128i	−0.253647	−	0.967297i	$-0.581630\pi$
$751$	14630.4	−	25340.6i	0.710880	−	1.23128i	0.964527	−	0.263984i	$-0.0850365\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−7465.77			−0.359877
$756$		0			0
$757$	1284.23			0.0616593			0.0308296	−	0.999525i	$-0.490185\pi$
$757$	1284.23			0.0616593			0.0308296	+	0.999525i	$0.490185\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	12668.9	−	21943.2i	0.603480	−	1.04526i	−0.388810	−	0.921318i	$-0.627114\pi$
$761$	12668.9	−	21943.2i	0.603480	−	1.04526i	0.992290	−	0.123940i	$-0.0395531\pi$
$762$		0			0
$763$	13511.4	+	23402.4i	0.641081	+	1.11038i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	6084.58	+	10538.8i	0.286443	+	0.496133i
$768$		0			0
$769$	−9619.20	+	16660.9i	−0.451076	+	0.781286i	−0.998453	−	0.0555997i	$-0.982293\pi$
$769$	−9619.20	+	16660.9i	−0.451076	+	0.781286i	0.547377	+	0.836886i	$0.315626\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−33366.9			−1.55255			−0.776276	−	0.630393i	$-0.782894\pi$
$773$	−33366.9			−1.55255			−0.776276	+	0.630393i	$0.782894\pi$
$774$		0			0
$775$	−7440.64			−0.344872
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−6982.29	+	12093.7i	−0.321138	+	0.556227i
$780$		0			0
$781$	−27494.8	−	47622.3i	−1.25972	−	2.18190i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−4150.39	−	7188.69i	−0.188705	−	0.326847i
$786$		0			0
$787$	4937.88	−	8552.66i	0.223655	−	0.387382i	−0.732260	−	0.681025i	$-0.761534\pi$
$787$	4937.88	−	8552.66i	0.223655	−	0.387382i	0.955915	+	0.293643i	$0.0948678\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	49417.3			2.22133
$792$		0			0
$793$	−17116.4			−0.766485
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−1321.96	+	2289.70i	−0.0587530	+	0.101763i	−0.893906	−	0.448255i	$-0.852046\pi$
$797$	−1321.96	+	2289.70i	−0.0587530	+	0.101763i	0.835153	+	0.550018i	$0.185379\pi$
$798$		0			0
$799$	1969.13	+	3410.63i	0.0871875	+	0.151013i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−21497.3	−	37234.4i	−0.944735	−	1.63633i
$804$		0			0
$805$	6866.77	−	11893.6i	0.300648	−	0.520738i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	23696.4			1.02982			0.514908	−	0.857246i	$-0.327826\pi$
$809$	23696.4			1.02982			0.514908	+	0.857246i	$0.327826\pi$
$810$		0			0
$811$	30745.5			1.33122			0.665611	−	0.746299i	$-0.268171\pi$
$811$	30745.5			1.33122			0.665611	+	0.746299i	$0.268171\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	8361.45	−	14482.4i	0.359373	−	0.622452i
$816$		0			0
$817$	−14265.6	−	24708.8i	−0.610882	−	1.05808i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−10022.4	−	17359.2i	−0.426045	−	0.737931i	0.570473	−	0.821317i	$-0.306760\pi$
$821$	−10022.4	−	17359.2i	−0.426045	−	0.737931i	−0.996517	+	0.0833854i	$0.973427\pi$
$822$		0			0
$823$	9131.45	−	15816.1i	0.386759	−	0.669886i	−0.605253	−	0.796033i	$-0.706928\pi$
$823$	9131.45	−	15816.1i	0.386759	−	0.669886i	0.992011	+	0.126147i	$0.0402613\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	20442.0			0.859538			0.429769	−	0.902939i	$-0.358595\pi$
$827$	20442.0			0.859538			0.429769	+	0.902939i	$0.358595\pi$
$828$		0			0
$829$	−8012.09			−0.335671			−0.167835	−	0.985815i	$-0.553678\pi$
$829$	−8012.09			−0.335671			−0.167835	+	0.985815i	$0.553678\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	2798.25	−	4846.71i	0.116391	−	0.201595i
$834$		0			0
$835$	−1102.79	−	1910.09i	−0.0457050	−	0.0791634i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	17785.4	+	30805.2i	0.731847	+	1.26760i	0.956093	+	0.293063i	$0.0946746\pi$
$839$	17785.4	+	30805.2i	0.731847	+	1.26760i	−0.224247	+	0.974532i	$0.571992\pi$
$840$		0			0
$841$	−25102.7	+	43479.2i	−1.02926	+	1.78274i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−1494.21			−0.0608313
$846$		0			0
$847$	41778.4			1.69483
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−16407.1	+	28417.9i	−0.660902	+	1.14472i
$852$		0			0
$853$	−21526.6	−	37285.2i	−0.864076	−	1.49662i	−0.867961	−	0.496631i	$-0.834570\pi$
$853$	−21526.6	−	37285.2i	−0.864076	−	1.49662i	0.00388525	−	0.999992i	$-0.498763\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	4512.43	+	7815.76i	0.179862	+	0.311530i	0.941833	−	0.336081i	$-0.109102\pi$
$857$	4512.43	+	7815.76i	0.179862	+	0.311530i	−0.761971	+	0.647611i	$0.775768\pi$
$858$		0			0
$859$	7844.20	−	13586.6i	0.311572	−	0.539659i	−0.667131	−	0.744941i	$-0.732478\pi$
$859$	7844.20	−	13586.6i	0.311572	−	0.539659i	0.978703	+	0.205282i	$0.0658110\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−46241.1			−1.82394			−0.911972	−	0.410252i	$-0.865441\pi$
$863$	−46241.1			−1.82394			−0.911972	+	0.410252i	$0.865441\pi$
$864$		0			0
$865$	16767.0			0.659068
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−23503.0	+	40708.5i	−0.917475	+	1.58911i
$870$		0			0
$871$	−8595.51	−	14887.9i	−0.334383	−	0.579168i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−1401.67	−	2427.76i	−0.0541543	−	0.0937979i
$876$		0			0
$877$	20068.2	−	34759.2i	0.772699	−	1.33835i	−0.163380	−	0.986563i	$-0.552240\pi$
$877$	20068.2	−	34759.2i	0.772699	−	1.33835i	0.936079	−	0.351790i	$-0.114427\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−38113.5			−1.45752			−0.728761	−	0.684768i	$-0.759904\pi$
$881$	−38113.5			−1.45752			−0.728761	+	0.684768i	$0.759904\pi$
$882$		0			0
$883$	18171.9			0.692563			0.346281	−	0.938131i	$-0.387444\pi$
$883$	18171.9			0.692563			0.346281	+	0.938131i	$0.387444\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−25132.6	+	43531.0i	−0.951377	+	1.64783i	−0.208927	+	0.977931i	$0.566997\pi$
$887$	−25132.6	+	43531.0i	−0.951377	+	1.64783i	−0.742450	+	0.669902i	$0.766336\pi$
$888$		0			0
$889$	−30874.4	−	53476.0i	−1.16478	−	2.01747i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−4341.50	−	7519.69i	−0.162691	−	0.281788i
$894$		0			0
$895$	−2058.62	+	3565.64i	−0.0768851	+	0.133169i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	81287.5			3.01567
$900$		0			0
$901$	−809.671			−0.0299379
$902$		0			0
$903$		0			0
$904$		0			0
$905$	4385.65	−	7596.17i	0.161087	−	0.279011i
$906$		0			0
$907$	−4552.99	−	7886.01i	−0.166681	−	0.288700i	0.770570	−	0.637355i	$-0.219972\pi$
$907$	−4552.99	−	7886.01i	−0.166681	−	0.288700i	−0.937251	+	0.348656i	$0.886638\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	16085.0	+	27860.0i	0.584983	+	1.01322i	0.994878	+	0.101088i	$0.0322323\pi$
$911$	16085.0	+	27860.0i	0.584983	+	1.01322i	−0.409894	+	0.912133i	$0.634434\pi$
$912$		0			0
$913$	14690.7	−	25445.0i	0.532519	−	0.922350i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−20892.5			−0.752379
$918$		0			0
$919$	16150.1			0.579699			0.289849	−	0.957072i	$-0.406395\pi$
$919$	16150.1			0.579699			0.289849	+	0.957072i	$0.406395\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	21196.1	−	36712.8i	0.755882	−	1.30923i
$924$		0			0
$925$	3349.06	+	5800.75i	0.119045	+	0.206192i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	16613.7	+	28775.8i	0.586738	+	1.01626i	0.994656	+	0.103241i	$0.0329213\pi$
$929$	16613.7	+	28775.8i	0.586738	+	1.01626i	−0.407919	+	0.913018i	$0.633745\pi$
$930$		0			0
$931$	−6169.52	+	10685.9i	−0.217184	+	0.376173i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	9886.62			0.345804
$936$		0			0
$937$	−43603.7			−1.52025			−0.760123	−	0.649779i	$-0.774861\pi$
$937$	−43603.7			−1.52025			−0.760123	+	0.649779i	$0.774861\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−15502.7	+	26851.4i	−0.537059	+	0.930213i	0.462002	+	0.886879i	$0.347131\pi$
$941$	−15502.7	+	26851.4i	−0.537059	+	0.930213i	−0.999061	+	0.0433342i	$0.986202\pi$
$942$		0			0
$943$	11085.7	+	19201.0i	0.382821	+	0.663065i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−2728.27	−	4725.51i	−0.0936187	−	0.162152i	0.815413	−	0.578880i	$-0.196510\pi$
$947$	−2728.27	−	4725.51i	−0.0936187	−	0.162152i	−0.909031	+	0.416728i	$0.863177\pi$
$948$		0			0
$949$	16572.6	−	28704.5i	0.566879	−	0.981863i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	7491.89			0.254655			0.127327	−	0.991861i	$-0.459360\pi$
$953$	7491.89			0.254655			0.127327	+	0.991861i	$0.459360\pi$
$954$		0			0
$955$	−5571.15			−0.188773
$956$		0			0
$957$		0			0
$958$		0			0
$959$	14515.2	−	25141.0i	0.488759	−	0.846555i
$960$		0			0
$961$	−29395.0	−	50913.6i	−0.986707	−	1.70903i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−7708.30	−	13351.2i	−0.257139	−	0.445377i
$966$		0			0
$967$	−19727.2	+	34168.5i	−0.656033	+	1.13628i	0.325600	+	0.945508i	$0.394434\pi$
$967$	−19727.2	+	34168.5i	−0.656033	+	1.13628i	−0.981634	+	0.190776i	$0.938900\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−9799.12			−0.323861			−0.161930	−	0.986802i	$-0.551772\pi$
$971$	−9799.12			−0.323861			−0.161930	+	0.986802i	$0.551772\pi$
$972$		0			0
$973$	21903.3			0.721674
$974$		0			0
$975$		0			0
$976$		0			0
$977$	2024.53	−	3506.59i	0.0662953	−	0.114827i	−0.830973	−	0.556313i	$-0.812215\pi$
$977$	2024.53	−	3506.59i	0.0662953	−	0.114827i	0.897268	+	0.441487i	$0.145549\pi$
$978$		0			0
$979$	33604.0	+	58203.8i	1.09703	+	1.90010i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−18243.5	−	31598.7i	−0.591941	−	1.02527i	−0.993971	−	0.109646i	$-0.965028\pi$
$983$	−18243.5	−	31598.7i	−0.591941	−	1.02527i	0.402030	−	0.915627i	$-0.368305\pi$
$984$		0			0
$985$	7421.72	−	12854.8i	0.240077	−	0.415825i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−45298.7			−1.45643
$990$		0			0
$991$	−20630.9			−0.661313			−0.330656	−	0.943751i	$-0.607270\pi$
$991$	−20630.9			−0.661313			−0.330656	+	0.943751i	$0.607270\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	8416.41	−	14577.6i	0.268159	−	0.464465i
$996$		0			0
$997$	6607.83	+	11445.1i	0.209902	+	0.363561i	0.951683	−	0.307081i	$-0.0993522\pi$
$997$	6607.83	+	11445.1i	0.209902	+	0.363561i	−0.741782	+	0.670642i	$0.766019\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.4.i.w.1081.1		12
3.2	odd	2		1620.4.i.x.1081.1		12
9.2	odd	6		1620.4.i.x.541.1		12
9.4	even	3		1620.4.a.j.1.6	yes	6
9.5	odd	6		1620.4.a.i.1.6	✓	6
9.7	even	3	inner	1620.4.i.w.541.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1620.4.a.i.1.6	✓	6	9.5	odd	6
1620.4.a.j.1.6	yes	6	9.4	even	3
1620.4.i.w.541.1		12	9.7	even	3	inner
1620.4.i.w.1081.1		12	1.1	even	1	trivial
1620.4.i.x.541.1		12	9.2	odd	6
1620.4.i.x.1081.1		12	3.2	odd	2

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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1620.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.50000 + 4.33013i) q^{5} +(-11.2133 - 19.4221i) q^{7} +O(q^{10})\)	\(q+(-2.50000 + 4.33013i) q^{5} +(-11.2133 - 19.4221i) q^{7} +(28.2573 + 48.9430i) q^{11} +(-21.7839 + 37.7309i) q^{13} -34.9879 q^{17} +77.1406 q^{19} +(61.2376 - 106.067i) q^{23} +(-12.5000 - 21.6506i) q^{25} +(136.560 + 236.529i) q^{29} +(148.813 - 257.751i) q^{31} +112.133 q^{35} -267.925 q^{37} +(-90.5138 + 156.774i) q^{41} +(-184.930 - 320.308i) q^{43} +(-56.2803 - 97.4803i) q^{47} +(-79.9776 + 138.525i) q^{49} +23.1415 q^{53} -282.573 q^{55} +(139.658 - 241.894i) q^{59} +(196.434 + 340.234i) q^{61} +(-108.920 - 188.654i) q^{65} +(-197.290 + 341.716i) q^{67} -973.017 q^{71} -760.770 q^{73} +(633.716 - 1097.63i) q^{77} +(415.876 + 720.319i) q^{79} +(-259.945 - 450.238i) q^{83} +(87.4698 - 151.502i) q^{85} +1189.22 q^{89} +977.081 q^{91} +(-192.852 + 334.029i) q^{95} +(419.859 + 727.217i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 30 q^{5} - 12 q^{7}+O(q^{10}) \) 12 * q - 30 * q^5 - 12 * q^7	\( 12 q - 30 q^{5} - 12 q^{7} + 84 q^{13} - 24 q^{17} - 228 q^{19} + 30 q^{23} - 150 q^{25} + 168 q^{29} + 324 q^{31} + 120 q^{35} - 984 q^{37} + 312 q^{41} + 156 q^{43} + 462 q^{47} + 588 q^{49} - 2028 q^{53} + 1008 q^{59} - 36 q^{61} + 420 q^{65} - 144 q^{67} - 2424 q^{71} - 1800 q^{73} + 672 q^{77} + 936 q^{79} + 288 q^{83} + 60 q^{85} + 240 q^{89} + 4572 q^{91} + 570 q^{95} + 1188 q^{97}+O(q^{100}) \) 12 * q - 30 * q^5 - 12 * q^7 + 84 * q^13 - 24 * q^17 - 228 * q^19 + 30 * q^23 - 150 * q^25 + 168 * q^29 + 324 * q^31 + 120 * q^35 - 984 * q^37 + 312 * q^41 + 156 * q^43 + 462 * q^47 + 588 * q^49 - 2028 * q^53 + 1008 * q^59 - 36 * q^61 + 420 * q^65 - 144 * q^67 - 2424 * q^71 - 1800 * q^73 + 672 * q^77 + 936 * q^79 + 288 * q^83 + 60 * q^85 + 240 * q^89 + 4572 * q^91 + 570 * q^95 + 1188 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more