LMFDB - Embedded newform 1620.4.i.f.1081.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1620,4,Mod(541,1620)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1620.541"); S:= CuspForms(chi, 4); N := Newforms(S);

sage: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")

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:traces = [2,0,0,0,-5,0,28] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$95.5830942093$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 60)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1081.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1620.1081
Dual form			1620.4.i.f.541.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.50000 + 4.33013i) q^{5} +(14.0000 + 24.2487i) q^{7} +(-12.0000 - 20.7846i) q^{11} +(35.0000 - 60.6218i) q^{13} -102.000 q^{17} +20.0000 q^{19} +(-36.0000 + 62.3538i) q^{23} +(-12.5000 - 21.6506i) q^{25} +(153.000 + 265.004i) q^{29} +(68.0000 - 117.779i) q^{31} -140.000 q^{35} -214.000 q^{37} +(-75.0000 + 129.904i) q^{41} +(146.000 + 252.879i) q^{43} +(-36.0000 - 62.3538i) q^{47} +(-220.500 + 381.917i) q^{49} +414.000 q^{53} +120.000 q^{55} +(-372.000 + 644.323i) q^{59} +(209.000 + 361.999i) q^{61} +(175.000 + 303.109i) q^{65} +(-94.0000 + 162.813i) q^{67} -480.000 q^{71} +434.000 q^{73} +(336.000 - 581.969i) q^{77} +(-676.000 - 1170.87i) q^{79} +(-306.000 - 530.008i) q^{83} +(255.000 - 441.673i) q^{85} +30.0000 q^{89} +1960.00 q^{91} +(-50.0000 + 86.6025i) q^{95} +(143.000 + 247.683i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 5 q^{5} + 28 q^{7} - 24 q^{11} + 70 q^{13} - 204 q^{17} + 40 q^{19} - 72 q^{23} - 25 q^{25} + 306 q^{29} + 136 q^{31} - 280 q^{35} - 428 q^{37} - 150 q^{41} + 292 q^{43} - 72 q^{47} - 441 q^{49} + 828 q^{53}+ \cdots + 286 q^{97}+O(q^{100}) $ 2 * q - 5 * q^5 + 28 * q^7 - 24 * q^11 + 70 * q^13 - 204 * q^17 + 40 * q^19 - 72 * q^23 - 25 * q^25 + 306 * q^29 + 136 * q^31 - 280 * q^35 - 428 * q^37 - 150 * q^41 + 292 * q^43 - 72 * q^47 - 441 * q^49 + 828 * q^53 + 240 * q^55 - 744 * q^59 + 418 * q^61 + 350 * q^65 - 188 * q^67 - 960 * q^71 + 868 * q^73 + 672 * q^77 - 1352 * q^79 - 612 * q^83 + 510 * q^85 + 60 * q^89 + 3920 * q^91 - 100 * q^95 + 286 * q^97

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
$7$	14.0000	+	24.2487i	0.755929	+	1.30931i	0.944911	+	0.327327i	$0.106148\pi$
$7$	14.0000	+	24.2487i	0.755929	+	1.30931i	−0.188982	+	0.981981i	$0.560519\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−12.0000	−	20.7846i	−0.328921	−	0.569709i	0.653377	−	0.757033i	$-0.273352\pi$
$11$	−12.0000	−	20.7846i	−0.328921	−	0.569709i	−0.982298	+	0.187324i	$0.940018\pi$
$12$		0			0
$13$	35.0000	−	60.6218i	0.746712	−	1.29334i	−0.202679	−	0.979245i	$-0.564965\pi$
$13$	35.0000	−	60.6218i	0.746712	−	1.29334i	0.949391	−	0.314098i	$-0.101702\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−102.000			−1.45521			−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−102.000			−1.45521			−0.727607	+	0.685994i	$0.759367\pi$
$18$		0			0
$19$	20.0000			0.241490			0.120745	−	0.992684i	$-0.461472\pi$
$19$	20.0000			0.241490			0.120745	+	0.992684i	$0.461472\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−36.0000	+	62.3538i	−0.326370	+	0.565290i	−0.981789	−	0.189976i	$-0.939159\pi$
$23$	−36.0000	+	62.3538i	−0.326370	+	0.565290i	0.655418	+	0.755266i	$0.272492\pi$
$24$		0			0
$25$	−12.5000	−	21.6506i	−0.100000	−	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	153.000	+	265.004i	0.979703	+	1.69690i	0.663450	+	0.748220i	$0.269091\pi$
$29$	153.000	+	265.004i	0.979703	+	1.69690i	0.316253	+	0.948675i	$0.397575\pi$
$30$		0			0
$31$	68.0000	−	117.779i	0.393973	−	0.682381i	−0.598997	−	0.800752i	$-0.704434\pi$
$31$	68.0000	−	117.779i	0.393973	−	0.682381i	0.992970	+	0.118370i	$0.0377670\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−140.000			−0.676123
$36$		0			0
$37$	−214.000			−0.950848			−0.475424	−	0.879757i	$-0.657705\pi$
$37$	−214.000			−0.950848			−0.475424	+	0.879757i	$0.657705\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−75.0000	+	129.904i	−0.285684	+	0.494819i	−0.972775	−	0.231753i	$-0.925554\pi$
$41$	−75.0000	+	129.904i	−0.285684	+	0.494819i	0.687091	+	0.726571i	$0.258887\pi$
$42$		0			0
$43$	146.000	+	252.879i	0.517786	+	0.896831i	0.999787	+	0.0206606i	$0.00657693\pi$
$43$	146.000	+	252.879i	0.517786	+	0.896831i	−0.482001	+	0.876171i	$0.660090\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−36.0000	−	62.3538i	−0.111726	−	0.193516i	0.804740	−	0.593627i	$-0.202305\pi$
$47$	−36.0000	−	62.3538i	−0.111726	−	0.193516i	−0.916466	+	0.400112i	$0.868971\pi$
$48$		0			0
$49$	−220.500	+	381.917i	−0.642857	+	1.11346i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	414.000			1.07297			0.536484	−	0.843911i	$-0.319752\pi$
$53$	414.000			1.07297			0.536484	+	0.843911i	$0.319752\pi$
$54$		0			0
$55$	120.000			0.294196
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−372.000	+	644.323i	−0.820852	+	1.42176i	0.0841964	+	0.996449i	$0.473168\pi$
$59$	−372.000	+	644.323i	−0.820852	+	1.42176i	−0.905048	+	0.425308i	$0.860166\pi$
$60$		0			0
$61$	209.000	+	361.999i	0.438684	+	0.759823i	0.997588	−	0.0694095i	$-0.0221115\pi$
$61$	209.000	+	361.999i	0.438684	+	0.759823i	−0.558905	+	0.829232i	$0.688778\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	175.000	+	303.109i	0.333940	+	0.578400i
$66$		0			0
$67$	−94.0000	+	162.813i	−0.171402	+	0.296877i	−0.938910	−	0.344162i	$-0.888163\pi$
$67$	−94.0000	+	162.813i	−0.171402	+	0.296877i	0.767508	+	0.641039i	$0.221496\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−480.000			−0.802331			−0.401166	−	0.916006i	$-0.631395\pi$
$71$	−480.000			−0.802331			−0.401166	+	0.916006i	$0.631395\pi$
$72$		0			0
$73$	434.000			0.695834			0.347917	−	0.937525i	$-0.386889\pi$
$73$	434.000			0.695834			0.347917	+	0.937525i	$0.386889\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	336.000	−	581.969i	0.497283	−	0.861319i
$78$		0			0
$79$	−676.000	−	1170.87i	−0.962733	−	1.66750i	−0.715585	−	0.698526i	$-0.753840\pi$
$79$	−676.000	−	1170.87i	−0.962733	−	1.66750i	−0.247148	−	0.968978i	$-0.579494\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−306.000	−	530.008i	−0.404673	−	0.700914i	0.589610	−	0.807688i	$-0.299281\pi$
$83$	−306.000	−	530.008i	−0.404673	−	0.700914i	−0.994283	+	0.106774i	$0.965948\pi$
$84$		0			0
$85$	255.000	−	441.673i	0.325396	−	0.563602i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	30.0000			0.0357303			0.0178651	−	0.999840i	$-0.494313\pi$
$89$	30.0000			0.0357303			0.0178651	+	0.999840i	$0.494313\pi$
$90$		0			0
$91$	1960.00			2.25784
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−50.0000	+	86.6025i	−0.0539989	+	0.0935288i
$96$		0			0
$97$	143.000	+	247.683i	0.149685	+	0.259262i	0.931111	−	0.364736i	$-0.118841\pi$
$97$	143.000	+	247.683i	0.149685	+	0.259262i	−0.781426	+	0.623998i	$0.785507\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−771.000	−	1335.41i	−0.759578	−	1.31563i	−0.943066	−	0.332606i	$-0.892072\pi$
$101$	−771.000	−	1335.41i	−0.759578	−	1.31563i	0.183488	−	0.983022i	$-0.441261\pi$
$102$		0			0
$103$	−586.000	+	1014.98i	−0.560585	+	0.970962i	0.436860	+	0.899530i	$0.356091\pi$
$103$	−586.000	+	1014.98i	−0.560585	+	0.970962i	−0.997445	+	0.0714329i	$0.977243\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1956.00			−1.76723			−0.883615	−	0.468214i	$-0.844898\pi$
$107$	−1956.00			−1.76723			−0.883615	+	0.468214i	$0.844898\pi$
$108$		0			0
$109$	−1858.00			−1.63270			−0.816349	−	0.577559i	$-0.804005\pi$
$109$	−1858.00			−1.63270			−0.816349	+	0.577559i	$0.804005\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	87.0000	−	150.688i	0.0724272	−	0.125448i	−0.827537	−	0.561411i	$-0.810259\pi$
$113$	87.0000	−	150.688i	0.0724272	−	0.125448i	0.899965	+	0.435963i	$0.143592\pi$
$114$		0			0
$115$	−180.000	−	311.769i	−0.145957	−	0.252805i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−1428.00	−	2473.37i	−1.10004	−	1.90532i
$120$		0			0
$121$	377.500	−	653.849i	0.283621	−	0.491247i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	125.000			0.0894427
$126$		0			0
$127$	−2068.00			−1.44492			−0.722462	−	0.691411i	$-0.756990\pi$
$127$	−2068.00			−1.44492			−0.722462	+	0.691411i	$0.756990\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	156.000	−	270.200i	0.104044	−	0.180210i	−0.809303	−	0.587391i	$-0.800155\pi$
$131$	156.000	−	270.200i	0.104044	−	0.180210i	0.913347	+	0.407181i	$0.133488\pi$
$132$		0			0
$133$	280.000	+	484.974i	0.182549	+	0.316185i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	1323.00	+	2291.50i	0.825048	+	1.42902i	0.901883	+	0.431981i	$0.142185\pi$
$137$	1323.00	+	2291.50i	0.825048	+	1.42902i	−0.0768354	+	0.997044i	$0.524482\pi$
$138$		0			0
$139$	638.000	−	1105.05i	0.389313	−	0.674309i	−0.603045	−	0.797707i	$-0.706046\pi$
$139$	638.000	−	1105.05i	0.389313	−	0.674309i	0.992357	+	0.123398i	$0.0393792\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1680.00			−0.982438
$144$		0			0
$145$	−1530.00			−0.876273
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1599.00	+	2769.55i	−0.879162	+	1.52275i	−0.0269006	+	0.999638i	$0.508564\pi$
$149$	−1599.00	+	2769.55i	−0.879162	+	1.52275i	−0.852262	+	0.523116i	$0.824770\pi$
$150$		0			0
$151$	380.000	+	658.179i	0.204794	+	0.354714i	0.950067	−	0.312045i	$-0.101014\pi$
$151$	380.000	+	658.179i	0.204794	+	0.354714i	−0.745273	+	0.666760i	$0.767681\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	340.000	+	588.897i	0.176190	+	0.305170i
$156$		0			0
$157$	83.0000	−	143.760i	0.0421919	−	0.0730784i	−0.844158	−	0.536094i	$-0.819899\pi$
$157$	83.0000	−	143.760i	0.0421919	−	0.0730784i	0.886350	+	0.463016i	$0.153233\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−2016.00			−0.986851
$162$		0			0
$163$	3020.00			1.45119			0.725597	−	0.688120i	$-0.241564\pi$
$163$	3020.00			1.45119			0.725597	+	0.688120i	$0.241564\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−492.000	+	852.169i	−0.227977	+	0.394867i	−0.957208	−	0.289400i	$-0.906544\pi$
$167$	−492.000	+	852.169i	−0.227977	+	0.394867i	0.729232	+	0.684267i	$0.239878\pi$
$168$		0			0
$169$	−1351.50	−	2340.87i	−0.615157	−	1.06548i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	981.000	+	1699.14i	0.431122	+	0.746725i	0.996970	−	0.0777846i	$-0.0247846\pi$
$173$	981.000	+	1699.14i	0.431122	+	0.746725i	−0.565849	+	0.824509i	$0.691451\pi$
$174$		0			0
$175$	350.000	−	606.218i	0.151186	−	0.261861i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−576.000			−0.240515			−0.120258	−	0.992743i	$-0.538372\pi$
$179$	−576.000			−0.240515			−0.120258	+	0.992743i	$0.538372\pi$
$180$		0			0
$181$	−1210.00			−0.496898			−0.248449	−	0.968645i	$-0.579921\pi$
$181$	−1210.00			−0.496898			−0.248449	+	0.968645i	$0.579921\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	535.000	−	926.647i	0.212616	−	0.368262i
$186$		0			0
$187$	1224.00	+	2120.03i	0.478651	+	0.829048i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	1692.00	+	2930.63i	0.640989	+	1.11022i	0.985212	+	0.171337i	$0.0548088\pi$
$191$	1692.00	+	2930.63i	0.640989	+	1.11022i	−0.344224	+	0.938888i	$0.611858\pi$
$192$		0			0
$193$	1019.00	−	1764.96i	0.380048	−	0.658262i	−0.611021	−	0.791614i	$-0.709241\pi$
$193$	1019.00	−	1764.96i	0.380048	−	0.658262i	0.991069	+	0.133352i	$0.0425742\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−4098.00			−1.48208			−0.741042	−	0.671459i	$-0.765668\pi$
$197$	−4098.00			−1.48208			−0.741042	+	0.671459i	$0.765668\pi$
$198$		0			0
$199$	−2248.00			−0.800786			−0.400393	−	0.916343i	$-0.631126\pi$
$199$	−2248.00			−0.800786			−0.400393	+	0.916343i	$0.631126\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−4284.00	+	7420.11i	−1.48117	+	2.56546i
$204$		0			0
$205$	−375.000	−	649.519i	−0.127762	−	0.221290i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−240.000	−	415.692i	−0.0794313	−	0.137579i
$210$		0			0
$211$	−1630.00	+	2823.24i	−0.531819	+	0.921138i	0.467491	+	0.883998i	$0.345158\pi$
$211$	−1630.00	+	2823.24i	−0.531819	+	0.921138i	−0.999310	+	0.0371398i	$0.988175\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−1460.00			−0.463122
$216$		0			0
$217$	3808.00			1.19126
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−3570.00	+	6183.42i	−1.08663	+	1.88209i
$222$		0			0
$223$	1490.00	+	2580.76i	0.447434	+	0.774978i	0.998218	−	0.0596693i	$-0.0190046\pi$
$223$	1490.00	+	2580.76i	0.447434	+	0.774978i	−0.550784	+	0.834648i	$0.685671\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−1590.00	−	2753.96i	−0.464899	−	0.805228i	0.534298	−	0.845296i	$-0.320576\pi$
$227$	−1590.00	−	2753.96i	−0.464899	−	0.805228i	−0.999197	+	0.0400678i	$0.987243\pi$
$228$		0			0
$229$	−1687.00	+	2921.97i	−0.486813	+	0.843184i	−0.999885	−	0.0151611i	$-0.995174\pi$
$229$	−1687.00	+	2921.97i	−0.486813	+	0.843184i	0.513072	+	0.858345i	$0.328507\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−1950.00			−0.548278			−0.274139	−	0.961690i	$-0.588393\pi$
$233$	−1950.00			−0.548278			−0.274139	+	0.961690i	$0.588393\pi$
$234$		0			0
$235$	360.000			0.0999311
$236$		0			0
$237$		0			0
$238$		0			0
$239$	1116.00	−	1932.97i	0.302042	−	0.523152i	−0.674556	−	0.738223i	$-0.735665\pi$
$239$	1116.00	−	1932.97i	0.302042	−	0.523152i	0.976598	+	0.215071i	$0.0689984\pi$
$240$		0			0
$241$	911.000	+	1577.90i	0.243497	+	0.421748i	0.961708	−	0.274077i	$-0.0883722\pi$
$241$	911.000	+	1577.90i	0.243497	+	0.421748i	−0.718211	+	0.695825i	$0.755039\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1102.50	−	1909.59i	−0.287494	−	0.497955i
$246$		0			0
$247$	700.000	−	1212.44i	0.180324	−	0.312330i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−1488.00			−0.374190			−0.187095	−	0.982342i	$-0.559907\pi$
$251$	−1488.00			−0.374190			−0.187095	+	0.982342i	$0.559907\pi$
$252$		0			0
$253$	1728.00			0.429401
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−1497.00	+	2592.88i	−0.363347	+	0.629336i	−0.988510	−	0.151159i	$-0.951700\pi$
$257$	−1497.00	+	2592.88i	−0.363347	+	0.629336i	0.625162	+	0.780495i	$0.285033\pi$
$258$		0			0
$259$	−2996.00	−	5189.22i	−0.718774	−	1.24495i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−1236.00	−	2140.81i	−0.289791	−	0.501933i	0.683969	−	0.729511i	$-0.260252\pi$
$263$	−1236.00	−	2140.81i	−0.289791	−	0.501933i	−0.973760	+	0.227579i	$0.926919\pi$
$264$		0			0
$265$	−1035.00	+	1792.67i	−0.239923	+	0.415558i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−3954.00			−0.896207			−0.448103	−	0.893982i	$-0.647900\pi$
$269$	−3954.00			−0.896207			−0.448103	+	0.893982i	$0.647900\pi$
$270$		0			0
$271$	−2176.00			−0.487759			−0.243879	−	0.969806i	$-0.578420\pi$
$271$	−2176.00			−0.487759			−0.243879	+	0.969806i	$0.578420\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−300.000	+	519.615i	−0.0657843	+	0.113942i
$276$		0			0
$277$	−517.000	−	895.470i	−0.112143	−	0.194237i	0.804491	−	0.593964i	$-0.202438\pi$
$277$	−517.000	−	895.470i	−0.112143	−	0.194237i	−0.916634	+	0.399728i	$0.869105\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−3327.00	−	5762.53i	−0.706307	−	1.22336i	−0.966218	−	0.257727i	$-0.917027\pi$
$281$	−3327.00	−	5762.53i	−0.706307	−	1.22336i	0.259911	−	0.965633i	$-0.416307\pi$
$282$		0			0
$283$	878.000	−	1520.74i	0.184423	−	0.319430i	−0.758959	−	0.651138i	$-0.774292\pi$
$283$	878.000	−	1520.74i	0.184423	−	0.319430i	0.943382	+	0.331709i	$0.107625\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−4200.00			−0.863826
$288$		0			0
$289$	5491.00			1.11765
$290$		0			0
$291$		0			0
$292$		0			0
$293$	1617.00	−	2800.73i	0.322410	−	0.558431i	−0.658575	−	0.752515i	$-0.728840\pi$
$293$	1617.00	−	2800.73i	0.322410	−	0.558431i	0.980985	+	0.194085i	$0.0621737\pi$
$294$		0			0
$295$	−1860.00	−	3221.61i	−0.367096	−	0.635829i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	2520.00	+	4364.77i	0.487409	+	0.844218i
$300$		0			0
$301$	−4088.00	+	7080.62i	−0.782819	+	1.35588i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−2090.00			−0.392371
$306$		0			0
$307$	2036.00			0.378504			0.189252	−	0.981929i	$-0.439394\pi$
$307$	2036.00			0.378504			0.189252	+	0.981929i	$0.439394\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−48.0000	+	83.1384i	−0.00875187	+	0.0151587i	−0.870368	−	0.492402i	$-0.836119\pi$
$311$	−48.0000	+	83.1384i	−0.00875187	+	0.0151587i	0.861616	+	0.507560i	$0.169453\pi$
$312$		0			0
$313$	−601.000	−	1040.96i	−0.108532	−	0.187983i	0.806644	−	0.591038i	$-0.201282\pi$
$313$	−601.000	−	1040.96i	−0.108532	−	0.187983i	−0.915176	+	0.403055i	$0.867948\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−1899.00	−	3289.16i	−0.336462	−	0.582769i	0.647303	−	0.762233i	$-0.275897\pi$
$317$	−1899.00	−	3289.16i	−0.336462	−	0.582769i	−0.983765	+	0.179464i	$0.942564\pi$
$318$		0			0
$319$	3672.00	−	6360.09i	0.644491	−	1.11629i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−2040.00			−0.351420
$324$		0			0
$325$	−1750.00			−0.298685
$326$		0			0
$327$		0			0
$328$		0			0
$329$	1008.00	−	1745.91i	0.168914	−	0.292568i
$330$		0			0
$331$	2834.00	+	4908.63i	0.470606	+	0.815114i	0.999435	−	0.0336145i	$-0.0107018\pi$
$331$	2834.00	+	4908.63i	0.470606	+	0.815114i	−0.528828	+	0.848729i	$0.677369\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−470.000	−	814.064i	−0.0766533	−	0.132767i
$336$		0			0
$337$	227.000	−	393.176i	0.0366928	−	0.0635538i	−0.847096	−	0.531440i	$-0.821651\pi$
$337$	227.000	−	393.176i	0.0366928	−	0.0635538i	0.883789	+	0.467886i	$0.154984\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−3264.00			−0.518345
$342$		0			0
$343$	−2744.00			−0.431959
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−2802.00	+	4853.21i	−0.433485	+	0.750818i	−0.997171	−	0.0751715i	$-0.976050\pi$
$347$	−2802.00	+	4853.21i	−0.433485	+	0.750818i	0.563686	+	0.825989i	$0.309383\pi$
$348$		0			0
$349$	5633.00	+	9756.64i	0.863976	+	1.49645i	0.868060	+	0.496458i	$0.165366\pi$
$349$	5633.00	+	9756.64i	0.863976	+	1.49645i	−0.00408463	+	0.999992i	$0.501300\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−3213.00	−	5565.08i	−0.484450	−	0.839091i	0.515391	−	0.856955i	$-0.327647\pi$
$353$	−3213.00	−	5565.08i	−0.484450	−	0.839091i	−0.999840	+	0.0178638i	$0.994313\pi$
$354$		0			0
$355$	1200.00	−	2078.46i	0.179407	−	0.310742i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−6936.00			−1.01969			−0.509844	−	0.860267i	$-0.670297\pi$
$359$	−6936.00			−1.01969			−0.509844	+	0.860267i	$0.670297\pi$
$360$		0			0
$361$	−6459.00			−0.941682
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−1085.00	+	1879.28i	−0.155593	+	0.269495i
$366$		0			0
$367$	194.000	+	336.018i	0.0275932	+	0.0477929i	0.879492	−	0.475913i	$-0.157882\pi$
$367$	194.000	+	336.018i	0.0275932	+	0.0477929i	−0.851899	+	0.523706i	$0.824549\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	5796.00	+	10039.0i	0.811087	+	1.40484i
$372$		0			0
$373$	4031.00	−	6981.90i	0.559564	−	0.969193i	−0.437969	−	0.898990i	$-0.644302\pi$
$373$	4031.00	−	6981.90i	0.559564	−	0.969193i	0.997533	−	0.0702027i	$-0.0223646\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	21420.0			2.92622
$378$		0			0
$379$	−3388.00			−0.459182			−0.229591	−	0.973287i	$-0.573739\pi$
$379$	−3388.00			−0.459182			−0.229591	+	0.973287i	$0.573739\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	3492.00	−	6048.32i	0.465882	−	0.806932i	−0.533359	−	0.845889i	$-0.679070\pi$
$383$	3492.00	−	6048.32i	0.465882	−	0.806932i	0.999241	+	0.0389576i	$0.0124037\pi$
$384$		0			0
$385$	1680.00	+	2909.85i	0.222392	+	0.385193i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−1263.00	−	2187.58i	−0.164619	−	0.285128i	0.771901	−	0.635743i	$-0.219306\pi$
$389$	−1263.00	−	2187.58i	−0.164619	−	0.285128i	−0.936520	+	0.350615i	$0.885973\pi$
$390$		0			0
$391$	3672.00	−	6360.09i	0.474939	−	0.822618i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	6760.00			0.861095
$396$		0			0
$397$	6146.00			0.776975			0.388487	−	0.921454i	$-0.372998\pi$
$397$	6146.00			0.776975			0.388487	+	0.921454i	$0.372998\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	4893.00	−	8474.92i	0.609339	−	1.05541i	−0.382011	−	0.924158i	$-0.624768\pi$
$401$	4893.00	−	8474.92i	0.609339	−	1.05541i	0.991350	−	0.131248i	$-0.0418983\pi$
$402$		0			0
$403$	−4760.00	−	8244.56i	−0.588368	−	1.01908i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	2568.00	+	4447.91i	0.312754	+	0.541706i
$408$		0			0
$409$	443.000	−	767.299i	0.0535573	−	0.0927640i	−0.838004	−	0.545664i	$-0.816277\pi$
$409$	443.000	−	767.299i	0.0535573	−	0.0927640i	0.891561	+	0.452900i	$0.149611\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−20832.0			−2.48202
$414$		0			0
$415$	3060.00			0.361951
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−5676.00	+	9831.12i	−0.661792	+	1.14626i	0.318353	+	0.947972i	$0.396870\pi$
$419$	−5676.00	+	9831.12i	−0.661792	+	1.14626i	−0.980144	+	0.198285i	$0.936463\pi$
$420$		0			0
$421$	−5095.00	−	8824.80i	−0.589822	−	1.02160i	−0.994255	−	0.107034i	$-0.965865\pi$
$421$	−5095.00	−	8824.80i	−0.589822	−	1.02160i	0.404433	−	0.914568i	$-0.367469\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	1275.00	+	2208.36i	0.145521	+	0.252050i
$426$		0			0
$427$	−5852.00	+	10136.0i	−0.663227	+	1.14874i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−2448.00			−0.273587			−0.136794	−	0.990600i	$-0.543680\pi$
$431$	−2448.00			−0.273587			−0.136794	+	0.990600i	$0.543680\pi$
$432$		0			0
$433$	−7078.00			−0.785559			−0.392779	−	0.919633i	$-0.628486\pi$
$433$	−7078.00			−0.785559			−0.392779	+	0.919633i	$0.628486\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−720.000	+	1247.08i	−0.0788153	+	0.136512i
$438$		0			0
$439$	9044.00	+	15664.7i	0.983250	+	1.70304i	0.649470	+	0.760387i	$0.274991\pi$
$439$	9044.00	+	15664.7i	0.983250	+	1.70304i	0.333779	+	0.942651i	$0.391676\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	1926.00	+	3335.93i	0.206562	+	0.357776i	0.950629	−	0.310329i	$-0.100439\pi$
$443$	1926.00	+	3335.93i	0.206562	+	0.357776i	−0.744067	+	0.668105i	$0.767106\pi$
$444$		0			0
$445$	−75.0000	+	129.904i	−0.00798953	+	0.0138383i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−6522.00			−0.685506			−0.342753	−	0.939426i	$-0.611359\pi$
$449$	−6522.00			−0.685506			−0.342753	+	0.939426i	$0.611359\pi$
$450$		0			0
$451$	3600.00			0.375870
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−4900.00	+	8487.05i	−0.504869	+	0.874459i
$456$		0			0
$457$	−1045.00	−	1809.99i	−0.106965	−	0.185269i	0.807574	−	0.589766i	$-0.200780\pi$
$457$	−1045.00	−	1809.99i	−0.106965	−	0.185269i	−0.914539	+	0.404497i	$0.867447\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−4947.00	−	8568.46i	−0.499793	−	0.865668i	0.500207	−	0.865906i	$-0.333257\pi$
$461$	−4947.00	−	8568.46i	−0.499793	−	0.865668i	−1.00000		0.000238537i	$0.999924\pi$
$462$		0			0
$463$	−1522.00	+	2636.18i	−0.152772	+	0.264609i	−0.932245	−	0.361826i	$-0.882153\pi$
$463$	−1522.00	+	2636.18i	−0.152772	+	0.264609i	0.779474	+	0.626435i	$0.215487\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−10236.0			−1.01427			−0.507137	−	0.861866i	$-0.669296\pi$
$467$	−10236.0			−1.01427			−0.507137	+	0.861866i	$0.669296\pi$
$468$		0			0
$469$	−5264.00			−0.518271
$470$		0			0
$471$		0			0
$472$		0			0
$473$	3504.00	−	6069.11i	0.340622	−	0.589974i
$474$		0			0
$475$	−250.000	−	433.013i	−0.0241490	−	0.0418273i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	5748.00	+	9955.83i	0.548294	+	0.949673i	0.998392	+	0.0566937i	$0.0180558\pi$
$479$	5748.00	+	9955.83i	0.548294	+	0.949673i	−0.450098	+	0.892979i	$0.648611\pi$
$480$		0			0
$481$	−7490.00	+	12973.1i	−0.710010	+	1.22977i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−1430.00			−0.133882
$486$		0			0
$487$	−15316.0			−1.42512			−0.712561	−	0.701610i	$-0.752465\pi$
$487$	−15316.0			−1.42512			−0.712561	+	0.701610i	$0.752465\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−5808.00	+	10059.8i	−0.533832	+	0.924624i	0.465387	+	0.885107i	$0.345915\pi$
$491$	−5808.00	+	10059.8i	−0.533832	+	0.924624i	−0.999219	+	0.0395165i	$0.987418\pi$
$492$		0			0
$493$	−15606.0	−	27030.4i	−1.42568	−	2.46935i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−6720.00	−	11639.4i	−0.606505	−	1.05050i
$498$		0			0
$499$	−7498.00	+	12986.9i	−0.672658	+	1.16508i	0.304489	+	0.952516i	$0.401514\pi$
$499$	−7498.00	+	12986.9i	−0.672658	+	1.16508i	−0.977147	+	0.212563i	$0.931819\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	21648.0			1.91896			0.959480	−	0.281778i	$-0.0909240\pi$
$503$	21648.0			1.91896			0.959480	+	0.281778i	$0.0909240\pi$
$504$		0			0
$505$	7710.00			0.679387
$506$		0			0
$507$		0			0
$508$		0			0
$509$	1689.00	−	2925.43i	0.147080	−	0.254750i	−0.783067	−	0.621937i	$-0.786346\pi$
$509$	1689.00	−	2925.43i	0.147080	−	0.254750i	0.930147	+	0.367187i	$0.119679\pi$
$510$		0			0
$511$	6076.00	+	10523.9i	0.526001	+	0.911060i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−2930.00	−	5074.91i	−0.250701	−	0.434228i
$516$		0			0
$517$	−864.000	+	1496.49i	−0.0734984	+	0.127303i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	16158.0			1.35872			0.679362	−	0.733804i	$-0.262257\pi$
$521$	16158.0			1.35872			0.679362	+	0.733804i	$0.262257\pi$
$522$		0			0
$523$	−76.0000			−0.00635420			−0.00317710	−	0.999995i	$-0.501011\pi$
$523$	−76.0000			−0.00635420			−0.00317710	+	0.999995i	$0.501011\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−6936.00	+	12013.5i	−0.573315	+	0.993010i
$528$		0			0
$529$	3491.50	+	6047.46i	0.286965	+	0.497038i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	5250.00	+	9093.27i	0.426647	+	0.738974i
$534$		0			0
$535$	4890.00	−	8469.73i	0.395165	−	0.684445i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	10584.0			0.845798
$540$		0			0
$541$	9278.00			0.737324			0.368662	−	0.929563i	$-0.379816\pi$
$541$	9278.00			0.737324			0.368662	+	0.929563i	$0.379816\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	4645.00	−	8045.38i	0.365082	−	0.632341i
$546$		0			0
$547$	−7282.00	−	12612.8i	−0.569206	−	0.985894i	−0.996645	−	0.0818497i	$-0.973917\pi$
$547$	−7282.00	−	12612.8i	−0.569206	−	0.985894i	0.427438	−	0.904044i	$-0.359416\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	3060.00	+	5300.08i	0.236589	+	0.409784i
$552$		0			0
$553$	18928.0	−	32784.3i	1.45552	−	2.52103i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−2154.00			−0.163856			−0.0819281	−	0.996638i	$-0.526108\pi$
$557$	−2154.00			−0.163856			−0.0819281	+	0.996638i	$0.526108\pi$
$558$		0			0
$559$	20440.0			1.54655
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−4350.00	+	7534.42i	−0.325632	+	0.564011i	−0.981640	−	0.190743i	$-0.938910\pi$
$563$	−4350.00	+	7534.42i	−0.325632	+	0.564011i	0.656008	+	0.754754i	$0.272244\pi$
$564$		0			0
$565$	435.000	+	753.442i	0.0323904	+	0.0561019i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	2097.00	+	3632.11i	0.154501	+	0.267603i	0.932877	−	0.360195i	$-0.117290\pi$
$569$	2097.00	+	3632.11i	0.154501	+	0.267603i	−0.778377	+	0.627798i	$0.783956\pi$
$570$		0			0
$571$	4010.00	−	6945.52i	0.293894	−	0.509039i	−0.680833	−	0.732438i	$-0.738382\pi$
$571$	4010.00	−	6945.52i	0.293894	−	0.509039i	0.974727	+	0.223400i	$0.0717155\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	1800.00			0.130548
$576$		0			0
$577$	−2686.00			−0.193795			−0.0968974	−	0.995294i	$-0.530892\pi$
$577$	−2686.00			−0.193795			−0.0968974	+	0.995294i	$0.530892\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	8568.00	−	14840.2i	0.611808	−	1.05968i
$582$		0			0
$583$	−4968.00	−	8604.83i	−0.352922	−	0.611279i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	1506.00	+	2608.47i	0.105893	+	0.183412i	0.914103	−	0.405483i	$-0.132897\pi$
$587$	1506.00	+	2608.47i	0.105893	+	0.183412i	−0.808210	+	0.588895i	$0.799563\pi$
$588$		0			0
$589$	1360.00	−	2355.59i	0.0951406	−	0.164788i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	15522.0			1.07489			0.537447	−	0.843298i	$-0.319389\pi$
$593$	15522.0			1.07489			0.537447	+	0.843298i	$0.319389\pi$
$594$		0			0
$595$	14280.0			0.983904
$596$		0			0
$597$		0			0
$598$		0			0
$599$	9612.00	−	16648.5i	0.655652	−	1.13562i	−0.326078	−	0.945343i	$-0.605727\pi$
$599$	9612.00	−	16648.5i	0.655652	−	1.13562i	0.981730	−	0.190280i	$-0.0609396\pi$
$600$		0			0
$601$	3251.00	+	5630.90i	0.220651	+	0.382178i	0.955006	−	0.296587i	$-0.0958486\pi$
$601$	3251.00	+	5630.90i	0.220651	+	0.382178i	−0.734355	+	0.678766i	$0.762515\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	1887.50	+	3269.25i	0.126839	+	0.219692i
$606$		0			0
$607$	−14698.0	+	25457.7i	−0.982823	+	1.70230i	−0.331585	+	0.943425i	$0.607583\pi$
$607$	−14698.0	+	25457.7i	−0.982823	+	1.70230i	−0.651238	+	0.758874i	$0.725750\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−5040.00			−0.333710
$612$		0			0
$613$	−10006.0			−0.659280			−0.329640	−	0.944107i	$-0.606927\pi$
$613$	−10006.0			−0.659280			−0.329640	+	0.944107i	$0.606927\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	11559.0	−	20020.8i	0.754210	−	1.30633i	−0.191556	−	0.981482i	$-0.561353\pi$
$617$	11559.0	−	20020.8i	0.754210	−	1.30633i	0.945766	−	0.324849i	$-0.105313\pi$
$618$		0			0
$619$	−7018.00	−	12155.5i	−0.455698	−	0.789293i	0.543030	−	0.839713i	$-0.317277\pi$
$619$	−7018.00	−	12155.5i	−0.455698	−	0.789293i	−0.998728	+	0.0504209i	$0.983944\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	420.000	+	727.461i	0.0270095	+	0.0467819i
$624$		0			0
$625$	−312.500	+	541.266i	−0.0200000	+	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	21828.0			1.38369
$630$		0			0
$631$	−4288.00			−0.270527			−0.135264	−	0.990810i	$-0.543188\pi$
$631$	−4288.00			−0.270527			−0.135264	+	0.990810i	$0.543188\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	5170.00	−	8954.70i	0.323095	−	0.559617i
$636$		0			0
$637$	15435.0	+	26734.2i	0.960058	+	1.66287i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	657.000	+	1137.96i	0.0404835	+	0.0701195i	0.885557	−	0.464530i	$-0.153777\pi$
$641$	657.000	+	1137.96i	0.0404835	+	0.0701195i	−0.845074	+	0.534650i	$0.820444\pi$
$642$		0			0
$643$	314.000	−	543.864i	0.0192581	−	0.0333560i	−0.856236	−	0.516585i	$-0.827203\pi$
$643$	314.000	−	543.864i	0.0192581	−	0.0333560i	0.875494	+	0.483229i	$0.160536\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	10944.0			0.664997			0.332498	−	0.943104i	$-0.392108\pi$
$647$	10944.0			0.664997			0.332498	+	0.943104i	$0.392108\pi$
$648$		0			0
$649$	17856.0			1.07998
$650$		0			0
$651$		0			0
$652$		0			0
$653$	549.000	−	950.896i	0.0329005	−	0.0569853i	−0.849106	−	0.528222i	$-0.822859\pi$
$653$	549.000	−	950.896i	0.0329005	−	0.0569853i	0.882007	+	0.471237i	$0.156192\pi$
$654$		0			0
$655$	780.000	+	1351.00i	0.0465300	+	0.0805922i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	156.000	+	270.200i	0.00922139	+	0.0159719i	0.870599	−	0.491993i	$-0.163731\pi$
$659$	156.000	+	270.200i	0.00922139	+	0.0159719i	−0.861378	+	0.507965i	$0.830398\pi$
$660$		0			0
$661$	−4339.00	+	7515.37i	−0.255322	+	0.442230i	−0.964983	−	0.262313i	$-0.915515\pi$
$661$	−4339.00	+	7515.37i	−0.255322	+	0.442230i	0.709661	+	0.704543i	$0.248848\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−2800.00			−0.163277
$666$		0			0
$667$	−22032.0			−1.27898
$668$		0			0
$669$		0			0
$670$		0			0
$671$	5016.00	−	8687.97i	0.288585	−	0.499844i
$672$		0			0
$673$	7235.00	+	12531.4i	0.414396	+	0.717756i	0.995365	−	0.0961705i	$-0.0306594\pi$
$673$	7235.00	+	12531.4i	0.414396	+	0.717756i	−0.580969	+	0.813926i	$0.697326\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−5919.00	−	10252.0i	−0.336020	−	0.582004i	0.647660	−	0.761929i	$-0.275748\pi$
$677$	−5919.00	−	10252.0i	−0.336020	−	0.582004i	−0.983680	+	0.179925i	$0.942414\pi$
$678$		0			0
$679$	−4004.00	+	6935.13i	−0.226303	+	0.391967i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	25548.0			1.43128			0.715642	−	0.698467i	$-0.246134\pi$
$683$	25548.0			1.43128			0.715642	+	0.698467i	$0.246134\pi$
$684$		0			0
$685$	−13230.0			−0.737945
$686$		0			0
$687$		0			0
$688$		0			0
$689$	14490.0	−	25097.4i	0.801197	−	1.38771i
$690$		0			0
$691$	9206.00	+	15945.3i	0.506820	+	0.877838i	0.999969	+	0.00789325i	$0.00251253\pi$
$691$	9206.00	+	15945.3i	0.506820	+	0.877838i	−0.493149	+	0.869945i	$0.664154\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	3190.00	+	5525.24i	0.174106	+	0.301560i
$696$		0			0
$697$	7650.00	−	13250.2i	0.415731	−	0.720067i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	8814.00			0.474893			0.237447	−	0.971401i	$-0.423690\pi$
$701$	8814.00			0.474893			0.237447	+	0.971401i	$0.423690\pi$
$702$		0			0
$703$	−4280.00			−0.229621
$704$		0			0
$705$		0			0
$706$		0			0
$707$	21588.0	−	37391.5i	1.14837	−	1.98904i
$708$		0			0
$709$	8657.00	+	14994.4i	0.458562	+	0.794253i	0.998885	−	0.0472049i	$-0.0150314\pi$
$709$	8657.00	+	14994.4i	0.458562	+	0.794253i	−0.540323	+	0.841458i	$0.681698\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	4896.00	+	8480.12i	0.257162	+	0.445418i
$714$		0			0
$715$	4200.00	−	7274.61i	0.219680	−	0.380497i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	768.000			0.0398353			0.0199176	−	0.999802i	$-0.493660\pi$
$719$	768.000			0.0398353			0.0199176	+	0.999802i	$0.493660\pi$
$720$		0			0
$721$	−32816.0			−1.69505
$722$		0			0
$723$		0			0
$724$		0			0
$725$	3825.00	−	6625.09i	0.195941	−	0.339379i
$726$		0			0
$727$	9098.00	+	15758.2i	0.464135	+	0.803905i	0.999162	−	0.0409295i	$-0.0130319\pi$
$727$	9098.00	+	15758.2i	0.464135	+	0.803905i	−0.535027	+	0.844835i	$0.679699\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−14892.0	−	25793.7i	−0.753489	−	1.30508i
$732$		0			0
$733$	9071.00	−	15711.4i	0.457087	−	0.791699i	−0.541718	−	0.840560i	$-0.682226\pi$
$733$	9071.00	−	15711.4i	0.457087	−	0.791699i	0.998806	+	0.0488616i	$0.0155593\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	4512.00			0.225511
$738$		0			0
$739$	−13660.0			−0.679961			−0.339981	−	0.940432i	$-0.610420\pi$
$739$	−13660.0			−0.679961			−0.339981	+	0.940432i	$0.610420\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−6384.00	+	11057.4i	−0.315217	+	0.545972i	−0.979484	−	0.201524i	$-0.935411\pi$
$743$	−6384.00	+	11057.4i	−0.315217	+	0.545972i	0.664267	+	0.747496i	$0.268744\pi$
$744$		0			0
$745$	−7995.00	−	13847.7i	−0.393173	−	0.680996i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−27384.0	−	47430.5i	−1.33590	−	2.31385i
$750$		0			0
$751$	−11476.0	+	19877.0i	−0.557610	+	0.965809i	0.440085	+	0.897956i	$0.354948\pi$
$751$	−11476.0	+	19877.0i	−0.557610	+	0.965809i	−0.997695	+	0.0678530i	$0.978385\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−3800.00			−0.183174
$756$		0			0
$757$	15818.0			0.759465			0.379732	−	0.925096i	$-0.376016\pi$
$757$	15818.0			0.759465			0.379732	+	0.925096i	$0.376016\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−9279.00	+	16071.7i	−0.442002	+	0.765570i	−0.997838	−	0.0657221i	$-0.979065\pi$
$761$	−9279.00	+	16071.7i	−0.442002	+	0.765570i	0.555836	+	0.831292i	$0.312398\pi$
$762$		0			0
$763$	−26012.0	−	45054.1i	−1.23420	−	2.13770i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	26040.0	+	45102.6i	1.22588	+	2.12329i
$768$		0			0
$769$	−7489.00	+	12971.3i	−0.351184	+	0.608268i	−0.986457	−	0.164019i	$-0.947554\pi$
$769$	−7489.00	+	12971.3i	−0.351184	+	0.608268i	0.635273	+	0.772287i	$0.280887\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−8946.00			−0.416255			−0.208128	−	0.978102i	$-0.566737\pi$
$773$	−8946.00			−0.416255			−0.208128	+	0.978102i	$0.566737\pi$
$774$		0			0
$775$	−3400.00			−0.157589
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−1500.00	+	2598.08i	−0.0689898	+	0.119494i
$780$		0			0
$781$	5760.00	+	9976.61i	0.263904	+	0.457095i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	415.000	+	718.801i	0.0188688	+	0.0326817i
$786$		0			0
$787$	9218.00	−	15966.0i	0.417517	−	0.723161i	−0.578172	−	0.815915i	$-0.696234\pi$
$787$	9218.00	−	15966.0i	0.417517	−	0.723161i	0.995689	+	0.0927538i	$0.0295669\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	4872.00			0.218999
$792$		0			0
$793$	29260.0			1.31028
$794$		0			0
$795$		0			0
$796$		0			0
$797$	8157.00	−	14128.3i	0.362529	−	0.627919i	−0.625847	−	0.779946i	$-0.715247\pi$
$797$	8157.00	−	14128.3i	0.362529	−	0.627919i	0.988376	+	0.152027i	$0.0485800\pi$
$798$		0			0
$799$	3672.00	+	6360.09i	0.162586	+	0.281607i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−5208.00	−	9020.52i	−0.228875	−	0.396422i
$804$		0			0
$805$	5040.00	−	8729.54i	0.220667	−	0.382206i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	25446.0			1.10585			0.552926	−	0.833231i	$-0.313511\pi$
$809$	25446.0			1.10585			0.552926	+	0.833231i	$0.313511\pi$
$810$		0			0
$811$	42740.0			1.85056			0.925280	−	0.379284i	$-0.123830\pi$
$811$	42740.0			1.85056			0.925280	+	0.379284i	$0.123830\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−7550.00	+	13077.0i	−0.324497	+	0.562045i
$816$		0			0
$817$	2920.00	+	5057.59i	0.125040	+	0.216576i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	14973.0	+	25934.0i	0.636494	+	1.10244i	0.986197	+	0.165579i	$0.0529492\pi$
$821$	14973.0	+	25934.0i	0.636494	+	1.10244i	−0.349703	+	0.936861i	$0.613717\pi$
$822$		0			0
$823$	16298.0	−	28229.0i	0.690295	−	1.19563i	−0.281447	−	0.959577i	$-0.590814\pi$
$823$	16298.0	−	28229.0i	0.690295	−	1.19563i	0.971741	−	0.236049i	$-0.0758525\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−3804.00			−0.159949			−0.0799746	−	0.996797i	$-0.525484\pi$
$827$	−3804.00			−0.159949			−0.0799746	+	0.996797i	$0.525484\pi$
$828$		0			0
$829$	3278.00			0.137334			0.0686669	−	0.997640i	$-0.478125\pi$
$829$	3278.00			0.137334			0.0686669	+	0.997640i	$0.478125\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	22491.0	−	38955.6i	0.935495	−	1.62032i
$834$		0			0
$835$	−2460.00	−	4260.84i	−0.101954	−	0.176590i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−2892.00	−	5009.09i	−0.119002	−	0.206118i	0.800370	−	0.599506i	$-0.204636\pi$
$839$	−2892.00	−	5009.09i	−0.119002	−	0.206118i	−0.919373	+	0.393388i	$0.871303\pi$
$840$		0			0
$841$	−34623.5	+	59969.7i	−1.41964	+	2.45888i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	13515.0			0.550213
$846$		0			0
$847$	21140.0			0.857590
$848$		0			0
$849$		0			0
$850$		0			0
$851$	7704.00	−	13343.7i	0.310329	−	0.537505i
$852$		0			0
$853$	−8653.00	−	14987.4i	−0.347331	−	0.601594i	0.638444	−	0.769669i	$-0.279579\pi$
$853$	−8653.00	−	14987.4i	−0.347331	−	0.601594i	−0.985774	+	0.168074i	$0.946245\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	15567.0	+	26962.8i	0.620488	+	1.07472i	0.989395	+	0.145251i	$0.0463990\pi$
$857$	15567.0	+	26962.8i	0.620488	+	1.07472i	−0.368906	+	0.929467i	$0.620268\pi$
$858$		0			0
$859$	5390.00	−	9335.75i	0.214091	−	0.370817i	−0.738900	−	0.673815i	$-0.764654\pi$
$859$	5390.00	−	9335.75i	0.214091	−	0.370817i	0.952991	+	0.302998i	$0.0979877\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−3456.00			−0.136319			−0.0681597	−	0.997674i	$-0.521713\pi$
$863$	−3456.00			−0.136319			−0.0681597	+	0.997674i	$0.521713\pi$
$864$		0			0
$865$	−9810.00			−0.385607
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−16224.0	+	28100.8i	−0.633327	+	1.09696i
$870$		0			0
$871$	6580.00	+	11396.9i	0.255976	+	0.443363i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1750.00	+	3031.09i	0.0676123	+	0.117108i
$876$		0			0
$877$	−1309.00	+	2267.25i	−0.0504011	+	0.0872973i	−0.890125	−	0.455716i	$-0.849383\pi$
$877$	−1309.00	+	2267.25i	−0.0504011	+	0.0872973i	0.839724	+	0.543013i	$0.182717\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	26550.0			1.01531			0.507657	−	0.861559i	$-0.330512\pi$
$881$	26550.0			1.01531			0.507657	+	0.861559i	$0.330512\pi$
$882$		0			0
$883$	27596.0			1.05173			0.525866	−	0.850567i	$-0.323741\pi$
$883$	27596.0			1.05173			0.525866	+	0.850567i	$0.323741\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−18924.0	+	32777.3i	−0.716354	+	1.24076i	0.246081	+	0.969249i	$0.420857\pi$
$887$	−18924.0	+	32777.3i	−0.716354	+	1.24076i	−0.962435	+	0.271512i	$0.912476\pi$
$888$		0			0
$889$	−28952.0	−	50146.3i	−1.09226	−	1.89185i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−720.000	−	1247.08i	−0.0269808	−	0.0467322i
$894$		0			0
$895$	1440.00	−	2494.15i	0.0537809	−	0.0931512i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	41616.0			1.54391
$900$		0			0
$901$	−42228.0			−1.56140
$902$		0			0
$903$		0			0
$904$		0			0
$905$	3025.00	−	5239.45i	0.111110	−	0.192448i
$906$		0			0
$907$	2402.00	+	4160.39i	0.0879351	+	0.152308i	0.906638	−	0.421909i	$-0.138640\pi$
$907$	2402.00	+	4160.39i	0.0879351	+	0.152308i	−0.818703	+	0.574217i	$0.805307\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	14304.0	+	24775.3i	0.520211	+	0.901033i	0.999724	+	0.0234976i	$0.00748021\pi$
$911$	14304.0	+	24775.3i	0.520211	+	0.901033i	−0.479512	+	0.877535i	$0.659186\pi$
$912$		0			0
$913$	−7344.00	+	12720.2i	−0.266211	+	0.461092i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	8736.00			0.314600
$918$		0			0
$919$	−40768.0			−1.46334			−0.731672	−	0.681657i	$-0.761259\pi$
$919$	−40768.0			−1.46334			−0.731672	+	0.681657i	$0.761259\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−16800.0	+	29098.5i	−0.599110	+	1.03769i
$924$		0			0
$925$	2675.00	+	4633.24i	0.0950848	+	0.164692i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	13821.0	+	23938.7i	0.488108	+	0.845428i	0.999906	−	0.0136777i	$-0.00435389\pi$
$929$	13821.0	+	23938.7i	0.488108	+	0.845428i	−0.511798	+	0.859106i	$0.671021\pi$
$930$		0			0
$931$	−4410.00	+	7638.34i	−0.155244	+	0.268890i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−12240.0			−0.428119
$936$		0			0
$937$	28106.0			0.979918			0.489959	−	0.871746i	$-0.337012\pi$
$937$	28106.0			0.979918			0.489959	+	0.871746i	$0.337012\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	7365.00	−	12756.6i	0.255146	−	0.441925i	−0.709789	−	0.704414i	$-0.751210\pi$
$941$	7365.00	−	12756.6i	0.255146	−	0.441925i	0.964935	+	0.262489i	$0.0845432\pi$
$942$		0			0
$943$	−5400.00	−	9353.07i	−0.186477	−	0.322988i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−4782.00	−	8282.67i	−0.164091	−	0.284214i	0.772241	−	0.635330i	$-0.219136\pi$
$947$	−4782.00	−	8282.67i	−0.164091	−	0.284214i	−0.936332	+	0.351116i	$0.885802\pi$
$948$		0			0
$949$	15190.0	−	26309.9i	0.519587	−	0.899951i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	53898.0			1.83203			0.916017	−	0.401141i	$-0.131386\pi$
$953$	53898.0			1.83203			0.916017	+	0.401141i	$0.131386\pi$
$954$		0			0
$955$	−16920.0			−0.573318
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−37044.0	+	64162.1i	−1.24735	+	2.16048i
$960$		0			0
$961$	5647.50	+	9781.76i	0.189571	+	0.328346i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	5095.00	+	8824.80i	0.169963	+	0.294384i
$966$		0			0
$967$	−7570.00	+	13111.6i	−0.251742	+	0.436030i	−0.964006	−	0.265882i	$-0.914337\pi$
$967$	−7570.00	+	13111.6i	−0.251742	+	0.436030i	0.712263	+	0.701912i	$0.247670\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−23808.0			−0.786854			−0.393427	−	0.919356i	$-0.628711\pi$
$971$	−23808.0			−0.786854			−0.393427	+	0.919356i	$0.628711\pi$
$972$		0			0
$973$	35728.0			1.17717
$974$		0			0
$975$		0			0
$976$		0			0
$977$	11547.0	−	20000.0i	0.378118	−	0.654920i	−0.612670	−	0.790338i	$-0.709905\pi$
$977$	11547.0	−	20000.0i	0.378118	−	0.654920i	0.990788	+	0.135419i	$0.0432380\pi$
$978$		0			0
$979$	−360.000	−	623.538i	−0.0117525	−	0.0203558i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	3792.00	+	6567.94i	0.123038	+	0.213107i	0.920964	−	0.389647i	$-0.127403\pi$
$983$	3792.00	+	6567.94i	0.123038	+	0.213107i	−0.797927	+	0.602755i	$0.794070\pi$
$984$		0			0
$985$	10245.0	−	17744.9i	0.331404	−	0.574008i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−21024.0			−0.675960
$990$		0			0
$991$	−26752.0			−0.857523			−0.428761	−	0.903418i	$-0.641050\pi$
$991$	−26752.0			−0.857523			−0.428761	+	0.903418i	$0.641050\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	5620.00	−	9734.13i	0.179061	−	0.310143i
$996$		0			0
$997$	−3889.00	−	6735.95i	−0.123536	−	0.213971i	0.797623	−	0.603156i	$-0.206090\pi$
$997$	−3889.00	−	6735.95i	−0.123536	−	0.213971i	−0.921160	+	0.389184i	$0.872757\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.4.i.f.1081.1		2
3.2	odd	2		1620.4.i.l.1081.1		2
9.2	odd	6		1620.4.i.l.541.1		2
9.4	even	3		180.4.a.d.1.1		1
9.5	odd	6		60.4.a.a.1.1	✓	1
9.7	even	3	inner	1620.4.i.f.541.1		2
36.23	even	6		240.4.a.i.1.1		1
36.31	odd	6		720.4.a.bb.1.1		1
45.4	even	6		900.4.a.q.1.1		1
45.13	odd	12		900.4.d.h.649.2		2
45.14	odd	6		300.4.a.i.1.1		1
45.22	odd	12		900.4.d.h.649.1		2
45.23	even	12		300.4.d.b.49.1		2
45.32	even	12		300.4.d.b.49.2		2
72.5	odd	6		960.4.a.bc.1.1		1
72.59	even	6		960.4.a.r.1.1		1
180.23	odd	12		1200.4.f.n.49.2		2
180.59	even	6		1200.4.a.a.1.1		1
180.167	odd	12		1200.4.f.n.49.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.4.a.a.1.1	✓	1	9.5	odd	6
180.4.a.d.1.1		1	9.4	even	3
240.4.a.i.1.1		1	36.23	even	6
300.4.a.i.1.1		1	45.14	odd	6
300.4.d.b.49.1		2	45.23	even	12
300.4.d.b.49.2		2	45.32	even	12
720.4.a.bb.1.1		1	36.31	odd	6
900.4.a.q.1.1		1	45.4	even	6
900.4.d.h.649.1		2	45.22	odd	12
900.4.d.h.649.2		2	45.13	odd	12
960.4.a.r.1.1		1	72.59	even	6
960.4.a.bc.1.1		1	72.5	odd	6
1200.4.a.a.1.1		1	180.59	even	6
1200.4.f.n.49.1		2	180.167	odd	12
1200.4.f.n.49.2		2	180.23	odd	12
1620.4.i.f.541.1		2	9.7	even	3	inner
1620.4.i.f.1081.1		2	1.1	even	1	trivial
1620.4.i.l.541.1		2	9.2	odd	6
1620.4.i.l.1081.1		2	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 $\F_{q}$
Belyi maps

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)

\(f(q)\)	\(=\)	\(q+(-2.50000 + 4.33013i) q^{5} +(14.0000 + 24.2487i) q^{7} +(-12.0000 - 20.7846i) q^{11} +(35.0000 - 60.6218i) q^{13} -102.000 q^{17} +20.0000 q^{19} +(-36.0000 + 62.3538i) q^{23} +(-12.5000 - 21.6506i) q^{25} +(153.000 + 265.004i) q^{29} +(68.0000 - 117.779i) q^{31} -140.000 q^{35} -214.000 q^{37} +(-75.0000 + 129.904i) q^{41} +(146.000 + 252.879i) q^{43} +(-36.0000 - 62.3538i) q^{47} +(-220.500 + 381.917i) q^{49} +414.000 q^{53} +120.000 q^{55} +(-372.000 + 644.323i) q^{59} +(209.000 + 361.999i) q^{61} +(175.000 + 303.109i) q^{65} +(-94.0000 + 162.813i) q^{67} -480.000 q^{71} +434.000 q^{73} +(336.000 - 581.969i) q^{77} +(-676.000 - 1170.87i) q^{79} +(-306.000 - 530.008i) q^{83} +(255.000 - 441.673i) q^{85} +30.0000 q^{89} +1960.00 q^{91} +(-50.0000 + 86.6025i) q^{95} +(143.000 + 247.683i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 5 q^{5} + 28 q^{7} - 24 q^{11} + 70 q^{13} - 204 q^{17} + 40 q^{19} - 72 q^{23} - 25 q^{25} + 306 q^{29} + 136 q^{31} - 280 q^{35} - 428 q^{37} - 150 q^{41} + 292 q^{43} - 72 q^{47} - 441 q^{49} + 828 q^{53}+ \cdots + 286 q^{97}+O(q^{100}) \) 2 * q - 5 * q^5 + 28 * q^7 - 24 * q^11 + 70 * q^13 - 204 * q^17 + 40 * q^19 - 72 * q^23 - 25 * q^25 + 306 * q^29 + 136 * q^31 - 280 * q^35 - 428 * q^37 - 150 * q^41 + 292 * q^43 - 72 * q^47 - 441 * q^49 + 828 * q^53 + 240 * q^55 - 744 * q^59 + 418 * q^61 + 350 * q^65 - 188 * q^67 - 960 * q^71 + 868 * q^73 + 672 * q^77 - 1352 * q^79 - 612 * q^83 + 510 * q^85 + 60 * q^89 + 3920 * q^91 - 100 * q^95 + 286 * q^97

Introduction

L-functions

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