LMFDB - Embedded newform 560.3.f.c.321.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [560,3,Mod(321,560)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(560, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("560.321");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 560 = 2^{4} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	560.f (of order $2$, degree $1$, 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:	$15.2588948042$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-5}, \sqrt{-13})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9x^{2} + 4 $ x^4 + 9*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 140)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			321.4
Root			$2.92081i$ of defining polynomial
Character	$\chi$	$=$	560.321
Dual form			560.3.f.c.321.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.92081i q^{3} -2.23607i q^{5} +(5.53113 + 4.29029i) q^{7} +0.468871 q^{9} +O(q^{10})$	$q+2.92081i q^{3} -2.23607i q^{5} +(5.53113 + 4.29029i) q^{7} +0.468871 q^{9} +11.5934 q^{11} -2.92081i q^{13} +6.53113 q^{15} -19.0762i q^{17} -1.73317i q^{19} +(-12.5311 + 16.1554i) q^{21} -0.937742 q^{23} -5.00000 q^{25} +27.6568i q^{27} +41.7179 q^{29} +3.83002i q^{31} +33.8621i q^{33} +(9.59339 - 12.3680i) q^{35} +53.4358 q^{37} +8.53113 q^{39} +28.4807i q^{41} -42.2490 q^{43} -1.04843i q^{45} -17.4283i q^{47} +(12.1868 + 47.4603i) q^{49} +55.7179 q^{51} -23.8755 q^{53} -25.9236i q^{55} +5.06226 q^{57} +95.9264i q^{59} +120.941i q^{61} +(2.59339 + 2.01159i) q^{63} -6.53113 q^{65} -108.374 q^{67} -2.73897i q^{69} +74.4981 q^{71} -80.8621i q^{73} -14.6040i q^{75} +(64.1245 + 49.7390i) q^{77} +14.5311 q^{79} -76.5603 q^{81} +115.827i q^{83} -42.6556 q^{85} +121.850i q^{87} -141.483i q^{89} +(12.5311 - 16.1554i) q^{91} -11.1868 q^{93} -3.87548 q^{95} -12.1435i q^{97} +5.43580 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 6 q^{7} + 18 q^{9}+O(q^{10}) $ 4 * q + 6 * q^7 + 18 * q^9	$ 4 q + 6 q^{7} + 18 q^{9} - 2 q^{11} + 10 q^{15} - 34 q^{21} - 36 q^{23} - 20 q^{25} + 54 q^{29} - 10 q^{35} - 12 q^{37} + 18 q^{39} - 40 q^{43} - 48 q^{49} + 110 q^{51} - 160 q^{53} - 12 q^{57} - 38 q^{63} - 10 q^{65} - 240 q^{67} + 40 q^{71} + 192 q^{77} + 42 q^{79} - 16 q^{81} - 90 q^{85} + 34 q^{91} + 52 q^{93} - 80 q^{95} - 204 q^{99}+O(q^{100}) $ 4 * q + 6 * q^7 + 18 * q^9 - 2 * q^11 + 10 * q^15 - 34 * q^21 - 36 * q^23 - 20 * q^25 + 54 * q^29 - 10 * q^35 - 12 * q^37 + 18 * q^39 - 40 * q^43 - 48 * q^49 + 110 * q^51 - 160 * q^53 - 12 * q^57 - 38 * q^63 - 10 * q^65 - 240 * q^67 + 40 * q^71 + 192 * q^77 + 42 * q^79 - 16 * q^81 - 90 * q^85 + 34 * q^91 + 52 * q^93 - 80 * q^95 - 204 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/560\mathbb{Z}\right)^\times$.

$n$	$241$	$337$	$351$	$421$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$3$			2.92081i			0.973603i	0.873513	+	0.486802i	$0.161837\pi$
$3$			2.92081i			0.973603i	−0.873513	+	0.486802i	$0.838163\pi$
$4$		0			0
$5$		−	2.23607i		−	0.447214i
$5$		−	2.23607i		−	0.447214i
$6$		0			0
$7$	5.53113	+	4.29029i	0.790161	+	0.612899i
$7$	5.53113	+	4.29029i	0.790161	+	0.612899i
$8$		0			0
$9$	0.468871			0.0520968
$10$		0			0
$11$	11.5934			1.05394			0.526972	−	0.849883i	$-0.323327\pi$
$11$	11.5934			1.05394			0.526972	+	0.849883i	$0.323327\pi$
$12$		0			0
$13$		−	2.92081i		−	0.224678i	−0.993670	−	0.112339i	$-0.964166\pi$
$13$		−	2.92081i		−	0.224678i	0.993670	−	0.112339i	$-0.0358342\pi$
$14$		0			0
$15$	6.53113			0.435409
$16$		0			0
$17$		−	19.0762i		−	1.12213i	−0.827772	−	0.561064i	$-0.810392\pi$
$17$		−	19.0762i		−	1.12213i	0.827772	−	0.561064i	$-0.189608\pi$
$18$		0			0
$19$		−	1.73317i		−	0.0912194i	−0.998959	−	0.0456097i	$-0.985477\pi$
$19$		−	1.73317i		−	0.0912194i	0.998959	−	0.0456097i	$-0.0145231\pi$
$20$		0			0
$21$	−12.5311	+	16.1554i	−0.596720	+	0.769304i
$22$		0			0
$23$	−0.937742			−0.0407714			−0.0203857	−	0.999792i	$-0.506489\pi$
$23$	−0.937742			−0.0407714			−0.0203857	+	0.999792i	$0.506489\pi$
$24$		0			0
$25$	−5.00000			−0.200000
$26$		0			0
$27$			27.6568i			1.02432i
$28$		0			0
$29$	41.7179			1.43855			0.719274	−	0.694726i	$-0.244475\pi$
$29$	41.7179			1.43855			0.719274	+	0.694726i	$0.244475\pi$
$30$		0			0
$31$			3.83002i			0.123549i	0.998090	+	0.0617746i	$0.0196760\pi$
$31$			3.83002i			0.123549i	−0.998090	+	0.0617746i	$0.980324\pi$
$32$		0			0
$33$			33.8621i			1.02612i
$34$		0			0
$35$	9.59339	−	12.3680i	0.274097	−	0.353371i
$36$		0			0
$37$	53.4358			1.44421			0.722105	−	0.691783i	$-0.243175\pi$
$37$	53.4358			1.44421			0.722105	+	0.691783i	$0.243175\pi$
$38$		0			0
$39$	8.53113			0.218747
$40$		0			0
$41$			28.4807i			0.694652i	0.937744	+	0.347326i	$0.112910\pi$
$41$			28.4807i			0.694652i	−0.937744	+	0.347326i	$0.887090\pi$
$42$		0			0
$43$	−42.2490			−0.982536			−0.491268	−	0.871009i	$-0.663466\pi$
$43$	−42.2490			−0.982536			−0.491268	+	0.871009i	$0.663466\pi$
$44$		0			0
$45$		−	1.04843i		−	0.0232984i
$46$		0			0
$47$		−	17.4283i		−	0.370814i	−0.982662	−	0.185407i	$-0.940640\pi$
$47$		−	17.4283i		−	0.370814i	0.982662	−	0.185407i	$-0.0593604\pi$
$48$		0			0
$49$	12.1868	+	47.4603i	0.248710	+	0.968578i
$50$		0			0
$51$	55.7179			1.09251
$52$		0			0
$53$	−23.8755			−0.450481			−0.225240	−	0.974303i	$-0.572317\pi$
$53$	−23.8755			−0.450481			−0.225240	+	0.974303i	$0.572317\pi$
$54$		0			0
$55$		−	25.9236i		−	0.471338i
$56$		0			0
$57$	5.06226			0.0888115
$58$		0			0
$59$			95.9264i			1.62587i	0.582353	+	0.812936i	$0.302132\pi$
$59$			95.9264i			1.62587i	−0.582353	+	0.812936i	$0.697868\pi$
$60$		0			0
$61$			120.941i			1.98264i	0.131484	+	0.991318i	$0.458026\pi$
$61$			120.941i			1.98264i	−0.131484	+	0.991318i	$0.541974\pi$
$62$		0			0
$63$	2.59339	+	2.01159i	0.0411649	+	0.0319301i
$64$		0			0
$65$	−6.53113			−0.100479
$66$		0			0
$67$	−108.374			−1.61752			−0.808758	−	0.588142i	$-0.799860\pi$
$67$	−108.374			−1.61752			−0.808758	+	0.588142i	$0.799860\pi$
$68$		0			0
$69$		−	2.73897i		−	0.0396952i
$70$		0			0
$71$	74.4981			1.04927			0.524634	−	0.851328i	$-0.324202\pi$
$71$	74.4981			1.04927			0.524634	+	0.851328i	$0.324202\pi$
$72$		0			0
$73$		−	80.8621i		−	1.10770i	−0.832616	−	0.553850i	$-0.813158\pi$
$73$		−	80.8621i		−	1.10770i	0.832616	−	0.553850i	$-0.186842\pi$
$74$		0			0
$75$		−	14.6040i		−	0.194721i
$76$		0			0
$77$	64.1245	+	49.7390i	0.832786	+	0.645961i
$78$		0			0
$79$	14.5311			0.183938			0.0919692	−	0.995762i	$-0.470684\pi$
$79$	14.5311			0.183938			0.0919692	+	0.995762i	$0.470684\pi$
$80$		0			0
$81$	−76.5603			−0.945189
$82$		0			0
$83$			115.827i			1.39550i	0.716341	+	0.697751i	$0.245816\pi$
$83$			115.827i			1.39550i	−0.716341	+	0.697751i	$0.754184\pi$
$84$		0			0
$85$	−42.6556			−0.501831
$86$		0			0
$87$			121.850i			1.40058i
$88$		0			0
$89$		−	141.483i		−	1.58970i	−0.606808	−	0.794849i	$-0.707550\pi$
$89$		−	141.483i		−	1.58970i	0.606808	−	0.794849i	$-0.292450\pi$
$90$		0			0
$91$	12.5311	−	16.1554i	0.137705	−	0.177532i
$92$		0			0
$93$	−11.1868			−0.120288
$94$		0			0
$95$	−3.87548			−0.0407946
$96$		0			0
$97$		−	12.1435i		−	0.125191i	−0.998039	−	0.0625954i	$-0.980062\pi$
$97$		−	12.1435i		−	0.125191i	0.998039	−	0.0625954i	$-0.0199378\pi$
$98$		0			0
$99$	5.43580			0.0549071
$100$		0			0
$101$		−	38.9650i		−	0.385792i	−0.981219	−	0.192896i	$-0.938212\pi$
$101$		−	38.9650i		−	0.385792i	0.981219	−	0.192896i	$-0.0617880\pi$
$102$		0			0
$103$		−	170.401i		−	1.65438i	−0.561920	−	0.827191i	$-0.689937\pi$
$103$		−	170.401i		−	1.65438i	0.561920	−	0.827191i	$-0.310063\pi$
$104$		0			0
$105$	36.1245	+	28.0205i	0.344043	+	0.266861i
$106$		0			0
$107$	58.4981			0.546711			0.273355	−	0.961913i	$-0.411866\pi$
$107$	58.4981			0.546711			0.273355	+	0.961913i	$0.411866\pi$
$108$		0			0
$109$	3.84242			0.0352515			0.0176258	−	0.999845i	$-0.494389\pi$
$109$	3.84242			0.0352515			0.0176258	+	0.999845i	$0.494389\pi$
$110$		0			0
$111$			156.076i			1.40609i
$112$		0			0
$113$	−121.934			−1.07906			−0.539530	−	0.841966i	$-0.681398\pi$
$113$	−121.934			−1.07906			−0.539530	+	0.841966i	$0.681398\pi$
$114$		0			0
$115$			2.09686i			0.0182335i
$116$		0			0
$117$		−	1.36948i		−	0.0117050i
$118$		0			0
$119$	81.8424	−	105.513i	0.687751	−	0.886662i
$120$		0			0
$121$	13.4066			0.110798
$122$		0			0
$123$	−83.1868			−0.676315
$124$		0			0
$125$			11.1803i			0.0894427i
$126$		0			0
$127$	135.626			1.06792			0.533962	−	0.845508i	$-0.320702\pi$
$127$	135.626			1.06792			0.533962	+	0.845508i	$0.320702\pi$
$128$		0			0
$129$		−	123.401i		−	0.956600i
$130$		0			0
$131$		−	129.328i		−	0.987239i	−0.869678	−	0.493619i	$-0.835674\pi$
$131$		−	129.328i		−	0.987239i	0.869678	−	0.493619i	$-0.164326\pi$
$132$		0			0
$133$	7.43580	−	9.58638i	0.0559083	−	0.0720781i
$134$		0			0
$135$	61.8424			0.458092
$136$		0			0
$137$	10.5642			0.0771109			0.0385555	−	0.999256i	$-0.487724\pi$
$137$	10.5642			0.0771109			0.0385555	+	0.999256i	$0.487724\pi$
$138$		0			0
$139$		−	233.580i		−	1.68043i	−0.542255	−	0.840214i	$-0.682429\pi$
$139$		−	233.580i		−	1.68043i	0.542255	−	0.840214i	$-0.317571\pi$
$140$		0			0
$141$	50.9047			0.361026
$142$		0			0
$143$		−	33.8621i		−	0.236798i
$144$		0			0
$145$		−	93.2841i		−	0.643338i
$146$		0			0
$147$	−138.623	+	35.5952i	−0.943011	+	0.242145i
$148$		0			0
$149$	242.996			1.63085			0.815423	−	0.578865i	$-0.196504\pi$
$149$	242.996			1.63085			0.815423	+	0.578865i	$0.196504\pi$
$150$		0			0
$151$	−160.025			−1.05977			−0.529885	−	0.848069i	$-0.677765\pi$
$151$	−160.025			−1.05977			−0.529885	+	0.848069i	$0.677765\pi$
$152$		0			0
$153$		−	8.94427i		−	0.0584593i
$154$		0			0
$155$	8.56420			0.0552529
$156$		0			0
$157$			22.1902i			0.141339i	0.997500	+	0.0706693i	$0.0225135\pi$
$157$			22.1902i			0.141339i	−0.997500	+	0.0706693i	$0.977486\pi$
$158$		0			0
$159$		−	69.7357i		−	0.438590i
$160$		0			0
$161$	−5.18677	−	4.02319i	−0.0322160	−	0.0249888i
$162$		0			0
$163$	−134.432			−0.824736			−0.412368	−	0.911017i	$-0.635298\pi$
$163$	−134.432			−0.824736			−0.412368	+	0.911017i	$0.635298\pi$
$164$		0			0
$165$	75.7179			0.458896
$166$		0			0
$167$			37.1352i			0.222367i	0.993800	+	0.111183i	$0.0354641\pi$
$167$			37.1352i			0.222367i	−0.993800	+	0.111183i	$0.964536\pi$
$168$		0			0
$169$	160.469			0.949520
$170$		0			0
$171$		−	0.812633i		−	0.00475224i
$172$		0			0
$173$			127.436i			0.736624i	0.929702	+	0.368312i	$0.120064\pi$
$173$			127.436i			0.736624i	−0.929702	+	0.368312i	$0.879936\pi$
$174$		0			0
$175$	−27.6556	−	21.4515i	−0.158032	−	0.122580i
$176$		0			0
$177$	−280.183			−1.58295
$178$		0			0
$179$	166.000			0.927374			0.463687	−	0.885999i	$-0.346526\pi$
$179$	166.000			0.927374			0.463687	+	0.885999i	$0.346526\pi$
$180$		0			0
$181$			136.625i			0.754832i	0.926044	+	0.377416i	$0.123187\pi$
$181$			136.625i			0.754832i	−0.926044	+	0.377416i	$0.876813\pi$
$182$		0			0
$183$	−353.245			−1.93030
$184$		0			0
$185$		−	119.486i		−	0.645871i
$186$		0			0
$187$		−	221.158i		−	1.18266i
$188$		0			0
$189$	−118.656	+	152.973i	−0.627808	+	0.809382i
$190$		0			0
$191$	−199.278			−1.04334			−0.521671	−	0.853147i	$-0.674691\pi$
$191$	−199.278			−1.04334			−0.521671	+	0.853147i	$0.674691\pi$
$192$		0			0
$193$	−256.307			−1.32802			−0.664009	−	0.747725i	$-0.731146\pi$
$193$	−256.307			−1.32802			−0.664009	+	0.747725i	$0.731146\pi$
$194$		0			0
$195$		−	19.0762i		−	0.0978266i
$196$		0			0
$197$	−286.249			−1.45304			−0.726520	−	0.687145i	$-0.758864\pi$
$197$	−286.249			−1.45304			−0.726520	+	0.687145i	$0.758864\pi$
$198$		0			0
$199$		−	214.685i		−	1.07882i	−0.842043	−	0.539410i	$-0.818647\pi$
$199$		−	214.685i		−	1.07882i	0.842043	−	0.539410i	$-0.181353\pi$
$200$		0			0
$201$		−	316.538i		−	1.57482i
$202$		0			0
$203$	230.747	+	178.982i	1.13669	+	0.881685i
$204$		0			0
$205$	63.6848			0.310658
$206$		0			0
$207$	−0.439680			−0.00212406
$208$		0			0
$209$		−	20.0933i		−	0.0961402i
$210$		0			0
$211$	−255.527			−1.21103			−0.605515	−	0.795834i	$-0.707033\pi$
$211$	−255.527			−1.21103			−0.605515	+	0.795834i	$0.707033\pi$
$212$		0			0
$213$			217.595i			1.02157i
$214$		0			0
$215$			94.4717i			0.439403i
$216$		0			0
$217$	−16.4319	+	21.1844i	−0.0757232	+	0.0976238i
$218$		0			0
$219$	236.183			1.07846
$220$		0			0
$221$	−55.7179			−0.252117
$222$		0			0
$223$			166.401i			0.746192i	0.927793	+	0.373096i	$0.121704\pi$
$223$			166.401i			0.746192i	−0.927793	+	0.373096i	$0.878296\pi$
$224$		0			0
$225$	−2.34436			−0.0104194
$226$		0			0
$227$		−	288.240i		−	1.26978i	−0.772603	−	0.634889i	$-0.781046\pi$
$227$		−	288.240i		−	1.26978i	0.772603	−	0.634889i	$-0.218954\pi$
$228$		0			0
$229$		−	379.234i		−	1.65604i	−0.560697	−	0.828021i	$-0.689467\pi$
$229$		−	379.234i		−	1.65604i	0.560697	−	0.828021i	$-0.310533\pi$
$230$		0			0
$231$	−145.278	+	187.296i	−0.628910	+	0.810803i
$232$		0			0
$233$	−118.432			−0.508292			−0.254146	−	0.967166i	$-0.581794\pi$
$233$	−118.432			−0.508292			−0.254146	+	0.967166i	$0.581794\pi$
$234$		0			0
$235$	−38.9708			−0.165833
$236$		0			0
$237$			42.4427i			0.179083i
$238$		0			0
$239$	−263.095			−1.10082			−0.550409	−	0.834895i	$-0.685528\pi$
$239$	−263.095			−1.10082			−0.550409	+	0.834895i	$0.685528\pi$
$240$		0			0
$241$		−	223.823i		−	0.928724i	−0.885645	−	0.464362i	$-0.846284\pi$
$241$		−	223.823i		−	0.928724i	0.885645	−	0.464362i	$-0.153716\pi$
$242$		0			0
$243$			25.2928i			0.104086i
$244$		0			0
$245$	106.125	−	27.2505i	0.433161	−	0.111226i
$246$		0			0
$247$	−5.06226			−0.0204950
$248$		0			0
$249$	−338.307			−1.35866
$250$		0			0
$251$			324.306i			1.29206i	0.763313	+	0.646029i	$0.223571\pi$
$251$			324.306i			1.29206i	−0.763313	+	0.646029i	$0.776429\pi$
$252$		0			0
$253$	−10.8716			−0.0429708
$254$		0			0
$255$		−	124.589i		−	0.488584i
$256$		0			0
$257$		−	160.247i		−	0.623529i	−0.950159	−	0.311764i	$-0.899080\pi$
$257$		−	160.247i		−	0.623529i	0.950159	−	0.311764i	$-0.100920\pi$
$258$		0			0
$259$	295.560	+	229.255i	1.14116	+	0.885155i
$260$		0			0
$261$	19.5603			0.0749438
$262$		0			0
$263$	95.8755			0.364546			0.182273	−	0.983248i	$-0.441655\pi$
$263$	95.8755			0.364546			0.182273	+	0.983248i	$0.441655\pi$
$264$		0			0
$265$			53.3872i			0.201461i
$266$		0			0
$267$	413.245			1.54773
$268$		0			0
$269$			16.8601i			0.0626770i	0.999509	+	0.0313385i	$0.00997698\pi$
$269$			16.8601i			0.0626770i	−0.999509	+	0.0313385i	$0.990023\pi$
$270$		0			0
$271$			78.1458i			0.288361i	0.989551	+	0.144180i	$0.0460546\pi$
$271$			78.1458i			0.288361i	−0.989551	+	0.144180i	$0.953945\pi$
$272$		0			0
$273$	47.1868	+	36.6010i	0.172845	+	0.134070i
$274$		0			0
$275$	−57.9669			−0.210789
$276$		0			0
$277$	314.813			1.13651			0.568255	−	0.822853i	$-0.307619\pi$
$277$	314.813			1.13651			0.568255	+	0.822853i	$0.307619\pi$
$278$		0			0
$279$			1.79579i			0.00643652i
$280$		0			0
$281$	195.278			0.694940			0.347470	−	0.937691i	$-0.387041\pi$
$281$	195.278			0.694940			0.347470	+	0.937691i	$0.387041\pi$
$282$		0			0
$283$			177.271i			0.626401i	0.949687	+	0.313201i	$0.101401\pi$
$283$			177.271i			0.626401i	−0.949687	+	0.313201i	$0.898599\pi$
$284$		0			0
$285$		−	11.3196i		−	0.0397177i
$286$		0			0
$287$	−122.191	+	157.531i	−0.425751	+	0.548887i
$288$		0			0
$289$	−74.9008			−0.259172
$290$		0			0
$291$	35.4689			0.121886
$292$		0			0
$293$			295.922i			1.00997i	0.863127	+	0.504987i	$0.168503\pi$
$293$			295.922i			1.00997i	−0.863127	+	0.504987i	$0.831497\pi$
$294$		0			0
$295$	214.498			0.727112
$296$		0			0
$297$			320.636i			1.07958i
$298$		0			0
$299$			2.73897i			0.00916042i
$300$		0			0
$301$	−233.685	−	181.261i	−0.776362	−	0.602195i
$302$		0			0
$303$	113.809			0.375608
$304$		0			0
$305$	270.432			0.886662
$306$		0			0
$307$		−	471.148i		−	1.53468i	−0.641238	−	0.767342i	$-0.721579\pi$
$307$		−	471.148i		−	1.53468i	0.641238	−	0.767342i	$-0.278421\pi$
$308$		0			0
$309$	497.710			1.61071
$310$		0			0
$311$		−	528.633i		−	1.69978i	−0.526958	−	0.849892i	$-0.676667\pi$
$311$		−	528.633i		−	1.69978i	0.526958	−	0.849892i	$-0.323333\pi$
$312$		0			0
$313$		−	311.135i		−	0.994040i	−0.867739	−	0.497020i	$-0.834428\pi$
$313$		−	311.135i		−	0.994040i	0.867739	−	0.497020i	$-0.165572\pi$
$314$		0			0
$315$	4.49806	−	5.79899i	0.0142796	−	0.0184095i
$316$		0			0
$317$	174.864			0.551621			0.275810	−	0.961212i	$-0.411054\pi$
$317$	174.864			0.551621			0.275810	+	0.961212i	$0.411054\pi$
$318$		0			0
$319$	483.652			1.51615
$320$		0			0
$321$			170.862i			0.532279i
$322$		0			0
$323$	−33.0623			−0.102360
$324$		0			0
$325$			14.6040i			0.0449355i
$326$		0			0
$327$			11.2230i			0.0343210i
$328$		0			0
$329$	74.7724	−	96.3980i	0.227272	−	0.293003i
$330$		0			0
$331$	−127.875			−0.386331			−0.193165	−	0.981166i	$-0.561875\pi$
$331$	−127.875			−0.386331			−0.193165	+	0.981166i	$0.561875\pi$
$332$		0			0
$333$	25.0545			0.0752388
$334$		0			0
$335$			242.331i			0.723375i
$336$		0			0
$337$	−649.934			−1.92859			−0.964294	−	0.264836i	$-0.914682\pi$
$337$	−649.934			−1.92859			−0.964294	+	0.264836i	$0.914682\pi$
$338$		0			0
$339$		−	356.146i		−	1.05058i
$340$		0			0
$341$			44.4030i			0.130214i
$342$		0			0
$343$	−136.212	+	314.794i	−0.397120	+	0.917767i
$344$		0			0
$345$	−6.12452			−0.0177522
$346$		0			0
$347$	−391.062			−1.12698			−0.563490	−	0.826123i	$-0.690542\pi$
$347$	−391.062			−1.12698			−0.563490	+	0.826123i	$0.690542\pi$
$348$		0			0
$349$			268.220i			0.768539i	0.923221	+	0.384270i	$0.125547\pi$
$349$			268.220i			0.768539i	−0.923221	+	0.384270i	$0.874453\pi$
$350$		0			0
$351$	80.7802			0.230143
$352$		0			0
$353$			590.100i			1.67167i	0.548979	+	0.835836i	$0.315017\pi$
$353$			590.100i			1.67167i	−0.548979	+	0.835836i	$0.684983\pi$
$354$		0			0
$355$		−	166.583i		−	0.469247i
$356$		0			0
$357$	308.183	+	239.046i	0.863257	+	0.669597i
$358$		0			0
$359$	−266.498			−0.742334			−0.371167	−	0.928566i	$-0.621042\pi$
$359$	−266.498			−0.742334			−0.371167	+	0.928566i	$0.621042\pi$
$360$		0			0
$361$	357.996			0.991679
$362$		0			0
$363$			39.1582i			0.107874i
$364$		0			0
$365$	−180.813			−0.495379
$366$		0			0
$367$			219.532i			0.598181i	0.954225	+	0.299090i	$0.0966832\pi$
$367$			219.532i			0.598181i	−0.954225	+	0.299090i	$0.903317\pi$
$368$		0			0
$369$			13.3538i			0.0361891i
$370$		0			0
$371$	−132.058	−	102.433i	−0.355953	−	0.276099i
$372$		0			0
$373$	337.685			0.905321			0.452661	−	0.891683i	$-0.350475\pi$
$373$	337.685			0.905321			0.452661	+	0.891683i	$0.350475\pi$
$374$		0			0
$375$	−32.6556			−0.0870817
$376$		0			0
$377$		−	121.850i		−	0.323210i
$378$		0			0
$379$	−37.8832			−0.0999558			−0.0499779	−	0.998750i	$-0.515915\pi$
$379$	−37.8832			−0.0999558			−0.0499779	+	0.998750i	$0.515915\pi$
$380$		0			0
$381$			396.139i			1.03974i
$382$		0			0
$383$			181.090i			0.472820i	0.971653	+	0.236410i	$0.0759709\pi$
$383$			181.090i			0.472820i	−0.971653	+	0.236410i	$0.924029\pi$
$384$		0			0
$385$	111.220	−	143.387i	0.288883	−	0.372433i
$386$		0			0
$387$	−19.8094			−0.0511870
$388$		0			0
$389$	568.325			1.46099			0.730495	−	0.682918i	$-0.239289\pi$
$389$	568.325			1.46099			0.730495	+	0.682918i	$0.239289\pi$
$390$		0			0
$391$			17.8885i			0.0457508i
$392$		0			0
$393$	377.743			0.961179
$394$		0			0
$395$		−	32.4926i		−	0.0822597i
$396$		0			0
$397$		−	699.358i		−	1.76161i	−0.473483	−	0.880803i	$-0.657003\pi$
$397$		−	699.358i		−	1.76161i	0.473483	−	0.880803i	$-0.342997\pi$
$398$		0			0
$399$	28.0000	+	21.7186i	0.0701754	+	0.0544325i
$400$		0			0
$401$	−502.465			−1.25303			−0.626515	−	0.779409i	$-0.715519\pi$
$401$	−502.465			−1.25303			−0.626515	+	0.779409i	$0.715519\pi$
$402$		0			0
$403$	11.1868			0.0277587
$404$		0			0
$405$			171.194i			0.422701i
$406$		0			0
$407$	619.502			1.52212
$408$		0			0
$409$			50.3925i			0.123209i	0.998101	+	0.0616045i	$0.0196217\pi$
$409$			50.3925i			0.123209i	−0.998101	+	0.0616045i	$0.980378\pi$
$410$		0			0
$411$			30.8560i			0.0750754i
$412$		0			0
$413$	−411.553	+	530.582i	−0.996495	+	1.28470i
$414$		0			0
$415$	258.996			0.624087
$416$		0			0
$417$	682.241			1.63607
$418$		0			0
$419$			545.924i			1.30292i	0.758682	+	0.651461i	$0.225844\pi$
$419$			545.924i			1.30292i	−0.758682	+	0.651461i	$0.774156\pi$
$420$		0			0
$421$	−581.453			−1.38112			−0.690562	−	0.723273i	$-0.742637\pi$
$421$	−581.453			−1.38112			−0.690562	+	0.723273i	$0.742637\pi$
$422$		0			0
$423$		−	8.17162i		−	0.0193182i
$424$		0			0
$425$			95.3809i			0.224426i
$426$		0			0
$427$	−518.872	+	668.939i	−1.21516	+	1.56660i
$428$		0			0
$429$	98.9047			0.230547
$430$		0			0
$431$	721.702			1.67448			0.837242	−	0.546833i	$-0.184167\pi$
$431$	721.702			1.67448			0.837242	+	0.546833i	$0.184167\pi$
$432$		0			0
$433$			374.182i			0.864162i	0.901835	+	0.432081i	$0.142221\pi$
$433$			374.182i			0.864162i	−0.901835	+	0.432081i	$0.857779\pi$
$434$		0			0
$435$	272.465			0.626356
$436$		0			0
$437$			1.62527i			0.00371914i
$438$		0			0
$439$		−	743.147i		−	1.69282i	−0.532533	−	0.846409i	$-0.678760\pi$
$439$		−	743.147i		−	1.69282i	0.532533	−	0.846409i	$-0.321240\pi$
$440$		0			0
$441$	5.71403	+	22.2528i	0.0129570	+	0.0504598i
$442$		0			0
$443$	48.8716			0.110320			0.0551598	−	0.998478i	$-0.482433\pi$
$443$	48.8716			0.110320			0.0551598	+	0.998478i	$0.482433\pi$
$444$		0			0
$445$	−316.366			−0.710934
$446$		0			0
$447$			709.745i			1.58780i
$448$		0			0
$449$	773.967			1.72376			0.861879	−	0.507115i	$-0.169288\pi$
$449$	773.967			1.72376			0.861879	+	0.507115i	$0.169288\pi$
$450$		0			0
$451$			330.188i			0.732124i
$452$		0			0
$453$		−	467.403i		−	1.03180i
$454$		0			0
$455$	−36.1245	−	28.0205i	−0.0793945	−	0.0615834i
$456$		0			0
$457$	−671.611			−1.46961			−0.734804	−	0.678279i	$-0.762726\pi$
$457$	−671.611			−1.46961			−0.734804	+	0.678279i	$0.762726\pi$
$458$		0			0
$459$	527.586			1.14942
$460$		0			0
$461$			841.324i			1.82500i	0.409080	+	0.912499i	$0.365850\pi$
$461$			841.324i			1.82500i	−0.409080	+	0.912499i	$0.634150\pi$
$462$		0			0
$463$	−158.696			−0.342757			−0.171378	−	0.985205i	$-0.554822\pi$
$463$	−158.696			−0.342757			−0.171378	+	0.985205i	$0.554822\pi$
$464$		0			0
$465$			25.0144i			0.0537944i
$466$		0			0
$467$			26.8668i			0.0575306i	0.999586	+	0.0287653i	$0.00915754\pi$
$467$			26.8668i			0.0575306i	−0.999586	+	0.0287653i	$0.990842\pi$
$468$		0			0
$469$	−599.428	−	464.954i	−1.27810	−	0.991374i
$470$		0			0
$471$	−64.8132			−0.137608
$472$		0			0
$473$	−489.809			−1.03554
$474$		0			0
$475$			8.66585i			0.0182439i
$476$		0			0
$477$	−11.1945			−0.0234686
$478$		0			0
$479$			89.4880i			0.186823i	0.995628	+	0.0934113i	$0.0297771\pi$
$479$			89.4880i			0.186823i	−0.995628	+	0.0934113i	$0.970223\pi$
$480$		0			0
$481$		−	156.076i		−	0.324482i
$482$		0			0
$483$	11.7510	−	15.1496i	0.0243291	−	0.0313656i
$484$		0			0
$485$	−27.1537			−0.0559870
$486$		0			0
$487$	−307.062			−0.630518			−0.315259	−	0.949006i	$-0.602091\pi$
$487$	−307.062			−0.630518			−0.315259	+	0.949006i	$0.602091\pi$
$488$		0			0
$489$		−	392.650i		−	0.802965i
$490$		0			0
$491$	195.461			0.398088			0.199044	−	0.979991i	$-0.436216\pi$
$491$	195.461			0.398088			0.199044	+	0.979991i	$0.436216\pi$
$492$		0			0
$493$		−	795.818i		−	1.61424i
$494$		0			0
$495$		−	12.1548i		−	0.0245552i
$496$		0			0
$497$	412.058	+	319.619i	0.829091	+	0.643096i
$498$		0			0
$499$	−491.710			−0.985391			−0.492696	−	0.870202i	$-0.663988\pi$
$499$	−491.710			−0.985391			−0.492696	+	0.870202i	$0.663988\pi$
$500$		0			0
$501$	−108.465			−0.216497
$502$		0			0
$503$		−	100.734i		−	0.200266i	−0.994974	−	0.100133i	$-0.968073\pi$
$503$		−	100.734i		−	0.200266i	0.994974	−	0.100133i	$-0.0319268\pi$
$504$		0			0
$505$	−87.1284			−0.172531
$506$		0			0
$507$			468.699i			0.924456i
$508$		0			0
$509$		−	132.840i		−	0.260982i	−0.991449	−	0.130491i	$-0.958345\pi$
$509$		−	132.840i		−	0.260982i	0.991449	−	0.130491i	$-0.0416554\pi$
$510$		0			0
$511$	346.922	−	447.259i	0.678908	−	0.875262i
$512$		0			0
$513$	47.9339			0.0934383
$514$		0			0
$515$	−381.029			−0.739863
$516$		0			0
$517$		−	202.053i		−	0.390818i
$518$		0			0
$519$	−372.216			−0.717179
$520$		0			0
$521$		−	402.685i		−	0.772909i	−0.922309	−	0.386454i	$-0.873700\pi$
$521$		−	402.685i		−	0.772909i	0.922309	−	0.386454i	$-0.126300\pi$
$522$		0			0
$523$			53.9893i			0.103230i	0.998667	+	0.0516150i	$0.0164369\pi$
$523$			53.9893i			0.103230i	−0.998667	+	0.0516150i	$0.983563\pi$
$524$		0			0
$525$	62.6556	−	80.7769i	0.119344	−	0.153861i
$526$		0			0
$527$	73.0623			0.138638
$528$		0			0
$529$	−528.121			−0.998338
$530$		0			0
$531$			44.9771i			0.0847027i
$532$		0			0
$533$	83.1868			0.156073
$534$		0			0
$535$		−	130.806i		−	0.244497i
$536$		0			0
$537$			484.854i			0.902895i
$538$		0			0
$539$	141.286	+	550.226i	0.262126	+	1.02083i
$540$		0			0
$541$	−362.831			−0.670667			−0.335333	−	0.942100i	$-0.608849\pi$
$541$	−362.831			−0.670667			−0.335333	+	0.942100i	$0.608849\pi$
$542$		0			0
$543$	−399.055			−0.734907
$544$		0			0
$545$		−	8.59191i		−	0.0157650i
$546$		0			0
$547$	−418.416			−0.764929			−0.382465	−	0.923970i	$-0.624925\pi$
$547$	−418.416			−0.764929			−0.382465	+	0.923970i	$0.624925\pi$
$548$		0			0
$549$			56.7057i			0.103289i
$550$		0			0
$551$		−	72.3042i		−	0.131224i
$552$		0			0
$553$	80.3735	+	62.3428i	0.145341	+	0.112736i
$554$		0			0
$555$	348.996			0.628822
$556$		0			0
$557$	321.319			0.576874			0.288437	−	0.957499i	$-0.406864\pi$
$557$	321.319			0.576874			0.288437	+	0.957499i	$0.406864\pi$
$558$		0			0
$559$			123.401i			0.220754i
$560$		0			0
$561$	645.959			1.15144
$562$		0			0
$563$		−	110.223i		−	0.195779i	−0.995197	−	0.0978894i	$-0.968791\pi$
$563$		−	110.223i		−	0.195779i	0.995197	−	0.0978894i	$-0.0312091\pi$
$564$		0			0
$565$			272.652i			0.482571i
$566$		0			0
$567$	−423.465	−	328.466i	−0.746852	−	0.579305i
$568$		0			0
$569$	396.607			0.697025			0.348512	−	0.937304i	$-0.386687\pi$
$569$	396.607			0.697025			0.348512	+	0.937304i	$0.386687\pi$
$570$		0			0
$571$	546.879			0.957757			0.478879	−	0.877881i	$-0.341043\pi$
$571$	546.879			0.957757			0.478879	+	0.877881i	$0.341043\pi$
$572$		0			0
$573$		−	582.054i		−	1.01580i
$574$		0			0
$575$	4.68871			0.00815428
$576$		0			0
$577$		−	78.1745i		−	0.135484i	−0.997703	−	0.0677422i	$-0.978420\pi$
$577$		−	78.1745i		−	0.135484i	0.997703	−	0.0677422i	$-0.0215795\pi$
$578$		0			0
$579$		−	748.625i		−	1.29296i
$580$		0			0
$581$	−496.930	+	640.652i	−0.855301	+	1.10267i
$582$		0			0
$583$	−276.798			−0.474782
$584$		0			0
$585$	−3.06226			−0.00523463
$586$		0			0
$587$		−	684.668i		−	1.16639i	−0.812334	−	0.583193i	$-0.801803\pi$
$587$		−	684.668i		−	1.16639i	0.812334	−	0.583193i	$-0.198197\pi$
$588$		0			0
$589$	6.63808			0.0112701
$590$		0			0
$591$		−	836.079i		−	1.41469i
$592$		0			0
$593$		−	114.917i		−	0.193790i	−0.995295	−	0.0968949i	$-0.969109\pi$
$593$		−	114.917i		−	0.193790i	0.995295	−	0.0968949i	$-0.0308911\pi$
$594$		0			0
$595$	−235.934	−	183.005i	−0.396528	−	0.307572i
$596$		0			0
$597$	627.055			1.05034
$598$		0			0
$599$	36.5389			0.0609998			0.0304999	−	0.999535i	$-0.490290\pi$
$599$	36.5389			0.0609998			0.0304999	+	0.999535i	$0.490290\pi$
$600$		0			0
$601$			493.924i			0.821837i	0.911672	+	0.410918i	$0.134792\pi$
$601$			493.924i			0.821837i	−0.911672	+	0.410918i	$0.865208\pi$
$602$		0			0
$603$	−50.8132			−0.0842674
$604$		0			0
$605$		−	29.9781i		−	0.0495506i
$606$		0			0
$607$		−	542.788i		−	0.894214i	−0.894481	−	0.447107i	$-0.852454\pi$
$607$		−	542.788i		−	0.894214i	0.894481	−	0.447107i	$-0.147546\pi$
$608$		0			0
$609$	−522.772	+	673.968i	−0.858411	+	1.10668i
$610$		0			0
$611$	−50.9047			−0.0833137
$612$		0			0
$613$	−676.490			−1.10357			−0.551787	−	0.833985i	$-0.686054\pi$
$613$	−676.490			−1.10357			−0.551787	+	0.833985i	$0.686054\pi$
$614$		0			0
$615$			186.011i			0.302457i
$616$		0			0
$617$	−89.0700			−0.144360			−0.0721799	−	0.997392i	$-0.522996\pi$
$617$	−89.0700			−0.144360			−0.0721799	+	0.997392i	$0.522996\pi$
$618$		0			0
$619$			959.861i			1.55066i	0.631554	+	0.775332i	$0.282418\pi$
$619$			959.861i			1.55066i	−0.631554	+	0.775332i	$0.717582\pi$
$620$		0			0
$621$		−	25.9349i		−	0.0417632i
$622$		0			0
$623$	607.004	−	782.561i	0.974324	−	1.25612i
$624$		0			0
$625$	25.0000			0.0400000
$626$		0			0
$627$	58.6887			0.0936024
$628$		0			0
$629$		−	1019.35i		−	1.62059i
$630$		0			0
$631$	−440.508			−0.698111			−0.349055	−	0.937102i	$-0.613497\pi$
$631$	−440.508			−0.698111			−0.349055	+	0.937102i	$0.613497\pi$
$632$		0			0
$633$		−	746.346i		−	1.17906i
$634$		0			0
$635$		−	303.270i		−	0.477591i
$636$		0			0
$637$	138.623	−	35.5952i	0.217618	−	0.0558795i
$638$		0			0
$639$	34.9300			0.0546635
$640$		0			0
$641$	448.490			0.699673			0.349836	−	0.936811i	$-0.386237\pi$
$641$	448.490			0.699673			0.349836	+	0.936811i	$0.386237\pi$
$642$		0			0
$643$		−	1209.52i		−	1.88106i	−0.339709	−	0.940531i	$-0.610329\pi$
$643$		−	1209.52i		−	1.88106i	0.339709	−	0.940531i	$-0.389671\pi$
$644$		0			0
$645$	−275.934			−0.427804
$646$		0			0
$647$		−	42.9029i		−	0.0663106i	−0.999450	−	0.0331553i	$-0.989444\pi$
$647$		−	42.9029i		−	0.0663106i	0.999450	−	0.0331553i	$-0.0105556\pi$
$648$		0			0
$649$			1112.11i			1.71358i
$650$		0			0
$651$	−61.8755	−	47.9945i	−0.0950468	−	0.0737243i
$652$		0			0
$653$	234.879			0.359693			0.179846	−	0.983695i	$-0.442440\pi$
$653$	234.879			0.359693			0.179846	+	0.983695i	$0.442440\pi$
$654$		0			0
$655$	−289.187			−0.441507
$656$		0			0
$657$		−	37.9139i		−	0.0577076i
$658$		0			0
$659$	−1090.32			−1.65450			−0.827251	−	0.561832i	$-0.810097\pi$
$659$	−1090.32			−1.65450			−0.827251	+	0.561832i	$0.810097\pi$
$660$		0			0
$661$		−	1200.50i		−	1.81619i	−0.418760	−	0.908097i	$-0.637535\pi$
$661$		−	1200.50i		−	1.81619i	0.418760	−	0.908097i	$-0.362465\pi$
$662$		0			0
$663$		−	162.741i		−	0.245462i
$664$		0			0
$665$	−21.4358	−	16.6270i	−0.0322343	−	0.0250030i
$666$		0			0
$667$	−39.1206			−0.0586516
$668$		0			0
$669$	−486.025			−0.726495
$670$		0			0
$671$			1402.11i			2.08959i
$672$		0			0
$673$	863.047			1.28239			0.641194	−	0.767379i	$-0.278439\pi$
$673$	863.047			1.28239			0.641194	+	0.767379i	$0.278439\pi$
$674$		0			0
$675$		−	138.284i		−	0.204865i
$676$		0			0
$677$		−	233.312i		−	0.344627i	−0.985042	−	0.172313i	$-0.944876\pi$
$677$		−	233.312i		−	0.344627i	0.985042	−	0.172313i	$-0.0551242\pi$
$678$		0			0
$679$	52.0992	−	67.1673i	0.0767293	−	0.0989209i
$680$		0			0
$681$	841.893			1.23626
$682$		0			0
$683$	−420.572			−0.615772			−0.307886	−	0.951423i	$-0.599621\pi$
$683$	−420.572			−0.615772			−0.307886	+	0.951423i	$0.599621\pi$
$684$		0			0
$685$		−	23.6223i		−	0.0344851i
$686$		0			0
$687$	1107.67			1.61233
$688$		0			0
$689$			69.7357i			0.101213i
$690$		0			0
$691$		−	523.604i		−	0.757748i	−0.925448	−	0.378874i	$-0.876311\pi$
$691$		−	523.604i		−	0.757748i	0.925448	−	0.378874i	$-0.123689\pi$
$692$		0			0
$693$	30.0661	+	23.3212i	0.0433855	+	0.0336525i
$694$		0			0
$695$	−522.300			−0.751510
$696$		0			0
$697$	543.304			0.779489
$698$		0			0
$699$		−	345.917i		−	0.494874i
$700$		0			0
$701$	−352.076			−0.502248			−0.251124	−	0.967955i	$-0.580800\pi$
$701$	−352.076			−0.502248			−0.251124	+	0.967955i	$0.580800\pi$
$702$		0			0
$703$		−	92.6133i		−	0.131740i
$704$		0			0
$705$		−	113.826i		−	0.161456i
$706$		0			0
$707$	167.171	−	215.520i	0.236452	−	0.304838i
$708$		0			0
$709$	−616.508			−0.869546			−0.434773	−	0.900540i	$-0.643171\pi$
$709$	−616.508			−0.869546			−0.434773	+	0.900540i	$0.643171\pi$
$710$		0			0
$711$	6.81323			0.00958260
$712$		0			0
$713$		−	3.59158i		−	0.00503727i
$714$		0			0
$715$	−75.7179			−0.105899
$716$		0			0
$717$		−	768.451i		−	1.07176i
$718$		0			0
$719$			667.786i			0.928770i	0.885633	+	0.464385i	$0.153725\pi$
$719$			667.786i			0.928770i	−0.885633	+	0.464385i	$0.846275\pi$
$720$		0			0
$721$	731.072	−	942.512i	1.01397	−	1.30723i
$722$		0			0
$723$	653.743			0.904209
$724$		0			0
$725$	−208.590			−0.287710
$726$		0			0
$727$		−	668.706i		−	0.919816i	−0.887967	−	0.459908i	$-0.847882\pi$
$727$		−	668.706i		−	0.919816i	0.887967	−	0.459908i	$-0.152118\pi$
$728$		0			0
$729$	−762.918			−1.04653
$730$		0			0
$731$			805.950i			1.10253i
$732$		0			0
$733$		−	147.574i		−	0.201329i	−0.994920	−	0.100665i	$-0.967903\pi$
$733$		−	147.574i		−	0.201329i	0.994920	−	0.100665i	$-0.0320969\pi$
$734$		0			0
$735$	79.5934	+	309.970i	0.108290	+	0.421727i
$736$		0			0
$737$	−1256.42			−1.70477
$738$		0			0
$739$	−278.333			−0.376634			−0.188317	−	0.982108i	$-0.560303\pi$
$739$	−278.333			−0.376634			−0.188317	+	0.982108i	$0.560303\pi$
$740$		0			0
$741$		−	14.7859i		−	0.0199540i
$742$		0			0
$743$	488.856			0.657949			0.328974	−	0.944339i	$-0.393297\pi$
$743$	488.856			0.657949			0.328974	+	0.944339i	$0.393297\pi$
$744$		0			0
$745$		−	543.356i		−	0.729337i
$746$		0			0
$747$			54.3077i			0.0727011i
$748$		0			0
$749$	323.560	+	250.974i	0.431990	+	0.335079i
$750$		0			0
$751$	−273.154			−0.363720			−0.181860	−	0.983324i	$-0.558212\pi$
$751$	−273.154			−0.363720			−0.181860	+	0.983324i	$0.558212\pi$
$752$		0			0
$753$	−947.237			−1.25795
$754$		0			0
$755$			357.827i			0.473944i
$756$		0			0
$757$	−1419.66			−1.87538			−0.937689	−	0.347475i	$-0.887039\pi$
$757$	−1419.66			−1.87538			−0.937689	+	0.347475i	$0.887039\pi$
$758$		0			0
$759$		−	31.7539i		−	0.0418365i
$760$		0			0
$761$			320.664i			0.421372i	0.977554	+	0.210686i	$0.0675698\pi$
$761$			320.664i			0.421372i	−0.977554	+	0.210686i	$0.932430\pi$
$762$		0			0
$763$	21.2529	+	16.4851i	0.0278544	+	0.0216056i
$764$		0			0
$765$	−20.0000			−0.0261438
$766$		0			0
$767$	280.183			0.365297
$768$		0			0
$769$			792.471i			1.03052i	0.857033	+	0.515261i	$0.172305\pi$
$769$			792.471i			1.03052i	−0.857033	+	0.515261i	$0.827695\pi$
$770$		0			0
$771$	468.051			0.607070
$772$		0			0
$773$		−	415.687i		−	0.537758i	−0.963174	−	0.268879i	$-0.913347\pi$
$773$		−	415.687i		−	0.537758i	0.963174	−	0.268879i	$-0.0866532\pi$
$774$		0			0
$775$		−	19.1501i		−	0.0247098i
$776$		0			0
$777$	−669.611	+	863.275i	−0.861790	+	1.11104i
$778$		0			0
$779$	49.3619			0.0633657
$780$		0			0
$781$	863.685			1.10587
$782$		0			0
$783$			1153.78i			1.47354i
$784$		0			0
$785$	49.6187			0.0632085
$786$		0			0
$787$			546.271i			0.694119i	0.937843	+	0.347059i	$0.112820\pi$
$787$			546.271i			0.694119i	−0.937843	+	0.347059i	$0.887180\pi$
$788$		0			0
$789$			280.034i			0.354923i
$790$		0			0
$791$	−674.432	−	523.132i	−0.852632	−	0.661355i
$792$		0			0
$793$	353.245			0.445454
$794$		0			0
$795$	−155.934			−0.196143
$796$		0			0
$797$		−	1453.29i		−	1.82345i	−0.410800	−	0.911725i	$-0.634751\pi$
$797$		−	1453.29i		−	1.82345i	0.410800	−	0.911725i	$-0.365249\pi$
$798$		0			0
$799$	−332.465			−0.416101
$800$		0			0
$801$		−	66.3373i		−	0.0828181i
$802$		0			0
$803$		−	937.466i		−	1.16745i
$804$		0			0
$805$	−8.99612	+	11.5980i	−0.0111753	+	0.0144074i
$806$		0			0
$807$	−49.2452			−0.0610225
$808$		0			0
$809$	−789.006			−0.975285			−0.487643	−	0.873043i	$-0.662143\pi$
$809$	−789.006			−0.975285			−0.487643	+	0.873043i	$0.662143\pi$
$810$		0			0
$811$		−	357.981i		−	0.441407i	−0.975341	−	0.220704i	$-0.929165\pi$
$811$		−	357.981i		−	0.441407i	0.975341	−	0.220704i	$-0.0708354\pi$
$812$		0			0
$813$	−228.249			−0.280749
$814$		0			0
$815$			300.599i			0.368833i
$816$		0			0
$817$			73.2247i			0.0896263i
$818$		0			0
$819$	5.87548	−	7.57479i	0.00717397	−	0.00924883i
$820$		0			0
$821$	−940.590			−1.14566			−0.572832	−	0.819673i	$-0.694155\pi$
$821$	−940.590			−1.14566			−0.572832	+	0.819673i	$0.694155\pi$
$822$		0			0
$823$	−972.607			−1.18178			−0.590891	−	0.806751i	$-0.701224\pi$
$823$	−972.607			−1.18178			−0.590891	+	0.806751i	$0.701224\pi$
$824$		0			0
$825$		−	169.310i		−	0.205225i
$826$		0			0
$827$	814.284			0.984624			0.492312	−	0.870419i	$-0.336152\pi$
$827$	814.284			0.984624			0.492312	+	0.870419i	$0.336152\pi$
$828$		0			0
$829$			580.849i			0.700662i	0.936626	+	0.350331i	$0.113931\pi$
$829$			580.849i			0.700662i	−0.936626	+	0.350331i	$0.886069\pi$
$830$		0			0
$831$			919.510i			1.10651i
$832$		0			0
$833$	905.362	−	232.477i	1.08687	−	0.279084i
$834$		0			0
$835$	83.0369			0.0994454
$836$		0			0
$837$	−105.926			−0.126554
$838$		0			0
$839$			106.047i			0.126397i	0.998001	+	0.0631985i	$0.0201301\pi$
$839$			106.047i			0.126397i	−0.998001	+	0.0631985i	$0.979870\pi$
$840$		0			0
$841$	899.383			1.06942
$842$		0			0
$843$			570.371i			0.676596i
$844$		0			0
$845$		−	358.819i		−	0.424638i
$846$		0			0
$847$	74.1537	+	57.5183i	0.0875486	+	0.0679083i
$848$		0			0
$849$	−517.776			−0.609866
$850$		0			0
$851$	−50.1090			−0.0588825
$852$		0			0
$853$			302.713i			0.354881i	0.984132	+	0.177440i	$0.0567817\pi$
$853$			302.713i			0.354881i	−0.984132	+	0.177440i	$0.943218\pi$
$854$		0			0
$855$	−1.81710			−0.00212527
$856$		0			0
$857$			1439.12i			1.67925i	0.543166	+	0.839625i	$0.317225\pi$
$857$			1439.12i			1.67925i	−0.543166	+	0.839625i	$0.682775\pi$
$858$		0			0
$859$			97.2333i			0.113194i	0.998397	+	0.0565968i	$0.0180250\pi$
$859$			97.2333i			0.113194i	−0.998397	+	0.0565968i	$0.981975\pi$
$860$		0			0
$861$	−460.117	−	356.896i	−0.534398	−	0.414513i
$862$		0			0
$863$	1256.84			1.45636			0.728181	−	0.685385i	$-0.240366\pi$
$863$	1256.84			1.45636			0.728181	+	0.685385i	$0.240366\pi$
$864$		0			0
$865$	284.955			0.329428
$866$		0			0
$867$		−	218.771i		−	0.252331i
$868$		0			0
$869$	168.465			0.193861
$870$		0			0
$871$			316.538i			0.363420i
$872$		0			0
$873$		−	5.69374i		−	0.00652204i
$874$		0			0
$875$	−47.9669	+	61.8399i	−0.0548194	+	0.0706742i
$876$		0			0
$877$	780.475			0.889937			0.444969	−	0.895546i	$-0.353215\pi$
$877$	780.475			0.889937			0.444969	+	0.895546i	$0.353215\pi$
$878$		0			0
$879$	−864.333			−0.983314
$880$		0			0
$881$		−	681.651i		−	0.773724i	−0.922138	−	0.386862i	$-0.873559\pi$
$881$		−	681.651i		−	0.773724i	0.922138	−	0.386862i	$-0.126441\pi$
$882$		0			0
$883$	−561.420			−0.635810			−0.317905	−	0.948123i	$-0.602979\pi$
$883$	−561.420			−0.635810			−0.317905	+	0.948123i	$0.602979\pi$
$884$		0			0
$885$			626.508i			0.707919i
$886$		0			0
$887$		−	118.185i		−	0.133241i	−0.997778	−	0.0666204i	$-0.978778\pi$
$887$		−	118.185i		−	0.133241i	0.997778	−	0.0666204i	$-0.0212216\pi$
$888$		0			0
$889$	750.167	+	581.877i	0.843833	+	0.654530i
$890$		0			0
$891$	−887.593			−0.996177
$892$		0			0
$893$	−30.2062			−0.0338255
$894$		0			0
$895$		−	371.187i		−	0.414734i
$896$		0			0
$897$	−8.00000			−0.00891862
$898$		0			0
$899$			159.781i			0.177731i
$900$		0			0
$901$			455.453i			0.505497i
$902$		0			0
$903$	529.428	−	682.549i	0.586299	−	0.755868i
$904$		0			0
$905$	305.502			0.337571
$906$		0			0
$907$	925.553			1.02045			0.510227	−	0.860040i	$-0.329561\pi$
$907$	925.553			1.02045			0.510227	+	0.860040i	$0.329561\pi$
$908$		0			0
$909$		−	18.2696i		−	0.0200985i
$910$		0			0
$911$	−960.607			−1.05445			−0.527227	−	0.849725i	$-0.676768\pi$
$911$	−960.607			−1.05445			−0.527227	+	0.849725i	$0.676768\pi$
$912$		0			0
$913$			1342.82i			1.47078i
$914$		0			0
$915$			789.880i			0.863257i
$916$		0			0
$917$	554.856	−	715.331i	0.605078	−	0.780078i
$918$		0			0
$919$	514.414			0.559754			0.279877	−	0.960036i	$-0.409706\pi$
$919$	514.414			0.559754			0.279877	+	0.960036i	$0.409706\pi$
$920$		0			0
$921$	1376.13			1.49417
$922$		0			0
$923$		−	217.595i		−	0.235747i
$924$		0			0
$925$	−267.179			−0.288842
$926$		0			0
$927$		−	79.8963i		−	0.0861880i
$928$		0			0
$929$		−	6.73951i		−	0.00725459i	−0.999993	−	0.00362729i	$-0.998845\pi$
$929$		−	6.73951i		−	0.00725459i	0.999993	−	0.00362729i	$-0.00115461\pi$
$930$		0			0
$931$	82.2568	−	21.1217i	0.0883532	−	0.0226872i
$932$		0			0
$933$	1544.04			1.65491
$934$		0			0
$935$	−494.523			−0.528902
$936$		0			0
$937$			1592.31i			1.69937i	0.527288	+	0.849687i	$0.323209\pi$
$937$			1592.31i			1.69937i	−0.527288	+	0.849687i	$0.676791\pi$
$938$		0			0
$939$	908.765			0.967800
$940$		0			0
$941$		−	17.4170i		−	0.0185090i	−0.999957	−	0.00925449i	$-0.997054\pi$
$941$		−	17.4170i		−	0.0185090i	0.999957	−	0.00925449i	$-0.00294584\pi$
$942$		0			0
$943$		−	26.7076i		−	0.0283219i
$944$		0			0
$945$	342.058	+	265.322i	0.361967	+	0.280764i
$946$		0			0
$947$	340.623			0.359686			0.179843	−	0.983695i	$-0.442441\pi$
$947$	340.623			0.359686			0.179843	+	0.983695i	$0.442441\pi$
$948$		0			0
$949$	−236.183			−0.248876
$950$		0			0
$951$			510.744i			0.537060i
$952$		0			0
$953$	−1474.48			−1.54720			−0.773601	−	0.633674i	$-0.781546\pi$
$953$	−1474.48			−1.54720			−0.773601	+	0.633674i	$0.781546\pi$
$954$		0			0
$955$			445.600i			0.466596i
$956$		0			0
$957$			1412.65i			1.47613i
$958$		0			0
$959$	58.4319	+	45.3235i	0.0609301	+	0.0472612i
$960$		0			0
$961$	946.331			0.984736
$962$		0			0
$963$	27.4281			0.0284819
$964$		0			0
$965$			573.121i			0.593908i
$966$		0			0
$967$	−1201.11			−1.24209			−0.621047	−	0.783773i	$-0.713292\pi$
$967$	−1201.11			−1.24209			−0.621047	+	0.783773i	$0.713292\pi$
$968$		0			0
$969$		−	96.5686i		−	0.0996580i
$970$		0			0
$971$			621.627i			0.640192i	0.947385	+	0.320096i	$0.103715\pi$
$971$			621.627i			0.640192i	−0.947385	+	0.320096i	$0.896285\pi$
$972$		0			0
$973$	1002.12	−	1291.96i	1.02993	−	1.32781i
$974$		0			0
$975$	−42.6556			−0.0437494
$976$		0			0
$977$	−282.747			−0.289403			−0.144702	−	0.989475i	$-0.546222\pi$
$977$	−282.747			−0.289403			−0.144702	+	0.989475i	$0.546222\pi$
$978$		0			0
$979$		−	1640.27i		−	1.67545i
$980$		0			0
$981$	1.80160			0.00183649
$982$		0			0
$983$		−	1201.00i		−	1.22177i	−0.791718	−	0.610887i	$-0.790813\pi$
$983$		−	1201.00i		−	1.22177i	0.791718	−	0.610887i	$-0.209187\pi$
$984$		0			0
$985$			640.072i			0.649820i
$986$		0			0
$987$	281.560	+	218.396i	0.285269	+	0.221273i
$988$		0			0
$989$	39.6187			0.0400594
$990$		0			0
$991$	897.105			0.905252			0.452626	−	0.891700i	$-0.350487\pi$
$991$	897.105			0.905252			0.452626	+	0.891700i	$0.350487\pi$
$992$		0			0
$993$		−	373.500i		−	0.376133i
$994$		0			0
$995$	−480.051			−0.482463
$996$		0			0
$997$			734.465i			0.736675i	0.929692	+	0.368337i	$0.120073\pi$
$997$			734.465i			0.736675i	−0.929692	+	0.368337i	$0.879927\pi$
$998$		0			0
$999$			1477.86i			1.47934i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	560.3.f.c.321.4		4
4.3	odd	2		140.3.d.a.41.1	✓	4
7.6	odd	2	inner	560.3.f.c.321.1		4
12.11	even	2		1260.3.j.a.181.3		4
20.3	even	4		700.3.h.b.349.8		8
20.7	even	4		700.3.h.b.349.1		8
20.19	odd	2		700.3.d.d.601.4		4
28.3	even	6		980.3.r.a.901.4		8
28.11	odd	6		980.3.r.a.901.1		8
28.19	even	6		980.3.r.a.521.1		8
28.23	odd	6		980.3.r.a.521.4		8
28.27	even	2		140.3.d.a.41.4	yes	4
84.83	odd	2		1260.3.j.a.181.1		4
140.27	odd	4		700.3.h.b.349.7		8
140.83	odd	4		700.3.h.b.349.2		8
140.139	even	2		700.3.d.d.601.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.3.d.a.41.1	✓	4	4.3	odd	2
140.3.d.a.41.4	yes	4	28.27	even	2
560.3.f.c.321.1		4	7.6	odd	2	inner
560.3.f.c.321.4		4	1.1	even	1	trivial
700.3.d.d.601.1		4	140.139	even	2
700.3.d.d.601.4		4	20.19	odd	2
700.3.h.b.349.1		8	20.7	even	4
700.3.h.b.349.2		8	140.83	odd	4
700.3.h.b.349.7		8	140.27	odd	4
700.3.h.b.349.8		8	20.3	even	4
980.3.r.a.521.1		8	28.19	even	6
980.3.r.a.521.4		8	28.23	odd	6
980.3.r.a.901.1		8	28.11	odd	6
980.3.r.a.901.4		8	28.3	even	6
1260.3.j.a.181.1		4	84.83	odd	2
1260.3.j.a.181.3		4	12.11	even	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:	\( N \)	\(=\)	\( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	560.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.92081i q^{3} -2.23607i q^{5} +(5.53113 + 4.29029i) q^{7} +0.468871 q^{9} +O(q^{10})\)	\(q+2.92081i q^{3} -2.23607i q^{5} +(5.53113 + 4.29029i) q^{7} +0.468871 q^{9} +11.5934 q^{11} -2.92081i q^{13} +6.53113 q^{15} -19.0762i q^{17} -1.73317i q^{19} +(-12.5311 + 16.1554i) q^{21} -0.937742 q^{23} -5.00000 q^{25} +27.6568i q^{27} +41.7179 q^{29} +3.83002i q^{31} +33.8621i q^{33} +(9.59339 - 12.3680i) q^{35} +53.4358 q^{37} +8.53113 q^{39} +28.4807i q^{41} -42.2490 q^{43} -1.04843i q^{45} -17.4283i q^{47} +(12.1868 + 47.4603i) q^{49} +55.7179 q^{51} -23.8755 q^{53} -25.9236i q^{55} +5.06226 q^{57} +95.9264i q^{59} +120.941i q^{61} +(2.59339 + 2.01159i) q^{63} -6.53113 q^{65} -108.374 q^{67} -2.73897i q^{69} +74.4981 q^{71} -80.8621i q^{73} -14.6040i q^{75} +(64.1245 + 49.7390i) q^{77} +14.5311 q^{79} -76.5603 q^{81} +115.827i q^{83} -42.6556 q^{85} +121.850i q^{87} -141.483i q^{89} +(12.5311 - 16.1554i) q^{91} -11.1868 q^{93} -3.87548 q^{95} -12.1435i q^{97} +5.43580 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 6 q^{7} + 18 q^{9}+O(q^{10}) \) 4 * q + 6 * q^7 + 18 * q^9	\( 4 q + 6 q^{7} + 18 q^{9} - 2 q^{11} + 10 q^{15} - 34 q^{21} - 36 q^{23} - 20 q^{25} + 54 q^{29} - 10 q^{35} - 12 q^{37} + 18 q^{39} - 40 q^{43} - 48 q^{49} + 110 q^{51} - 160 q^{53} - 12 q^{57} - 38 q^{63} - 10 q^{65} - 240 q^{67} + 40 q^{71} + 192 q^{77} + 42 q^{79} - 16 q^{81} - 90 q^{85} + 34 q^{91} + 52 q^{93} - 80 q^{95} - 204 q^{99}+O(q^{100}) \) 4 * q + 6 * q^7 + 18 * q^9 - 2 * q^11 + 10 * q^15 - 34 * q^21 - 36 * q^23 - 20 * q^25 + 54 * q^29 - 10 * q^35 - 12 * q^37 + 18 * q^39 - 40 * q^43 - 48 * q^49 + 110 * q^51 - 160 * q^53 - 12 * q^57 - 38 * q^63 - 10 * q^65 - 240 * q^67 + 40 * q^71 + 192 * q^77 + 42 * q^79 - 16 * q^81 - 90 * q^85 + 34 * q^91 + 52 * q^93 - 80 * q^95 - 204 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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