LMFDB - Embedded newform 720.3.l.c.161.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [720,3,Mod(161,720)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(720, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("720.161");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 720 = 2^{4} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	720.l (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:	$19.6185790339$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 9 $ x^4 - 4*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 180)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.4
Root			$-1.58114 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	720.161
Dual form			720.3.l.c.161.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.23607i q^{5} +13.4868 q^{7} +O(q^{10})$	$q+2.23607i q^{5} +13.4868 q^{7} -17.6590i q^{11} -7.48683 q^{13} -16.9706i q^{17} +10.9737 q^{19} -21.9017i q^{23} -5.00000 q^{25} +47.3575i q^{29} +16.9737 q^{31} +30.1575i q^{35} -5.53950 q^{37} -66.3936i q^{41} -38.9737 q^{43} +32.5642i q^{47} +132.895 q^{49} -11.2392i q^{53} +39.4868 q^{55} -31.8757i q^{59} +46.9210 q^{61} -16.7411i q^{65} +76.0000 q^{67} -77.7445i q^{71} +94.9210 q^{73} -238.165i q^{77} +6.92100 q^{79} -62.1509i q^{83} +37.9473 q^{85} +62.2626i q^{89} -100.974 q^{91} +24.5379i q^{95} -124.974 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 16 q^{7}+O(q^{10}) $ 4 * q + 16 * q^7	$ 4 q + 16 q^{7} + 8 q^{13} - 32 q^{19} - 20 q^{25} - 8 q^{31} - 136 q^{37} - 80 q^{43} + 228 q^{49} + 120 q^{55} - 40 q^{61} + 304 q^{67} + 152 q^{73} - 200 q^{79} - 328 q^{91} - 424 q^{97}+O(q^{100}) $ 4 * q + 16 * q^7 + 8 * q^13 - 32 * q^19 - 20 * q^25 - 8 * q^31 - 136 * q^37 - 80 * q^43 + 228 * q^49 + 120 * q^55 - 40 * q^61 + 304 * q^67 + 152 * q^73 - 200 * q^79 - 328 * q^91 - 424 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/720\mathbb{Z}\right)^\times$.

$n$	$181$	$271$	$577$	$641$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	2.23607i	0.447214i
$5$	2.23607i	0.447214i
$6$	0	0
$7$	13.4868	1.92669	0.963345	−	0.268265i	$-0.0864502\pi$
$7$	13.4868	1.92669	0.963345	+	0.268265i	$0.0864502\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 17.6590i	− 1.60537i	−0.596405	−	0.802684i	$-0.703405\pi$
$11$	− 17.6590i	− 1.60537i	0.596405	−	0.802684i	$-0.296595\pi$
$12$	0	0
$13$	−7.48683	−0.575910	−0.287955	−	0.957644i	$-0.592975\pi$
$13$	−7.48683	−0.575910	−0.287955	+	0.957644i	$0.592975\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 16.9706i	− 0.998268i	−0.866525	−	0.499134i	$-0.833651\pi$
$17$	− 16.9706i	− 0.998268i	0.866525	−	0.499134i	$-0.166349\pi$
$18$	0	0
$19$	10.9737	0.577561	0.288781	−	0.957395i	$-0.406750\pi$
$19$	10.9737	0.577561	0.288781	+	0.957395i	$0.406750\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 21.9017i	− 0.952247i	−0.879378	−	0.476124i	$-0.842041\pi$
$23$	− 21.9017i	− 0.952247i	0.879378	−	0.476124i	$-0.157959\pi$
$24$	0	0
$25$	−5.00000	−0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	47.3575i	1.63302i	0.577332	+	0.816509i	$0.304094\pi$
$29$	47.3575i	1.63302i	−0.577332	+	0.816509i	$0.695906\pi$
$30$	0	0
$31$	16.9737	0.547538	0.273769	−	0.961796i	$-0.411730\pi$
$31$	16.9737	0.547538	0.273769	+	0.961796i	$0.411730\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	30.1575i	0.861642i
$36$	0	0
$37$	−5.53950	−0.149716	−0.0748581	−	0.997194i	$-0.523850\pi$
$37$	−5.53950	−0.149716	−0.0748581	+	0.997194i	$0.523850\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 66.3936i	− 1.61935i	−0.586875	−	0.809677i	$-0.699642\pi$
$41$	− 66.3936i	− 1.61935i	0.586875	−	0.809677i	$-0.300358\pi$
$42$	0	0
$43$	−38.9737	−0.906364	−0.453182	−	0.891418i	$-0.649711\pi$
$43$	−38.9737	−0.906364	−0.453182	+	0.891418i	$0.649711\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	32.5642i	0.692854i	0.938077	+	0.346427i	$0.112605\pi$
$47$	32.5642i	0.692854i	−0.938077	+	0.346427i	$0.887395\pi$
$48$	0	0
$49$	132.895	2.71214
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 11.2392i	− 0.212061i	−0.994363	−	0.106030i	$-0.966186\pi$
$53$	− 11.2392i	− 0.212061i	0.994363	−	0.106030i	$-0.0338141\pi$
$54$	0	0
$55$	39.4868	0.717942
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 31.8757i	− 0.540266i	−0.962823	−	0.270133i	$-0.912932\pi$
$59$	− 31.8757i	− 0.540266i	0.962823	−	0.270133i	$-0.0870676\pi$
$60$	0	0
$61$	46.9210	0.769197	0.384598	−	0.923084i	$-0.374340\pi$
$61$	46.9210	0.769197	0.384598	+	0.923084i	$0.374340\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 16.7411i	− 0.257555i
$66$	0	0
$67$	76.0000	1.13433	0.567164	−	0.823605i	$-0.308040\pi$
$67$	76.0000	1.13433	0.567164	+	0.823605i	$0.308040\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 77.7445i	− 1.09499i	−0.836808	−	0.547497i	$-0.815581\pi$
$71$	− 77.7445i	− 1.09499i	0.836808	−	0.547497i	$-0.184419\pi$
$72$	0	0
$73$	94.9210	1.30029	0.650144	−	0.759811i	$-0.274709\pi$
$73$	94.9210	1.30029	0.650144	+	0.759811i	$0.274709\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 238.165i	− 3.09305i
$78$	0	0
$79$	6.92100	0.0876076	0.0438038	−	0.999040i	$-0.486052\pi$
$79$	6.92100	0.0876076	0.0438038	+	0.999040i	$0.486052\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 62.1509i	− 0.748806i	−0.927266	−	0.374403i	$-0.877848\pi$
$83$	− 62.1509i	− 0.748806i	0.927266	−	0.374403i	$-0.122152\pi$
$84$	0	0
$85$	37.9473	0.446439
$86$	0	0
$87$	0	0
$88$	0	0
$89$	62.2626i	0.699580i	0.936828	+	0.349790i	$0.113747\pi$
$89$	62.2626i	0.699580i	−0.936828	+	0.349790i	$0.886253\pi$
$90$	0	0
$91$	−100.974	−1.10960
$92$	0	0
$93$	0	0
$94$	0	0
$95$	24.5379i	0.258293i
$96$	0	0
$97$	−124.974	−1.28839	−0.644194	−	0.764862i	$-0.722807\pi$
$97$	−124.974	−1.28839	−0.644194	+	0.764862i	$0.722807\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	112.262i	1.11151i	0.831347	+	0.555754i	$0.187571\pi$
$101$	112.262i	1.11151i	−0.831347	+	0.555754i	$0.812429\pi$
$102$	0	0
$103$	170.302	1.65342	0.826711	−	0.562627i	$-0.190209\pi$
$103$	170.302	1.65342	0.826711	+	0.562627i	$0.190209\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	93.3381i	0.872319i	0.899869	+	0.436159i	$0.143662\pi$
$107$	93.3381i	0.872319i	−0.899869	+	0.436159i	$0.856338\pi$
$108$	0	0
$109$	−68.8683	−0.631820	−0.315910	−	0.948789i	$-0.602310\pi$
$109$	−68.8683	−0.631820	−0.315910	+	0.948789i	$0.602310\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	110.309i	0.976183i	0.872793	+	0.488091i	$0.162307\pi$
$113$	110.309i	0.976183i	−0.872793	+	0.488091i	$0.837693\pi$
$114$	0	0
$115$	48.9737	0.425858
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 228.879i	− 1.92335i
$120$	0	0
$121$	−190.842	−1.57721
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 11.1803i	− 0.0894427i
$126$	0	0
$127$	65.3815	0.514815	0.257407	−	0.966303i	$-0.417132\pi$
$127$	65.3815	0.514815	0.257407	+	0.966303i	$0.417132\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	47.2458i	0.360655i	0.983607	+	0.180328i	$0.0577158\pi$
$131$	47.2458i	0.360655i	−0.983607	+	0.180328i	$0.942284\pi$
$132$	0	0
$133$	148.000	1.11278
$134$	0	0
$135$	0	0
$136$	0	0
$137$	216.039i	1.57693i	0.615079	+	0.788465i	$0.289124\pi$
$137$	216.039i	1.57693i	−0.615079	+	0.788465i	$0.710876\pi$
$138$	0	0
$139$	−211.842	−1.52404	−0.762022	−	0.647552i	$-0.775793\pi$
$139$	−211.842	−1.52404	−0.762022	+	0.647552i	$0.775793\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	132.210i	0.924548i
$144$	0	0
$145$	−105.895	−0.730308
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 92.5379i	− 0.621060i	−0.950564	−	0.310530i	$-0.899494\pi$
$149$	− 92.5379i	− 0.621060i	0.950564	−	0.310530i	$-0.100506\pi$
$150$	0	0
$151$	123.842	0.820146	0.410073	−	0.912053i	$-0.365503\pi$
$151$	123.842	0.820146	0.410073	+	0.912053i	$0.365503\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	37.9543i	0.244866i
$156$	0	0
$157$	−31.4868	−0.200553	−0.100277	−	0.994960i	$-0.531973\pi$
$157$	−31.4868	−0.200553	−0.100277	+	0.994960i	$0.531973\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 295.384i	− 1.83469i
$162$	0	0
$163$	101.132	0.620440	0.310220	−	0.950665i	$-0.399597\pi$
$163$	101.132	0.620440	0.310220	+	0.950665i	$0.399597\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 147.804i	− 0.885054i	−0.896755	−	0.442527i	$-0.854082\pi$
$167$	− 147.804i	− 0.885054i	0.896755	−	0.442527i	$-0.145918\pi$
$168$	0	0
$169$	−112.947	−0.668327
$170$	0	0
$171$	0	0
$172$	0	0
$173$	332.079i	1.91953i	0.280797	+	0.959767i	$0.409401\pi$
$173$	332.079i	1.91953i	−0.280797	+	0.959767i	$0.590599\pi$
$174$	0	0
$175$	−67.4342	−0.385338
$176$	0	0
$177$	0	0
$178$	0	0
$179$	45.8688i	0.256250i	0.991758	+	0.128125i	$0.0408959\pi$
$179$	45.8688i	0.256250i	−0.991758	+	0.128125i	$0.959104\pi$
$180$	0	0
$181$	132.868	0.734079	0.367040	−	0.930205i	$-0.380371\pi$
$181$	132.868	0.734079	0.367040	+	0.930205i	$0.380371\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 12.3867i	− 0.0669551i
$186$	0	0
$187$	−299.684	−1.60259
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 163.974i	− 0.858504i	−0.903185	−	0.429252i	$-0.858777\pi$
$191$	− 163.974i	− 0.858504i	0.903185	−	0.429252i	$-0.141223\pi$
$192$	0	0
$193$	110.000	0.569948	0.284974	−	0.958535i	$-0.408015\pi$
$193$	110.000	0.569948	0.284974	+	0.958535i	$0.408015\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 211.108i	− 1.07162i	−0.844340	−	0.535808i	$-0.820007\pi$
$197$	− 211.108i	− 1.07162i	0.844340	−	0.535808i	$-0.179993\pi$
$198$	0	0
$199$	18.1053	0.0909816	0.0454908	−	0.998965i	$-0.485515\pi$
$199$	18.1053	0.0909816	0.0454908	+	0.998965i	$0.485515\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	638.703i	3.14632i
$204$	0	0
$205$	148.460	0.724198
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 193.785i	− 0.927199i
$210$	0	0
$211$	−341.579	−1.61886	−0.809428	−	0.587219i	$-0.800223\pi$
$211$	−341.579	−1.61886	−0.809428	+	0.587219i	$0.800223\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 87.1478i	− 0.405338i
$216$	0	0
$217$	228.921	1.05494
$218$	0	0
$219$	0	0
$220$	0	0
$221$	127.056i	0.574913i
$222$	0	0
$223$	−59.3288	−0.266049	−0.133024	−	0.991113i	$-0.542469\pi$
$223$	−59.3288	−0.266049	−0.133024	+	0.991113i	$0.542469\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	115.593i	0.509221i	0.967044	+	0.254610i	$0.0819472\pi$
$227$	115.593i	0.509221i	−0.967044	+	0.254610i	$0.918053\pi$
$228$	0	0
$229$	−253.895	−1.10871	−0.554355	−	0.832280i	$-0.687035\pi$
$229$	−253.895	−1.10871	−0.554355	+	0.832280i	$0.687035\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	206.401i	0.885840i	0.896561	+	0.442920i	$0.146057\pi$
$233$	206.401i	0.885840i	−0.896561	+	0.442920i	$0.853943\pi$
$234$	0	0
$235$	−72.8157	−0.309854
$236$	0	0
$237$	0	0
$238$	0	0
$239$	346.296i	1.44894i	0.689308	+	0.724469i	$0.257915\pi$
$239$	346.296i	1.44894i	−0.689308	+	0.724469i	$0.742085\pi$
$240$	0	0
$241$	−182.053	−0.755405	−0.377703	−	0.925927i	$-0.623286\pi$
$241$	−182.053	−0.755405	−0.377703	+	0.925927i	$0.623286\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	297.162i	1.21290i
$246$	0	0
$247$	−82.1580	−0.332623
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 140.807i	− 0.560985i	−0.959856	−	0.280493i	$-0.909502\pi$
$251$	− 140.807i	− 0.560985i	0.959856	−	0.280493i	$-0.0904979\pi$
$252$	0	0
$253$	−386.763	−1.52871
$254$	0	0
$255$	0	0
$256$	0	0
$257$	292.630i	1.13864i	0.822116	+	0.569320i	$0.192793\pi$
$257$	292.630i	1.13864i	−0.822116	+	0.569320i	$0.807207\pi$
$258$	0	0
$259$	−74.7103	−0.288457
$260$	0	0
$261$	0	0
$262$	0	0
$263$	270.952i	1.03024i	0.857119	+	0.515118i	$0.172252\pi$
$263$	270.952i	1.03024i	−0.857119	+	0.515118i	$0.827748\pi$
$264$	0	0
$265$	25.1317	0.0948365
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 275.083i	− 1.02261i	−0.859398	−	0.511307i	$-0.829162\pi$
$269$	− 275.083i	− 1.02261i	0.859398	−	0.511307i	$-0.170838\pi$
$270$	0	0
$271$	−248.158	−0.915712	−0.457856	−	0.889026i	$-0.651383\pi$
$271$	−248.158	−0.915712	−0.457856	+	0.889026i	$0.651383\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	88.2952i	0.321074i
$276$	0	0
$277$	−342.039	−1.23480	−0.617399	−	0.786650i	$-0.711814\pi$
$277$	−342.039	−1.23480	−0.617399	+	0.786650i	$0.711814\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 352.139i	− 1.25316i	−0.779355	−	0.626582i	$-0.784453\pi$
$281$	− 352.139i	− 1.25316i	0.779355	−	0.626582i	$-0.215547\pi$
$282$	0	0
$283$	−249.631	−0.882089	−0.441045	−	0.897485i	$-0.645392\pi$
$283$	−249.631	−0.882089	−0.441045	+	0.897485i	$0.645392\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 895.439i	− 3.12000i
$288$	0	0
$289$	1.00000	0.00346021
$290$	0	0
$291$	0	0
$292$	0	0
$293$	37.4953i	0.127970i	0.997951	+	0.0639851i	$0.0203810\pi$
$293$	37.4953i	0.127970i	−0.997951	+	0.0639851i	$0.979619\pi$
$294$	0	0
$295$	71.2762	0.241614
$296$	0	0
$297$	0	0
$298$	0	0
$299$	163.974i	0.548409i
$300$	0	0
$301$	−525.631	−1.74628
$302$	0	0
$303$	0	0
$304$	0	0
$305$	104.919i	0.343995i
$306$	0	0
$307$	−457.842	−1.49134	−0.745671	−	0.666314i	$-0.767871\pi$
$307$	−457.842	−1.49134	−0.745671	+	0.666314i	$0.767871\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 8.48528i	− 0.0272839i	−0.999907	−	0.0136419i	$-0.995658\pi$
$311$	− 8.48528i	− 0.0272839i	0.999907	−	0.0136419i	$-0.00434250\pi$
$312$	0	0
$313$	−1.57866	−0.00504363	−0.00252181	−	0.999997i	$-0.500803\pi$
$313$	−1.57866	−0.00504363	−0.00252181	+	0.999997i	$0.500803\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	529.195i	1.66938i	0.550717	+	0.834692i	$0.314354\pi$
$317$	529.195i	1.66938i	−0.550717	+	0.834692i	$0.685646\pi$
$318$	0	0
$319$	836.289	2.62160
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 186.229i	− 0.576561i
$324$	0	0
$325$	37.4342	0.115182
$326$	0	0
$327$	0	0
$328$	0	0
$329$	439.187i	1.33492i
$330$	0	0
$331$	390.763	1.18055	0.590276	−	0.807201i	$-0.299019\pi$
$331$	390.763	1.18055	0.590276	+	0.807201i	$0.299019\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	169.941i	0.507287i
$336$	0	0
$337$	414.710	1.23059	0.615297	−	0.788295i	$-0.289036\pi$
$337$	414.710	1.23059	0.615297	+	0.788295i	$0.289036\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 299.739i	− 0.878999i
$342$	0	0
$343$	1131.47	3.29876
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 128.656i	− 0.370767i	−0.982666	−	0.185384i	$-0.940647\pi$
$347$	− 128.656i	− 0.370767i	0.982666	−	0.185384i	$-0.0593528\pi$
$348$	0	0
$349$	−37.0790	−0.106244	−0.0531218	−	0.998588i	$-0.516917\pi$
$349$	−37.0790	−0.106244	−0.0531218	+	0.998588i	$0.516917\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 190.584i	− 0.539897i	−0.962875	−	0.269949i	$-0.912993\pi$
$353$	− 190.584i	− 0.539897i	0.962875	−	0.269949i	$-0.0870067\pi$
$354$	0	0
$355$	173.842	0.489696
$356$	0	0
$357$	0	0
$358$	0	0
$359$	254.782i	0.709699i	0.934923	+	0.354849i	$0.115468\pi$
$359$	254.782i	0.709699i	−0.934923	+	0.354849i	$0.884532\pi$
$360$	0	0
$361$	−240.579	−0.666423
$362$	0	0
$363$	0	0
$364$	0	0
$365$	212.250i	0.581506i
$366$	0	0
$367$	−300.460	−0.818693	−0.409347	−	0.912379i	$-0.634243\pi$
$367$	−300.460	−0.818693	−0.409347	+	0.912379i	$0.634243\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 151.582i	− 0.408576i
$372$	0	0
$373$	704.092	1.88765	0.943823	−	0.330452i	$-0.107201\pi$
$373$	704.092	1.88765	0.943823	+	0.330452i	$0.107201\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 354.558i	− 0.940472i
$378$	0	0
$379$	−333.789	−0.880711	−0.440355	−	0.897824i	$-0.645148\pi$
$379$	−333.789	−0.880711	−0.440355	+	0.897824i	$0.645148\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 260.996i	− 0.681453i	−0.940163	−	0.340726i	$-0.889327\pi$
$383$	− 260.996i	− 0.681453i	0.940163	−	0.340726i	$-0.110673\pi$
$384$	0	0
$385$	532.552	1.38325
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 35.8949i	− 0.0922747i	−0.998935	−	0.0461373i	$-0.985309\pi$
$389$	− 35.8949i	− 0.0922747i	0.998935	−	0.0461373i	$-0.0146912\pi$
$390$	0	0
$391$	−371.684	−0.950598
$392$	0	0
$393$	0	0
$394$	0	0
$395$	15.4758i	0.0391793i
$396$	0	0
$397$	−90.3552	−0.227595	−0.113797	−	0.993504i	$-0.536301\pi$
$397$	−90.3552	−0.227595	−0.113797	+	0.993504i	$0.536301\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	43.6917i	0.108957i	0.998515	+	0.0544784i	$0.0173496\pi$
$401$	43.6917i	0.108957i	−0.998515	+	0.0544784i	$0.982650\pi$
$402$	0	0
$403$	−127.079	−0.315333
$404$	0	0
$405$	0	0
$406$	0	0
$407$	97.8223i	0.240350i
$408$	0	0
$409$	−260.053	−0.635826	−0.317913	−	0.948120i	$-0.602982\pi$
$409$	−260.053	−0.635826	−0.317913	+	0.948120i	$0.602982\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 429.902i	− 1.04092i
$414$	0	0
$415$	138.974	0.334876
$416$	0	0
$417$	0	0
$418$	0	0
$419$	263.285i	0.628366i	0.949362	+	0.314183i	$0.101731\pi$
$419$	263.285i	0.628366i	−0.949362	+	0.314183i	$0.898269\pi$
$420$	0	0
$421$	367.210	0.872233	0.436116	−	0.899890i	$-0.356354\pi$
$421$	367.210	0.872233	0.436116	+	0.899890i	$0.356354\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	84.8528i	0.199654i
$426$	0	0
$427$	632.816	1.48200
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 690.768i	− 1.60271i	−0.598189	−	0.801355i	$-0.704113\pi$
$431$	− 690.768i	− 1.60271i	0.598189	−	0.801355i	$-0.295887\pi$
$432$	0	0
$433$	−117.500	−0.271363	−0.135682	−	0.990752i	$-0.543322\pi$
$433$	−117.500	−0.271363	−0.135682	+	0.990752i	$0.543322\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 240.342i	− 0.549981i
$438$	0	0
$439$	−50.0000	−0.113895	−0.0569476	−	0.998377i	$-0.518137\pi$
$439$	−50.0000	−0.113895	−0.0569476	+	0.998377i	$0.518137\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	526.794i	1.18915i	0.804040	+	0.594576i	$0.202680\pi$
$443$	526.794i	1.18915i	−0.804040	+	0.594576i	$0.797320\pi$
$444$	0	0
$445$	−139.223	−0.312862
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 321.882i	− 0.716887i	−0.933552	−	0.358443i	$-0.883308\pi$
$449$	− 321.882i	− 0.716887i	0.933552	−	0.358443i	$-0.116692\pi$
$450$	0	0
$451$	−1172.45	−2.59966
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 225.784i	− 0.496229i
$456$	0	0
$457$	−433.079	−0.947656	−0.473828	−	0.880617i	$-0.657128\pi$
$457$	−433.079	−0.947656	−0.473828	+	0.880617i	$0.657128\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 486.768i	− 1.05590i	−0.849277	−	0.527948i	$-0.822961\pi$
$461$	− 486.768i	− 1.05590i	0.849277	−	0.527948i	$-0.177039\pi$
$462$	0	0
$463$	−423.223	−0.914090	−0.457045	−	0.889444i	$-0.651092\pi$
$463$	−423.223	−0.914090	−0.457045	+	0.889444i	$0.651092\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	295.831i	0.633472i	0.948514	+	0.316736i	$0.102587\pi$
$467$	295.831i	0.633472i	−0.948514	+	0.316736i	$0.897413\pi$
$468$	0	0
$469$	1025.00	2.18550
$470$	0	0
$471$	0	0
$472$	0	0
$473$	688.238i	1.45505i
$474$	0	0
$475$	−54.8683	−0.115512
$476$	0	0
$477$	0	0
$478$	0	0
$479$	266.728i	0.556843i	0.960459	+	0.278421i	$0.0898112\pi$
$479$	266.728i	0.556843i	−0.960459	+	0.278421i	$0.910189\pi$
$480$	0	0
$481$	41.4733	0.0862231
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 279.450i	− 0.576185i
$486$	0	0
$487$	−247.750	−0.508727	−0.254364	−	0.967109i	$-0.581866\pi$
$487$	−247.750	−0.508727	−0.254364	+	0.967109i	$0.581866\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 170.394i	− 0.347035i	−0.984831	−	0.173517i	$-0.944487\pi$
$491$	− 170.394i	− 0.347035i	0.984831	−	0.173517i	$-0.0555133\pi$
$492$	0	0
$493$	803.684	1.63019
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 1048.53i	− 2.10971i
$498$	0	0
$499$	−47.3950	−0.0949800	−0.0474900	−	0.998872i	$-0.515122\pi$
$499$	−47.3950	−0.0949800	−0.0474900	+	0.998872i	$0.515122\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	321.064i	0.638298i	0.947705	+	0.319149i	$0.103397\pi$
$503$	321.064i	0.638298i	−0.947705	+	0.319149i	$0.896603\pi$
$504$	0	0
$505$	−251.026	−0.497082
$506$	0	0
$507$	0	0
$508$	0	0
$509$	122.125i	0.239931i	0.992778	+	0.119965i	$0.0382783\pi$
$509$	122.125i	0.239931i	−0.992778	+	0.119965i	$0.961722\pi$
$510$	0	0
$511$	1280.18	2.50525
$512$	0	0
$513$	0	0
$514$	0	0
$515$	380.808i	0.739433i
$516$	0	0
$517$	575.052	1.11229
$518$	0	0
$519$	0	0
$520$	0	0
$521$	79.4567i	0.152508i	0.997088	+	0.0762540i	$0.0242960\pi$
$521$	79.4567i	0.152508i	−0.997088	+	0.0762540i	$0.975704\pi$
$522$	0	0
$523$	864.605	1.65316	0.826582	−	0.562816i	$-0.190282\pi$
$523$	864.605	1.65316	0.826582	+	0.562816i	$0.190282\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 288.053i	− 0.546589i
$528$	0	0
$529$	49.3160	0.0932250
$530$	0	0
$531$	0	0
$532$	0	0
$533$	497.077i	0.932603i
$534$	0	0
$535$	−208.710	−0.390113
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 2346.79i	− 4.35398i
$540$	0	0
$541$	33.6057	0.0621177	0.0310589	−	0.999518i	$-0.490112\pi$
$541$	33.6057	0.0621177	0.0310589	+	0.999518i	$0.490112\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 153.994i	− 0.282558i
$546$	0	0
$547$	559.079	1.02208	0.511041	−	0.859556i	$-0.329260\pi$
$547$	559.079	1.02208	0.511041	+	0.859556i	$0.329260\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	519.686i	0.943168i
$552$	0	0
$553$	93.3423	0.168793
$554$	0	0
$555$	0	0
$556$	0	0
$557$	310.308i	0.557105i	0.960421	+	0.278553i	$0.0898547\pi$
$557$	310.308i	0.557105i	−0.960421	+	0.278553i	$0.910145\pi$
$558$	0	0
$559$	291.789	0.521984
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 566.690i	− 1.00655i	−0.864125	−	0.503277i	$-0.832128\pi$
$563$	− 566.690i	− 1.00655i	0.864125	−	0.503277i	$-0.167872\pi$
$564$	0	0
$565$	−246.658	−0.436562
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 134.052i	− 0.235593i	−0.993038	−	0.117796i	$-0.962417\pi$
$569$	− 134.052i	− 0.235593i	0.993038	−	0.117796i	$-0.0375830\pi$
$570$	0	0
$571$	661.920	1.15923	0.579615	−	0.814890i	$-0.303203\pi$
$571$	661.920	1.15923	0.579615	+	0.814890i	$0.303203\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	109.508i	0.190449i
$576$	0	0
$577$	76.7630	0.133038	0.0665191	−	0.997785i	$-0.478811\pi$
$577$	76.7630	0.133038	0.0665191	+	0.997785i	$0.478811\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 838.219i	− 1.44272i
$582$	0	0
$583$	−198.474	−0.340436
$584$	0	0
$585$	0	0
$586$	0	0
$587$	684.554i	1.16619i	0.812404	+	0.583095i	$0.198159\pi$
$587$	684.554i	1.16619i	−0.812404	+	0.583095i	$0.801841\pi$
$588$	0	0
$589$	186.263	0.316237
$590$	0	0
$591$	0	0
$592$	0	0
$593$	592.146i	0.998560i	0.866441	+	0.499280i	$0.166402\pi$
$593$	592.146i	0.998560i	−0.866441	+	0.499280i	$0.833598\pi$
$594$	0	0
$595$	511.789	0.860150
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 1012.76i	− 1.69075i	−0.534169	−	0.845377i	$-0.679376\pi$
$599$	− 1012.76i	− 1.69075i	0.534169	−	0.845377i	$-0.320624\pi$
$600$	0	0
$601$	323.579	0.538400	0.269200	−	0.963084i	$-0.413241\pi$
$601$	323.579	0.538400	0.269200	+	0.963084i	$0.413241\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 426.736i	− 0.705348i
$606$	0	0
$607$	799.828	1.31767	0.658837	−	0.752285i	$-0.271049\pi$
$607$	799.828	1.31767	0.658837	+	0.752285i	$0.271049\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 243.802i	− 0.399022i
$612$	0	0
$613$	−680.302	−1.10979	−0.554896	−	0.831920i	$-0.687242\pi$
$613$	−680.302	−1.10979	−0.554896	+	0.831920i	$0.687242\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 264.868i	− 0.429283i	−0.976693	−	0.214641i	$-0.931142\pi$
$617$	− 264.868i	− 0.429283i	0.976693	−	0.214641i	$-0.0688583\pi$
$618$	0	0
$619$	−535.842	−0.865658	−0.432829	−	0.901476i	$-0.642485\pi$
$619$	−535.842	−0.865658	−0.432829	+	0.901476i	$0.642485\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	839.726i	1.34787i
$624$	0	0
$625$	25.0000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	94.0085i	0.149457i
$630$	0	0
$631$	−307.026	−0.486571	−0.243286	−	0.969955i	$-0.578225\pi$
$631$	−307.026	−0.486571	−0.243286	+	0.969955i	$0.578225\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	146.197i	0.230232i
$636$	0	0
$637$	−994.960	−1.56195
$638$	0	0
$639$	0	0
$640$	0	0
$641$	1246.33i	1.94435i	0.234248	+	0.972177i	$0.424737\pi$
$641$	1246.33i	1.94435i	−0.234248	+	0.972177i	$0.575263\pi$
$642$	0	0
$643$	−155.395	−0.241672	−0.120836	−	0.992672i	$-0.538557\pi$
$643$	−155.395	−0.241672	−0.120836	+	0.992672i	$0.538557\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 142.649i	− 0.220478i	−0.993905	−	0.110239i	$-0.964838\pi$
$647$	− 142.649i	− 0.220478i	0.993905	−	0.110239i	$-0.0351617\pi$
$648$	0	0
$649$	−562.894	−0.867325
$650$	0	0
$651$	0	0
$652$	0	0
$653$	924.579i	1.41589i	0.706266	+	0.707947i	$0.250378\pi$
$653$	924.579i	1.41589i	−0.706266	+	0.707947i	$0.749622\pi$
$654$	0	0
$655$	−105.645	−0.161290
$656$	0	0
$657$	0	0
$658$	0	0
$659$	352.046i	0.534212i	0.963667	+	0.267106i	$0.0860674\pi$
$659$	352.046i	0.534212i	−0.963667	+	0.267106i	$0.913933\pi$
$660$	0	0
$661$	−642.921	−0.972649	−0.486325	−	0.873778i	$-0.661663\pi$
$661$	−642.921	−0.972649	−0.486325	+	0.873778i	$0.661663\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	330.938i	0.497651i
$666$	0	0
$667$	1037.21	1.55504
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 828.580i	− 1.23484i
$672$	0	0
$673$	982.605	1.46004	0.730019	−	0.683427i	$-0.239511\pi$
$673$	982.605	1.46004	0.730019	+	0.683427i	$0.239511\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 386.769i	− 0.571298i	−0.958334	−	0.285649i	$-0.907791\pi$
$677$	− 386.769i	− 0.571298i	0.958334	−	0.285649i	$-0.0922092\pi$
$678$	0	0
$679$	−1685.50	−2.48233
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 504.539i	− 0.738710i	−0.929288	−	0.369355i	$-0.879579\pi$
$683$	− 504.539i	− 0.738710i	0.929288	−	0.369355i	$-0.120421\pi$
$684$	0	0
$685$	−483.079	−0.705225
$686$	0	0
$687$	0	0
$688$	0	0
$689$	84.1462i	0.122128i
$690$	0	0
$691$	271.079	0.392300	0.196150	−	0.980574i	$-0.437156\pi$
$691$	271.079	0.392300	0.196150	+	0.980574i	$0.437156\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 473.693i	− 0.681573i
$696$	0	0
$697$	−1126.74	−1.61655
$698$	0	0
$699$	0	0
$700$	0	0
$701$	358.559i	0.511496i	0.966743	+	0.255748i	$0.0823218\pi$
$701$	358.559i	0.511496i	−0.966743	+	0.255748i	$0.917678\pi$
$702$	0	0
$703$	−60.7886	−0.0864703
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1514.06i	2.14153i
$708$	0	0
$709$	−733.579	−1.03467	−0.517333	−	0.855784i	$-0.673075\pi$
$709$	−733.579	−1.03467	−0.517333	+	0.855784i	$0.673075\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 371.752i	− 0.521391i
$714$	0	0
$715$	−295.631	−0.413470
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1108.37i	1.54155i	0.637110	+	0.770773i	$0.280130\pi$
$719$	1108.37i	1.54155i	−0.637110	+	0.770773i	$0.719870\pi$
$720$	0	0
$721$	2296.84	3.18563
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 236.788i	− 0.326604i
$726$	0	0
$727$	−1265.04	−1.74008	−0.870040	−	0.492981i	$-0.835907\pi$
$727$	−1265.04	−1.74008	−0.870040	+	0.492981i	$0.835907\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	661.405i	0.904795i
$732$	0	0
$733$	−586.749	−0.800477	−0.400238	−	0.916411i	$-0.631073\pi$
$733$	−586.749	−0.800477	−0.400238	+	0.916411i	$0.631073\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 1342.09i	− 1.82101i
$738$	0	0
$739$	215.973	0.292250	0.146125	−	0.989266i	$-0.453320\pi$
$739$	215.973	0.292250	0.146125	+	0.989266i	$0.453320\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	599.961i	0.807484i	0.914873	+	0.403742i	$0.132291\pi$
$743$	599.961i	0.807484i	−0.914873	+	0.403742i	$0.867709\pi$
$744$	0	0
$745$	206.921	0.277746
$746$	0	0
$747$	0	0
$748$	0	0
$749$	1258.84i	1.68069i
$750$	0	0
$751$	−813.657	−1.08343	−0.541716	−	0.840562i	$-0.682225\pi$
$751$	−813.657	−1.08343	−0.541716	+	0.840562i	$0.682225\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	276.919i	0.366780i
$756$	0	0
$757$	1070.43	1.41405	0.707023	−	0.707190i	$-0.250038\pi$
$757$	1070.43	1.41405	0.707023	+	0.707190i	$0.250038\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	654.892i	0.860567i	0.902694	+	0.430284i	$0.141586\pi$
$761$	654.892i	0.860567i	−0.902694	+	0.430284i	$0.858414\pi$
$762$	0	0
$763$	−928.816	−1.21732
$764$	0	0
$765$	0	0
$766$	0	0
$767$	238.648i	0.311144i
$768$	0	0
$769$	682.631	0.887686	0.443843	−	0.896105i	$-0.353615\pi$
$769$	682.631	0.887686	0.443843	+	0.896105i	$0.353615\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 770.690i	− 0.997012i	−0.866886	−	0.498506i	$-0.833882\pi$
$773$	− 770.690i	− 0.997012i	0.866886	−	0.498506i	$-0.166118\pi$
$774$	0	0
$775$	−84.8683	−0.109508
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 728.581i	− 0.935277i
$780$	0	0
$781$	−1372.89	−1.75787
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 70.4067i	− 0.0896901i
$786$	0	0
$787$	253.869	0.322578	0.161289	−	0.986907i	$-0.448435\pi$
$787$	253.869	0.322578	0.161289	+	0.986907i	$0.448435\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	1487.71i	1.88080i
$792$	0	0
$793$	−351.290	−0.442988
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1055.86i	1.32479i	0.749154	+	0.662396i	$0.230460\pi$
$797$	1055.86i	1.32479i	−0.749154	+	0.662396i	$0.769540\pi$
$798$	0	0
$799$	552.632	0.691655
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 1676.21i	− 2.08744i
$804$	0	0
$805$	660.500	0.820496
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 506.698i	− 0.626326i	−0.949699	−	0.313163i	$-0.898611\pi$
$809$	− 506.698i	− 0.626326i	0.949699	−	0.313163i	$-0.101389\pi$
$810$	0	0
$811$	−537.473	−0.662729	−0.331365	−	0.943503i	$-0.607509\pi$
$811$	−537.473	−0.662729	−0.331365	+	0.943503i	$0.607509\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	226.137i	0.277469i
$816$	0	0
$817$	−427.684	−0.523481
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 257.219i	− 0.313299i	−0.987654	−	0.156650i	$-0.949931\pi$
$821$	− 257.219i	− 0.313299i	0.987654	−	0.156650i	$-0.0500694\pi$
$822$	0	0
$823$	367.013	0.445945	0.222973	−	0.974825i	$-0.428424\pi$
$823$	367.013	0.445945	0.222973	+	0.974825i	$0.428424\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 122.254i	− 0.147829i	−0.997265	−	0.0739144i	$-0.976451\pi$
$827$	− 122.254i	− 0.147829i	0.997265	−	0.0739144i	$-0.0235492\pi$
$828$	0	0
$829$	335.237	0.404387	0.202194	−	0.979346i	$-0.435193\pi$
$829$	335.237	0.404387	0.202194	+	0.979346i	$0.435193\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 2255.30i	− 2.70744i
$834$	0	0
$835$	330.500	0.395808
$836$	0	0
$837$	0	0
$838$	0	0
$839$	774.021i	0.922552i	0.887257	+	0.461276i	$0.152608\pi$
$839$	774.021i	0.922552i	−0.887257	+	0.461276i	$0.847392\pi$
$840$	0	0
$841$	−1401.74	−1.66675
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 252.558i	− 0.298885i
$846$	0	0
$847$	−2573.85	−3.03879
$848$	0	0
$849$	0	0
$850$	0	0
$851$	121.324i	0.142567i
$852$	0	0
$853$	764.591	0.896356	0.448178	−	0.893944i	$-0.352073\pi$
$853$	764.591	0.896356	0.448178	+	0.893944i	$0.352073\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1516.15i	1.76913i	0.466413	+	0.884567i	$0.345546\pi$
$857$	1516.15i	1.76913i	−0.466413	+	0.884567i	$0.654454\pi$
$858$	0	0
$859$	−433.684	−0.504871	−0.252435	−	0.967614i	$-0.581232\pi$
$859$	−433.684	−0.504871	−0.252435	+	0.967614i	$0.581232\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 1255.50i	− 1.45481i	−0.686206	−	0.727407i	$-0.740725\pi$
$863$	− 1255.50i	− 1.45481i	0.686206	−	0.727407i	$-0.259275\pi$
$864$	0	0
$865$	−742.552	−0.858442
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 122.218i	− 0.140642i
$870$	0	0
$871$	−568.999	−0.653271
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 150.787i	− 0.172328i
$876$	0	0
$877$	1003.59	1.14435	0.572173	−	0.820133i	$-0.306100\pi$
$877$	1003.59	1.14435	0.572173	+	0.820133i	$0.306100\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 737.214i	− 0.836792i	−0.908265	−	0.418396i	$-0.862592\pi$
$881$	− 737.214i	− 0.836792i	0.908265	−	0.418396i	$-0.137408\pi$
$882$	0	0
$883$	−336.394	−0.380968	−0.190484	−	0.981690i	$-0.561006\pi$
$883$	−336.394	−0.380968	−0.190484	+	0.981690i	$0.561006\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 793.132i	− 0.894174i	−0.894491	−	0.447087i	$-0.852461\pi$
$887$	− 793.132i	− 0.894174i	0.894491	−	0.447087i	$-0.147539\pi$
$888$	0	0
$889$	881.789	0.991889
$890$	0	0
$891$	0	0
$892$	0	0
$893$	357.348i	0.400166i
$894$	0	0
$895$	−102.566	−0.114599
$896$	0	0
$897$	0	0
$898$	0	0
$899$	803.831i	0.894139i
$900$	0	0
$901$	−190.736	−0.211694
$902$	0	0
$903$	0	0
$904$	0	0
$905$	297.103i	0.328290i
$906$	0	0
$907$	1079.66	1.19036	0.595180	−	0.803592i	$-0.297081\pi$
$907$	1079.66	1.19036	0.595180	+	0.803592i	$0.297081\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1473.69i	1.61766i	0.588045	+	0.808828i	$0.299898\pi$
$911$	1473.69i	1.61766i	−0.588045	+	0.808828i	$0.700102\pi$
$912$	0	0
$913$	−1097.53	−1.20211
$914$	0	0
$915$	0	0
$916$	0	0
$917$	637.196i	0.694871i
$918$	0	0
$919$	1602.92	1.74420	0.872101	−	0.489327i	$-0.162757\pi$
$919$	1602.92	1.74420	0.872101	+	0.489327i	$0.162757\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	582.060i	0.630618i
$924$	0	0
$925$	27.6975	0.0299432
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1033.98i	1.11300i	0.830848	+	0.556499i	$0.187856\pi$
$929$	1033.98i	1.11300i	−0.830848	+	0.556499i	$0.812144\pi$
$930$	0	0
$931$	1458.34	1.56642
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 670.114i	− 0.716699i
$936$	0	0
$937$	−1258.05	−1.34264	−0.671319	−	0.741169i	$-0.734272\pi$
$937$	−1258.05	−1.34264	−0.671319	+	0.741169i	$0.734272\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	762.875i	0.810707i	0.914160	+	0.405353i	$0.132852\pi$
$941$	762.875i	0.810707i	−0.914160	+	0.405353i	$0.867148\pi$
$942$	0	0
$943$	−1454.13	−1.54203
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 1023.55i	− 1.08084i	−0.841396	−	0.540419i	$-0.818266\pi$
$947$	− 1023.55i	− 1.08084i	0.841396	−	0.540419i	$-0.181734\pi$
$948$	0	0
$949$	−710.658	−0.748849
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 63.7513i	− 0.0668954i	−0.999440	−	0.0334477i	$-0.989351\pi$
$953$	− 63.7513i	− 0.0668954i	0.999440	−	0.0334477i	$-0.0106487\pi$
$954$	0	0
$955$	366.658	0.383935
$956$	0	0
$957$	0	0
$958$	0	0
$959$	2913.69i	3.03826i
$960$	0	0
$961$	−672.895	−0.700203
$962$	0	0
$963$	0	0
$964$	0	0
$965$	245.967i	0.254889i
$966$	0	0
$967$	505.986	0.523254	0.261627	−	0.965169i	$-0.415741\pi$
$967$	505.986	0.523254	0.261627	+	0.965169i	$0.415741\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 1803.51i	− 1.85738i	−0.370862	−	0.928688i	$-0.620938\pi$
$971$	− 1803.51i	− 1.85738i	0.370862	−	0.928688i	$-0.379062\pi$
$972$	0	0
$973$	−2857.08	−2.93636
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1844.54i	1.88797i	0.329994	+	0.943983i	$0.392953\pi$
$977$	1844.54i	1.88797i	−0.329994	+	0.943983i	$0.607047\pi$
$978$	0	0
$979$	1099.50	1.12308
$980$	0	0
$981$	0	0
$982$	0	0
$983$	592.146i	0.602386i	0.953563	+	0.301193i	$0.0973849\pi$
$983$	592.146i	0.602386i	−0.953563	+	0.301193i	$0.902615\pi$
$984$	0	0
$985$	472.053	0.479241
$986$	0	0
$987$	0	0
$988$	0	0
$989$	853.589i	0.863083i
$990$	0	0
$991$	−961.684	−0.970418	−0.485209	−	0.874398i	$-0.661256\pi$
$991$	−961.684	−0.970418	−0.485209	+	0.874398i	$0.661256\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	40.4848i	0.0406882i
$996$	0	0
$997$	899.670	0.902378	0.451189	−	0.892429i	$-0.351000\pi$
$997$	899.670	0.902378	0.451189	+	0.892429i	$0.351000\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	720.3.l.c.161.4		4
3.2	odd	2	inner	720.3.l.c.161.2		4
4.3	odd	2		180.3.g.a.161.3	yes	4
5.2	odd	4		3600.3.c.k.449.7		8
5.3	odd	4		3600.3.c.k.449.1		8
5.4	even	2		3600.3.l.n.1601.1		4
8.3	odd	2		2880.3.l.b.1601.1		4
8.5	even	2		2880.3.l.f.1601.2		4
12.11	even	2		180.3.g.a.161.1	✓	4
15.2	even	4		3600.3.c.k.449.8		8
15.8	even	4		3600.3.c.k.449.2		8
15.14	odd	2		3600.3.l.n.1601.2		4
20.3	even	4		900.3.b.b.449.8		8
20.7	even	4		900.3.b.b.449.2		8
20.19	odd	2		900.3.g.d.701.4		4
24.5	odd	2		2880.3.l.f.1601.4		4
24.11	even	2		2880.3.l.b.1601.3		4
36.7	odd	6		1620.3.o.f.701.2		8
36.11	even	6		1620.3.o.f.701.4		8
36.23	even	6		1620.3.o.f.1241.2		8
36.31	odd	6		1620.3.o.f.1241.4		8
60.23	odd	4		900.3.b.b.449.7		8
60.47	odd	4		900.3.b.b.449.1		8
60.59	even	2		900.3.g.d.701.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.3.g.a.161.1	✓	4	12.11	even	2
180.3.g.a.161.3	yes	4	4.3	odd	2
720.3.l.c.161.2		4	3.2	odd	2	inner
720.3.l.c.161.4		4	1.1	even	1	trivial
900.3.b.b.449.1		8	60.47	odd	4
900.3.b.b.449.2		8	20.7	even	4
900.3.b.b.449.7		8	60.23	odd	4
900.3.b.b.449.8		8	20.3	even	4
900.3.g.d.701.3		4	60.59	even	2
900.3.g.d.701.4		4	20.19	odd	2
1620.3.o.f.701.2		8	36.7	odd	6
1620.3.o.f.701.4		8	36.11	even	6
1620.3.o.f.1241.2		8	36.23	even	6
1620.3.o.f.1241.4		8	36.31	odd	6
2880.3.l.b.1601.1		4	8.3	odd	2
2880.3.l.b.1601.3		4	24.11	even	2
2880.3.l.f.1601.2		4	8.5	even	2
2880.3.l.f.1601.4		4	24.5	odd	2
3600.3.c.k.449.1		8	5.3	odd	4
3600.3.c.k.449.2		8	15.8	even	4
3600.3.c.k.449.7		8	5.2	odd	4
3600.3.c.k.449.8		8	15.2	even	4
3600.3.l.n.1601.1		4	5.4	even	2
3600.3.l.n.1601.2		4	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	720.l (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.23607i q^{5} +13.4868 q^{7} +O(q^{10})\)	\(q+2.23607i q^{5} +13.4868 q^{7} -17.6590i q^{11} -7.48683 q^{13} -16.9706i q^{17} +10.9737 q^{19} -21.9017i q^{23} -5.00000 q^{25} +47.3575i q^{29} +16.9737 q^{31} +30.1575i q^{35} -5.53950 q^{37} -66.3936i q^{41} -38.9737 q^{43} +32.5642i q^{47} +132.895 q^{49} -11.2392i q^{53} +39.4868 q^{55} -31.8757i q^{59} +46.9210 q^{61} -16.7411i q^{65} +76.0000 q^{67} -77.7445i q^{71} +94.9210 q^{73} -238.165i q^{77} +6.92100 q^{79} -62.1509i q^{83} +37.9473 q^{85} +62.2626i q^{89} -100.974 q^{91} +24.5379i q^{95} -124.974 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 16 q^{7}+O(q^{10}) \) 4 * q + 16 * q^7	\( 4 q + 16 q^{7} + 8 q^{13} - 32 q^{19} - 20 q^{25} - 8 q^{31} - 136 q^{37} - 80 q^{43} + 228 q^{49} + 120 q^{55} - 40 q^{61} + 304 q^{67} + 152 q^{73} - 200 q^{79} - 328 q^{91} - 424 q^{97}+O(q^{100}) \) 4 * q + 16 * q^7 + 8 * q^13 - 32 * q^19 - 20 * q^25 - 8 * q^31 - 136 * q^37 - 80 * q^43 + 228 * q^49 + 120 * q^55 - 40 * q^61 + 304 * q^67 + 152 * q^73 - 200 * q^79 - 328 * q^91 - 424 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more