LMFDB - Embedded newform 288.8.f.a.143.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [288,8,Mod(143,288)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(288, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("288.143");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 288 = 2^{5} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	288.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:	$89.9668873394$
Analytic rank:	$0$
Dimension:	$28$
Twist minimal:	no (minimal twist has level 72)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			143.16
Character	$\chi$	$=$	288.143
Dual form			288.8.f.a.143.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+93.0830 q^{5} +1025.06i q^{7} +O(q^{10})$	$q+93.0830 q^{5} +1025.06i q^{7} -1545.61i q^{11} -5677.41i q^{13} +15703.6i q^{17} -33939.0 q^{19} +30068.2 q^{23} -69460.6 q^{25} -99025.6 q^{29} +152785. i q^{31} +95416.0i q^{35} -336316. i q^{37} -663449. i q^{41} +707879. q^{43} +402012. q^{47} -227213. q^{49} -662469. q^{53} -143870. i q^{55} +1.74805e6i q^{59} -321803. i q^{61} -528470. i q^{65} +464913. q^{67} -3.53880e6 q^{71} -3.41741e6 q^{73} +1.58435e6 q^{77} -7.60430e6i q^{79} +8.11070e6i q^{83} +1.46174e6i q^{85} -1.00923e7i q^{89} +5.81971e6 q^{91} -3.15915e6 q^{95} -5.88229e6 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q+O(q^{10}) $ 28 * q	$ 28 q + 121168 q^{19} + 437500 q^{25} - 1505696 q^{43} - 2272076 q^{49} + 776272 q^{67} - 2534128 q^{73} + 3406992 q^{91} - 26311456 q^{97}+O(q^{100}) $ 28 * q + 121168 * q^19 + 437500 * q^25 - 1505696 * q^43 - 2272076 * q^49 + 776272 * q^67 - 2534128 * q^73 + 3406992 * q^91 - 26311456 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/288\mathbb{Z}\right)^\times$.

$n$	$37$	$65$	$127$
$\chi(n)$	$-1$	$-1$	$-1$

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
$5$	93.0830	0.333024	0.166512	−	0.986039i	$-0.446750\pi$
$5$	93.0830	0.333024	0.166512	+	0.986039i	$0.446750\pi$
$6$	0	0
$7$	1025.06i	1.12956i	0.825243	+	0.564778i	$0.191038\pi$
$7$	1025.06i	1.12956i	−0.825243	+	0.564778i	$0.808962\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 1545.61i	− 0.350126i	−0.984557	−	0.175063i	$-0.943987\pi$
$11$	− 1545.61i	− 0.350126i	0.984557	−	0.175063i	$-0.0560130\pi$
$12$	0	0
$13$	− 5677.41i	− 0.716719i	−0.933584	−	0.358359i	$-0.883336\pi$
$13$	− 5677.41i	− 0.716719i	0.933584	−	0.358359i	$-0.116664\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	15703.6i	0.775227i	0.921822	+	0.387613i	$0.126700\pi$
$17$	15703.6i	0.775227i	−0.921822	+	0.387613i	$0.873300\pi$
$18$	0	0
$19$	−33939.0	−1.13517	−0.567586	−	0.823314i	$-0.692123\pi$
$19$	−33939.0	−1.13517	−0.567586	+	0.823314i	$0.692123\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	30068.2	0.515300	0.257650	−	0.966238i	$-0.417052\pi$
$23$	30068.2	0.515300	0.257650	+	0.966238i	$0.417052\pi$
$24$	0	0
$25$	−69460.6	−0.889095
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−99025.6	−0.753971	−0.376985	−	0.926219i	$-0.623039\pi$
$29$	−99025.6	−0.753971	−0.376985	+	0.926219i	$0.623039\pi$
$30$	0	0
$31$	152785.i	0.921115i	0.887630	+	0.460558i	$0.152351\pi$
$31$	152785.i	0.921115i	−0.887630	+	0.460558i	$0.847649\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	95416.0i	0.376169i
$36$	0	0
$37$	− 336316.i	− 1.09154i	−0.837933	−	0.545772i	$-0.816236\pi$
$37$	− 336316.i	− 1.09154i	0.837933	−	0.545772i	$-0.183764\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 663449.i	− 1.50336i	−0.659527	−	0.751681i	$-0.729243\pi$
$41$	− 663449.i	− 1.50336i	0.659527	−	0.751681i	$-0.270757\pi$
$42$	0	0
$43$	707879.	1.35775	0.678874	−	0.734255i	$-0.262468\pi$
$43$	707879.	1.35775	0.678874	+	0.734255i	$0.262468\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	402012.	0.564802	0.282401	−	0.959296i	$-0.408869\pi$
$47$	402012.	0.564802	0.282401	+	0.959296i	$0.408869\pi$
$48$	0	0
$49$	−227213.	−0.275896
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−662469.	−0.611223	−0.305612	−	0.952156i	$-0.598861\pi$
$53$	−662469.	−0.611223	−0.305612	+	0.952156i	$0.598861\pi$
$54$	0	0
$55$	− 143870.i	− 0.116600i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	1.74805e6i	1.10808i	0.832489	+	0.554041i	$0.186915\pi$
$59$	1.74805e6i	1.10808i	−0.832489	+	0.554041i	$0.813085\pi$
$60$	0	0
$61$	− 321803.i	− 0.181525i	−0.995873	−	0.0907624i	$-0.971070\pi$
$61$	− 321803.i	− 0.181525i	0.995873	−	0.0907624i	$-0.0289304\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 528470.i	− 0.238684i
$66$	0	0
$67$	464913.	0.188847	0.0944234	−	0.995532i	$-0.469899\pi$
$67$	464913.	0.188847	0.0944234	+	0.995532i	$0.469899\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−3.53880e6	−1.17342	−0.586708	−	0.809799i	$-0.699576\pi$
$71$	−3.53880e6	−1.17342	−0.586708	+	0.809799i	$0.699576\pi$
$72$	0	0
$73$	−3.41741e6	−1.02817	−0.514087	−	0.857738i	$-0.671869\pi$
$73$	−3.41741e6	−1.02817	−0.514087	+	0.857738i	$0.671869\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.58435e6	0.395487
$78$	0	0
$79$	− 7.60430e6i	− 1.73526i	−0.497210	−	0.867630i	$-0.665642\pi$
$79$	− 7.60430e6i	− 1.73526i	0.497210	−	0.867630i	$-0.334358\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	8.11070e6i	1.55699i	0.627653	+	0.778494i	$0.284016\pi$
$83$	8.11070e6i	1.55699i	−0.627653	+	0.778494i	$0.715984\pi$
$84$	0	0
$85$	1.46174e6i	0.258169i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 1.00923e7i	− 1.51749i	−0.651387	−	0.758746i	$-0.725813\pi$
$89$	− 1.00923e7i	− 1.51749i	0.651387	−	0.758746i	$-0.274187\pi$
$90$	0	0
$91$	5.81971e6	0.809574
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−3.15915e6	−0.378039
$96$	0	0
$97$	−5.88229e6	−0.654403	−0.327201	−	0.944955i	$-0.606106\pi$
$97$	−5.88229e6	−0.654403	−0.327201	+	0.944955i	$0.606106\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1.32223e7	−1.27697	−0.638486	−	0.769633i	$-0.720439\pi$
$101$	−1.32223e7	−1.27697	−0.638486	+	0.769633i	$0.720439\pi$
$102$	0	0
$103$	− 8.86665e6i	− 0.799520i	−0.916620	−	0.399760i	$-0.869093\pi$
$103$	− 8.86665e6i	− 0.799520i	0.916620	−	0.399760i	$-0.130907\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 4.71318e6i	− 0.371938i	−0.982556	−	0.185969i	$-0.940458\pi$
$107$	− 4.71318e6i	− 0.371938i	0.982556	−	0.185969i	$-0.0595424\pi$
$108$	0	0
$109$	− 2.63965e7i	− 1.95233i	−0.217027	−	0.976166i	$-0.569636\pi$
$109$	− 2.63965e7i	− 1.95233i	0.217027	−	0.976166i	$-0.430364\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 3.41021e6i	− 0.222334i	−0.993802	−	0.111167i	$-0.964541\pi$
$113$	− 3.41021e6i	− 0.222334i	0.993802	−	0.111167i	$-0.0354589\pi$
$114$	0	0
$115$	2.79884e6	0.171607
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1.60972e7	−0.875662
$120$	0	0
$121$	1.70983e7	0.877412
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.37377e7	−0.629114
$126$	0	0
$127$	− 9.87499e6i	− 0.427783i	−0.976857	−	0.213892i	$-0.931386\pi$
$127$	− 9.87499e6i	− 0.427783i	0.976857	−	0.213892i	$-0.0686139\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 2.16683e7i	− 0.842124i	−0.907032	−	0.421062i	$-0.861658\pi$
$131$	− 2.16683e7i	− 0.842124i	0.907032	−	0.421062i	$-0.138342\pi$
$132$	0	0
$133$	− 3.47897e7i	− 1.28224i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	5.12379e7i	1.70243i	0.524818	+	0.851214i	$0.324133\pi$
$137$	5.12379e7i	1.70243i	−0.524818	+	0.851214i	$0.675867\pi$
$138$	0	0
$139$	−3.92028e7	−1.23813	−0.619063	−	0.785341i	$-0.712488\pi$
$139$	−3.92028e7	−1.23813	−0.619063	+	0.785341i	$0.712488\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−8.77505e6	−0.250942
$144$	0	0
$145$	−9.21760e6	−0.251090
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−7.23566e7	−1.79195	−0.895975	−	0.444104i	$-0.853522\pi$
$149$	−7.23566e7	−1.79195	−0.895975	+	0.444104i	$0.853522\pi$
$150$	0	0
$151$	− 5.61796e7i	− 1.32788i	−0.747786	−	0.663940i	$-0.768883\pi$
$151$	− 5.61796e7i	− 1.32788i	0.747786	−	0.663940i	$-0.231117\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1.42217e7i	0.306753i
$156$	0	0
$157$	− 8.29634e7i	− 1.71095i	−0.517843	−	0.855476i	$-0.673265\pi$
$157$	− 8.29634e7i	− 1.71095i	0.517843	−	0.855476i	$-0.326735\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	3.08218e7i	0.582060i
$162$	0	0
$163$	−1.32708e7	−0.240017	−0.120008	−	0.992773i	$-0.538292\pi$
$163$	−1.32708e7	−0.240017	−0.120008	+	0.992773i	$0.538292\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−6.62945e7	−1.10146	−0.550732	−	0.834682i	$-0.685651\pi$
$167$	−6.62945e7	−1.10146	−0.550732	+	0.834682i	$0.685651\pi$
$168$	0	0
$169$	3.05155e7	0.486314
$170$	0	0
$171$	0	0
$172$	0	0
$173$	5.99909e7	0.880895	0.440448	−	0.897778i	$-0.354820\pi$
$173$	5.99909e7	0.880895	0.440448	+	0.897778i	$0.354820\pi$
$174$	0	0
$175$	− 7.12015e7i	− 1.00428i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 8.01095e7i	− 1.04399i	−0.852947	−	0.521997i	$-0.825187\pi$
$179$	− 8.01095e7i	− 1.04399i	0.852947	−	0.521997i	$-0.174813\pi$
$180$	0	0
$181$	3.83641e7i	0.480894i	0.970662	+	0.240447i	$0.0772941\pi$
$181$	3.83641e7i	0.480894i	−0.970662	+	0.240447i	$0.922706\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 3.13053e7i	− 0.363510i
$186$	0	0
$187$	2.42716e7	0.271427
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1.26246e8	−1.31100	−0.655499	−	0.755196i	$-0.727542\pi$
$191$	−1.26246e8	−1.31100	−0.655499	+	0.755196i	$0.727542\pi$
$192$	0	0
$193$	1.20271e8	1.20423	0.602115	−	0.798409i	$-0.294325\pi$
$193$	1.20271e8	1.20423	0.602115	+	0.798409i	$0.294325\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1.23903e8	−1.15465	−0.577326	−	0.816514i	$-0.695904\pi$
$197$	−1.23903e8	−1.15465	−0.577326	+	0.816514i	$0.695904\pi$
$198$	0	0
$199$	− 1.68027e8i	− 1.51145i	−0.654888	−	0.755726i	$-0.727284\pi$
$199$	− 1.68027e8i	− 1.51145i	0.654888	−	0.755726i	$-0.272716\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 1.01508e8i	− 0.851652i
$204$	0	0
$205$	− 6.17558e7i	− 0.500655i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	5.24564e7i	0.397454i
$210$	0	0
$211$	−8.24919e7	−0.604537	−0.302268	−	0.953223i	$-0.597744\pi$
$211$	−8.24919e7	−0.604537	−0.302268	+	0.953223i	$0.597744\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	6.58914e7	0.452162
$216$	0	0
$217$	−1.56614e8	−1.04045
$218$	0	0
$219$	0	0
$220$	0	0
$221$	8.91560e7	0.555620
$222$	0	0
$223$	1.10837e8i	0.669296i	0.942343	+	0.334648i	$0.108617\pi$
$223$	1.10837e8i	0.669296i	−0.942343	+	0.334648i	$0.891383\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2.44393e8i	1.38675i	0.720576	+	0.693376i	$0.243877\pi$
$227$	2.44393e8i	1.38675i	−0.720576	+	0.693376i	$0.756123\pi$
$228$	0	0
$229$	− 2.49235e7i	− 0.137146i	−0.997646	−	0.0685732i	$-0.978155\pi$
$229$	− 2.49235e7i	− 0.137146i	0.997646	−	0.0685732i	$-0.0218447\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 8.59209e7i	− 0.444993i	−0.974934	−	0.222496i	$-0.928579\pi$
$233$	− 8.59209e7i	− 0.444993i	0.974934	−	0.222496i	$-0.0714206\pi$
$234$	0	0
$235$	3.74204e7	0.188092
$236$	0	0
$237$	0	0
$238$	0	0
$239$	1.03442e8	0.490122	0.245061	−	0.969508i	$-0.421192\pi$
$239$	1.03442e8	0.490122	0.245061	+	0.969508i	$0.421192\pi$
$240$	0	0
$241$	−2.12373e8	−0.977325	−0.488662	−	0.872473i	$-0.662515\pi$
$241$	−2.12373e8	−0.977325	−0.488662	+	0.872473i	$0.662515\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−2.11496e7	−0.0918801
$246$	0	0
$247$	1.92686e8i	0.813599i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	2.88318e8i	1.15084i	0.817859	+	0.575418i	$0.195161\pi$
$251$	2.88318e8i	1.15084i	−0.817859	+	0.575418i	$0.804839\pi$
$252$	0	0
$253$	− 4.64736e7i	− 0.180420i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 3.06607e8i	− 1.12672i	−0.826211	−	0.563361i	$-0.809508\pi$
$257$	− 3.06607e8i	− 1.12672i	0.826211	−	0.563361i	$-0.190492\pi$
$258$	0	0
$259$	3.44745e8	1.23296
$260$	0	0
$261$	0	0
$262$	0	0
$263$	7.85030e7	0.266098	0.133049	−	0.991109i	$-0.457523\pi$
$263$	7.85030e7	0.266098	0.133049	+	0.991109i	$0.457523\pi$
$264$	0	0
$265$	−6.16645e7	−0.203552
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−2.64506e8	−0.828519	−0.414259	−	0.910159i	$-0.635959\pi$
$269$	−2.64506e8	−0.828519	−0.414259	+	0.910159i	$0.635959\pi$
$270$	0	0
$271$	3.60573e8i	1.10053i	0.834991	+	0.550263i	$0.185473\pi$
$271$	3.60573e8i	1.10053i	−0.834991	+	0.550263i	$0.814527\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.07359e8i	0.311296i
$276$	0	0
$277$	2.46978e8i	0.698198i	0.937086	+	0.349099i	$0.113512\pi$
$277$	2.46978e8i	0.698198i	−0.937086	+	0.349099i	$0.886488\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 1.16347e8i	− 0.312812i	−0.987693	−	0.156406i	$-0.950009\pi$
$281$	− 1.16347e8i	− 0.312812i	0.987693	−	0.156406i	$-0.0499908\pi$
$282$	0	0
$283$	3.05548e8	0.801358	0.400679	−	0.916219i	$-0.368774\pi$
$283$	3.05548e8	0.801358	0.400679	+	0.916219i	$0.368774\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	6.80077e8	1.69813
$288$	0	0
$289$	1.63735e8	0.399023
$290$	0	0
$291$	0	0
$292$	0	0
$293$	7.14442e8	1.65932	0.829661	−	0.558268i	$-0.188534\pi$
$293$	7.14442e8	1.65932	0.829661	+	0.558268i	$0.188534\pi$
$294$	0	0
$295$	1.62714e8i	0.369018i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 1.70709e8i	− 0.369325i
$300$	0	0
$301$	7.25621e8i	1.53365i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 2.99544e7i	− 0.0604521i
$306$	0	0
$307$	−3.95766e8	−0.780647	−0.390323	−	0.920678i	$-0.627637\pi$
$307$	−3.95766e8	−0.780647	−0.390323	+	0.920678i	$0.627637\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−4.17018e8	−0.786129	−0.393064	−	0.919511i	$-0.628585\pi$
$311$	−4.17018e8	−0.786129	−0.393064	+	0.919511i	$0.628585\pi$
$312$	0	0
$313$	2.16149e8	0.398426	0.199213	−	0.979956i	$-0.436161\pi$
$313$	2.16149e8	0.398426	0.199213	+	0.979956i	$0.436161\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	6.69525e8	1.18048	0.590241	−	0.807227i	$-0.299033\pi$
$317$	6.69525e8	1.18048	0.590241	+	0.807227i	$0.299033\pi$
$318$	0	0
$319$	1.53055e8i	0.263985i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 5.32966e8i	− 0.880016i
$324$	0	0
$325$	3.94356e8i	0.637231i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	4.12088e8i	0.637975i
$330$	0	0
$331$	−2.27775e8	−0.345230	−0.172615	−	0.984989i	$-0.555222\pi$
$331$	−2.27775e8	−0.345230	−0.172615	+	0.984989i	$0.555222\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	4.32755e7	0.0628905
$336$	0	0
$337$	−3.09334e8	−0.440274	−0.220137	−	0.975469i	$-0.570650\pi$
$337$	−3.09334e8	−0.440274	−0.220137	+	0.975469i	$0.570650\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	2.36145e8	0.322507
$342$	0	0
$343$	6.11277e8i	0.817915i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	4.22835e8i	0.543272i	0.962400	+	0.271636i	$0.0875647\pi$
$347$	4.22835e8i	0.543272i	−0.962400	+	0.271636i	$0.912435\pi$
$348$	0	0
$349$	1.23470e9i	1.55479i	0.629011	+	0.777396i	$0.283460\pi$
$349$	1.23470e9i	1.55479i	−0.629011	+	0.777396i	$0.716540\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	1.17813e8i	0.142554i	0.997457	+	0.0712772i	$0.0227075\pi$
$353$	1.17813e8i	0.142554i	−0.997457	+	0.0712772i	$0.977292\pi$
$354$	0	0
$355$	−3.29402e8	−0.390775
$356$	0	0
$357$	0	0
$358$	0	0
$359$	1.21083e9	1.38119	0.690595	−	0.723242i	$-0.257349\pi$
$359$	1.21083e9	1.38119	0.690595	+	0.723242i	$0.257349\pi$
$360$	0	0
$361$	2.57987e8	0.288617
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−3.18102e8	−0.342406
$366$	0	0
$367$	2.02237e8i	0.213565i	0.994282	+	0.106782i	$0.0340548\pi$
$367$	2.02237e8i	0.213565i	−0.994282	+	0.106782i	$0.965945\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 6.79072e8i	− 0.690411i
$372$	0	0
$373$	− 1.43859e8i	− 0.143534i	−0.997421	−	0.0717671i	$-0.977136\pi$
$373$	− 1.43859e8i	− 0.143534i	0.997421	−	0.0717671i	$-0.0228638\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	5.62209e8i	0.540385i
$378$	0	0
$379$	−8.77228e8	−0.827705	−0.413852	−	0.910344i	$-0.635817\pi$
$379$	−8.77228e8	−0.827705	−0.413852	+	0.910344i	$0.635817\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−2.04366e8	−0.185871	−0.0929356	−	0.995672i	$-0.529625\pi$
$383$	−2.04366e8	−0.185871	−0.0929356	+	0.995672i	$0.529625\pi$
$384$	0	0
$385$	1.47476e8	0.131707
$386$	0	0
$387$	0	0
$388$	0	0
$389$	1.58576e9	1.36589	0.682943	−	0.730472i	$-0.260700\pi$
$389$	1.58576e9	1.36589	0.682943	+	0.730472i	$0.260700\pi$
$390$	0	0
$391$	4.72180e8i	0.399474i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 7.07831e8i	− 0.577883i
$396$	0	0
$397$	− 1.83002e8i	− 0.146787i	−0.997303	−	0.0733935i	$-0.976617\pi$
$397$	− 1.83002e8i	− 0.146787i	0.997303	−	0.0733935i	$-0.0233829\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	1.57329e9i	1.21844i	0.793001	+	0.609220i	$0.208517\pi$
$401$	1.57329e9i	1.21844i	−0.793001	+	0.609220i	$0.791483\pi$
$402$	0	0
$403$	8.67422e8	0.660181
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−5.19813e8	−0.382179
$408$	0	0
$409$	1.66645e9	1.20437	0.602187	−	0.798355i	$-0.294296\pi$
$409$	1.66645e9	1.20437	0.602187	+	0.798355i	$0.294296\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−1.79186e9	−1.25164
$414$	0	0
$415$	7.54968e8i	0.518514i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 8.92118e8i	− 0.592479i	−0.955114	−	0.296240i	$-0.904267\pi$
$419$	− 8.92118e8i	− 0.592479i	0.955114	−	0.296240i	$-0.0957327\pi$
$420$	0	0
$421$	− 1.98849e9i	− 1.29878i	−0.760455	−	0.649391i	$-0.775024\pi$
$421$	− 1.98849e9i	− 1.29878i	0.760455	−	0.649391i	$-0.224976\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 1.09078e9i	− 0.689250i
$426$	0	0
$427$	3.29869e8	0.205042
$428$	0	0
$429$	0	0
$430$	0	0
$431$	9.29190e8	0.559029	0.279514	−	0.960142i	$-0.409827\pi$
$431$	9.29190e8	0.559029	0.279514	+	0.960142i	$0.409827\pi$
$432$	0	0
$433$	1.65923e9	0.982197	0.491099	−	0.871104i	$-0.336595\pi$
$433$	1.65923e9	0.982197	0.491099	+	0.871104i	$0.336595\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1.02049e9	−0.584954
$438$	0	0
$439$	− 9.17987e7i	− 0.0517858i	−0.999665	−	0.0258929i	$-0.991757\pi$
$439$	− 9.17987e7i	− 0.0517858i	0.999665	−	0.0258929i	$-0.00824288\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 7.47135e8i	− 0.408306i	−0.978939	−	0.204153i	$-0.934556\pi$
$443$	− 7.47135e8i	− 0.408306i	0.978939	−	0.204153i	$-0.0654441\pi$
$444$	0	0
$445$	− 9.39424e8i	− 0.505361i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	3.67451e8i	0.191574i	0.995402	+	0.0957871i	$0.0305368\pi$
$449$	3.67451e8i	0.191574i	−0.995402	+	0.0957871i	$0.969463\pi$
$450$	0	0
$451$	−1.02543e9	−0.526367
$452$	0	0
$453$	0	0
$454$	0	0
$455$	5.41716e8	0.269607
$456$	0	0
$457$	−3.88980e9	−1.90643	−0.953215	−	0.302292i	$-0.902248\pi$
$457$	−3.88980e9	−1.90643	−0.953215	+	0.302292i	$0.902248\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−7.45672e8	−0.354482	−0.177241	−	0.984167i	$-0.556717\pi$
$461$	−7.45672e8	−0.354482	−0.177241	+	0.984167i	$0.556717\pi$
$462$	0	0
$463$	− 1.76294e9i	− 0.825473i	−0.910850	−	0.412737i	$-0.864573\pi$
$463$	− 1.76294e9i	− 0.825473i	0.910850	−	0.412737i	$-0.135427\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	2.95161e9i	1.34106i	0.741881	+	0.670531i	$0.233934\pi$
$467$	2.95161e9i	1.34106i	−0.741881	+	0.670531i	$0.766066\pi$
$468$	0	0
$469$	4.76565e8i	0.213313i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 1.09410e9i	− 0.475383i
$474$	0	0
$475$	2.35742e9	1.00928
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−4.12030e9	−1.71299	−0.856495	−	0.516156i	$-0.827363\pi$
$479$	−4.12030e9	−1.71299	−0.856495	+	0.516156i	$0.827363\pi$
$480$	0	0
$481$	−1.90941e9	−0.782331
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−5.47541e8	−0.217932
$486$	0	0
$487$	− 4.42855e9i	− 1.73744i	−0.495304	−	0.868720i	$-0.664943\pi$
$487$	− 4.42855e9i	− 1.73744i	0.495304	−	0.868720i	$-0.335057\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	4.43647e9i	1.69142i	0.533641	+	0.845711i	$0.320823\pi$
$491$	4.43647e9i	1.69142i	−0.533641	+	0.845711i	$0.679177\pi$
$492$	0	0
$493$	− 1.55506e9i	− 0.584498i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 3.62750e9i	− 1.32544i
$498$	0	0
$499$	−3.41428e9	−1.23012	−0.615060	−	0.788481i	$-0.710868\pi$
$499$	−3.41428e9	−1.23012	−0.615060	+	0.788481i	$0.710868\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−2.56937e9	−0.900201	−0.450100	−	0.892978i	$-0.648612\pi$
$503$	−2.56937e9	−0.900201	−0.450100	+	0.892978i	$0.648612\pi$
$504$	0	0
$505$	−1.23077e9	−0.425262
$506$	0	0
$507$	0	0
$508$	0	0
$509$	1.06039e9	0.356414	0.178207	−	0.983993i	$-0.442970\pi$
$509$	1.06039e9	0.356414	0.178207	+	0.983993i	$0.442970\pi$
$510$	0	0
$511$	− 3.50306e9i	− 1.16138i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 8.25334e8i	− 0.266259i
$516$	0	0
$517$	− 6.21352e8i	− 0.197752i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 4.18471e8i	− 0.129638i	−0.997897	−	0.0648191i	$-0.979353\pi$
$521$	− 4.18471e8i	− 0.129638i	0.997897	−	0.0648191i	$-0.0206470\pi$
$522$	0	0
$523$	3.84414e9	1.17502	0.587508	−	0.809218i	$-0.300109\pi$
$523$	3.84414e9	1.17502	0.587508	+	0.809218i	$0.300109\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−2.39927e9	−0.714073
$528$	0	0
$529$	−2.50073e9	−0.734466
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−3.76667e9	−1.07749
$534$	0	0
$535$	− 4.38717e8i	− 0.123864i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	3.51181e8i	0.0965986i
$540$	0	0
$541$	− 3.23668e9i	− 0.878838i	−0.898282	−	0.439419i	$-0.855184\pi$
$541$	− 3.23668e9i	− 0.878838i	0.898282	−	0.439419i	$-0.144816\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 2.45706e9i	− 0.650173i
$546$	0	0
$547$	4.94425e9	1.29165	0.645824	−	0.763486i	$-0.276514\pi$
$547$	4.94425e9	1.29165	0.645824	+	0.763486i	$0.276514\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	3.36083e9	0.855887
$552$	0	0
$553$	7.79490e9	1.96007
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1.48349e9	0.363740	0.181870	−	0.983323i	$-0.441785\pi$
$557$	1.48349e9	0.363740	0.181870	+	0.983323i	$0.441785\pi$
$558$	0	0
$559$	− 4.01892e9i	− 0.973123i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 4.65548e9i	− 1.09947i	−0.835338	−	0.549737i	$-0.814728\pi$
$563$	− 4.65548e9i	− 1.09947i	0.835338	−	0.549737i	$-0.185272\pi$
$564$	0	0
$565$	− 3.17433e8i	− 0.0740426i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	4.03603e9i	0.918462i	0.888317	+	0.459231i	$0.151875\pi$
$569$	4.03603e9i	0.918462i	−0.888317	+	0.459231i	$0.848125\pi$
$570$	0	0
$571$	−4.33338e9	−0.974094	−0.487047	−	0.873376i	$-0.661926\pi$
$571$	−4.33338e9	−0.974094	−0.487047	+	0.873376i	$0.661926\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−2.08855e9	−0.458150
$576$	0	0
$577$	1.65483e9	0.358623	0.179311	−	0.983792i	$-0.442613\pi$
$577$	1.65483e9	0.358623	0.179311	+	0.983792i	$0.442613\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−8.31398e9	−1.75870
$582$	0	0
$583$	1.02392e9i	0.214005i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	5.30362e9i	1.08228i	0.840933	+	0.541140i	$0.182007\pi$
$587$	5.30362e9i	1.08228i	−0.840933	+	0.541140i	$0.817993\pi$
$588$	0	0
$589$	− 5.18537e9i	− 1.04563i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	7.54656e7i	0.0148613i	0.999972	+	0.00743066i	$0.00236527\pi$
$593$	7.54656e7i	0.0148613i	−0.999972	+	0.00743066i	$0.997635\pi$
$594$	0	0
$595$	−1.49838e9	−0.291616
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−7.13988e9	−1.35737	−0.678683	−	0.734431i	$-0.737449\pi$
$599$	−7.13988e9	−1.35737	−0.678683	+	0.734431i	$0.737449\pi$
$600$	0	0
$601$	−9.21234e9	−1.73105	−0.865524	−	0.500867i	$-0.833015\pi$
$601$	−9.21234e9	−1.73105	−0.865524	+	0.500867i	$0.833015\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	1.59156e9	0.292199
$606$	0	0
$607$	2.53391e9i	0.459866i	0.973206	+	0.229933i	$0.0738507\pi$
$607$	2.53391e9i	0.459866i	−0.973206	+	0.229933i	$0.926149\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 2.28239e9i	− 0.404804i
$612$	0	0
$613$	− 2.83258e9i	− 0.496674i	−0.968674	−	0.248337i	$-0.920116\pi$
$613$	− 2.83258e9i	− 0.496674i	0.968674	−	0.248337i	$-0.0798840\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1.09456e9i	0.187604i	0.995591	+	0.0938018i	$0.0299020\pi$
$617$	1.09456e9i	0.187604i	−0.995591	+	0.0938018i	$0.970098\pi$
$618$	0	0
$619$	3.22342e9	0.546260	0.273130	−	0.961977i	$-0.411941\pi$
$619$	3.22342e9	0.546260	0.273130	+	0.961977i	$0.411941\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1.03453e10	1.71409
$624$	0	0
$625$	4.14786e9	0.679585
$626$	0	0
$627$	0	0
$628$	0	0
$629$	5.28138e9	0.846195
$630$	0	0
$631$	8.31648e9i	1.31776i	0.752248	+	0.658880i	$0.228970\pi$
$631$	8.31648e9i	1.31776i	−0.752248	+	0.658880i	$0.771030\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 9.19193e8i	− 0.142462i
$636$	0	0
$637$	1.28998e9i	0.197740i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	8.71195e9i	1.30651i	0.757138	+	0.653255i	$0.226597\pi$
$641$	8.71195e9i	1.30651i	−0.757138	+	0.653255i	$0.773403\pi$
$642$	0	0
$643$	−9.96578e9	−1.47834	−0.739168	−	0.673522i	$-0.764781\pi$
$643$	−9.96578e9	−1.47834	−0.739168	+	0.673522i	$0.764781\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	4.53985e9	0.658987	0.329493	−	0.944158i	$-0.393122\pi$
$647$	4.53985e9	0.658987	0.329493	+	0.944158i	$0.393122\pi$
$648$	0	0
$649$	2.70180e9	0.387969
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−2.90772e9	−0.408655	−0.204328	−	0.978903i	$-0.565501\pi$
$653$	−2.90772e9	−0.408655	−0.204328	+	0.978903i	$0.565501\pi$
$654$	0	0
$655$	− 2.01695e9i	− 0.280447i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 3.96130e9i	− 0.539186i	−0.962974	−	0.269593i	$-0.913111\pi$
$659$	− 3.96130e9i	− 0.539186i	0.962974	−	0.269593i	$-0.0868892\pi$
$660$	0	0
$661$	4.86484e9i	0.655184i	0.944819	+	0.327592i	$0.106237\pi$
$661$	4.86484e9i	0.655184i	−0.944819	+	0.327592i	$0.893763\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 3.23833e9i	− 0.427017i
$666$	0	0
$667$	−2.97752e9	−0.388521
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−4.97381e8	−0.0635566
$672$	0	0
$673$	−2.43436e9	−0.307845	−0.153923	−	0.988083i	$-0.549191\pi$
$673$	−2.43436e9	−0.307845	−0.153923	+	0.988083i	$0.549191\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−1.30568e9	−0.161725	−0.0808624	−	0.996725i	$-0.525767\pi$
$677$	−1.30568e9	−0.161725	−0.0808624	+	0.996725i	$0.525767\pi$
$678$	0	0
$679$	− 6.02972e9i	− 0.739184i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 7.27394e9i	− 0.873569i	−0.899566	−	0.436785i	$-0.856117\pi$
$683$	− 7.27394e9i	− 0.873569i	0.899566	−	0.436785i	$-0.143883\pi$
$684$	0	0
$685$	4.76937e9i	0.566949i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	3.76111e9i	0.438075i
$690$	0	0
$691$	2.01649e9	0.232500	0.116250	−	0.993220i	$-0.462913\pi$
$691$	2.01649e9	0.232500	0.116250	+	0.993220i	$0.462913\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−3.64911e9	−0.412326
$696$	0	0
$697$	1.04185e10	1.16545
$698$	0	0
$699$	0	0
$700$	0	0
$701$	1.51344e10	1.65940	0.829700	−	0.558210i	$-0.188512\pi$
$701$	1.51344e10	1.65940	0.829700	+	0.558210i	$0.188512\pi$
$702$	0	0
$703$	1.14142e10i	1.23909i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 1.35537e10i	− 1.44241i
$708$	0	0
$709$	1.47287e10i	1.55204i	0.630708	+	0.776020i	$0.282765\pi$
$709$	1.47287e10i	1.55204i	−0.630708	+	0.776020i	$0.717235\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	4.59396e9i	0.474650i
$714$	0	0
$715$	−8.16808e8	−0.0835697
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−2.99832e9	−0.300834	−0.150417	−	0.988623i	$-0.548062\pi$
$719$	−2.99832e9	−0.300834	−0.150417	+	0.988623i	$0.548062\pi$
$720$	0	0
$721$	9.08888e9	0.903103
$722$	0	0
$723$	0	0
$724$	0	0
$725$	6.87837e9	0.670352
$726$	0	0
$727$	9.00906e9i	0.869579i	0.900532	+	0.434789i	$0.143177\pi$
$727$	9.00906e9i	0.869579i	−0.900532	+	0.434789i	$0.856823\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	1.11163e10i	1.05256i
$732$	0	0
$733$	8.89008e9i	0.833761i	0.908961	+	0.416880i	$0.136877\pi$
$733$	8.89008e9i	0.833761i	−0.908961	+	0.416880i	$0.863123\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 7.18573e8i	− 0.0661202i
$738$	0	0
$739$	−8.53349e9	−0.777806	−0.388903	−	0.921279i	$-0.627146\pi$
$739$	−8.53349e9	−0.777806	−0.388903	+	0.921279i	$0.627146\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1.12936e10	1.01011	0.505057	−	0.863086i	$-0.331472\pi$
$743$	1.12936e10	1.01011	0.505057	+	0.863086i	$0.331472\pi$
$744$	0	0
$745$	−6.73516e9	−0.596762
$746$	0	0
$747$	0	0
$748$	0	0
$749$	4.83131e9	0.420125
$750$	0	0
$751$	6.47122e9i	0.557502i	0.960363	+	0.278751i	$0.0899204\pi$
$751$	6.47122e9i	0.557502i	−0.960363	+	0.278751i	$0.910080\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 5.22936e9i	− 0.442216i
$756$	0	0
$757$	4.70372e9i	0.394100i	0.980393	+	0.197050i	$0.0631361\pi$
$757$	4.70372e9i	0.394100i	−0.980393	+	0.197050i	$0.936864\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 9.44372e9i	− 0.776778i	−0.921496	−	0.388389i	$-0.873032\pi$
$761$	− 9.44372e9i	− 0.776778i	0.921496	−	0.388389i	$-0.126968\pi$
$762$	0	0
$763$	2.70581e10	2.20527
$764$	0	0
$765$	0	0
$766$	0	0
$767$	9.92441e9	0.794183
$768$	0	0
$769$	−5.19374e9	−0.411849	−0.205925	−	0.978568i	$-0.566020\pi$
$769$	−5.19374e9	−0.411849	−0.205925	+	0.978568i	$0.566020\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	1.43367e10	1.11640	0.558202	−	0.829705i	$-0.311491\pi$
$773$	1.43367e10	1.11640	0.558202	+	0.829705i	$0.311491\pi$
$774$	0	0
$775$	− 1.06125e10i	− 0.818959i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	2.25168e10i	1.70658i
$780$	0	0
$781$	5.46960e9i	0.410844i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 7.72248e9i	− 0.569788i
$786$	0	0
$787$	−6.32430e9	−0.462489	−0.231244	−	0.972896i	$-0.574280\pi$
$787$	−6.32430e9	−0.462489	−0.231244	+	0.972896i	$0.574280\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	3.49568e9	0.251139
$792$	0	0
$793$	−1.82701e9	−0.130102
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−1.99261e10	−1.39418	−0.697091	−	0.716983i	$-0.745523\pi$
$797$	−1.99261e10	−1.39418	−0.697091	+	0.716983i	$0.745523\pi$
$798$	0	0
$799$	6.31304e9i	0.437850i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	5.28197e9i	0.359991i
$804$	0	0
$805$	2.86899e9i	0.193840i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 4.18784e9i	− 0.278080i	−0.990287	−	0.139040i	$-0.955598\pi$
$809$	− 4.18784e9i	− 0.278080i	0.990287	−	0.139040i	$-0.0444017\pi$
$810$	0	0
$811$	3.77264e9	0.248355	0.124177	−	0.992260i	$-0.460371\pi$
$811$	3.77264e9	0.248355	0.124177	+	0.992260i	$0.460371\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−1.23529e9	−0.0799314
$816$	0	0
$817$	−2.40247e10	−1.54128
$818$	0	0
$819$	0	0
$820$	0	0
$821$	8.35665e9	0.527025	0.263512	−	0.964656i	$-0.415119\pi$
$821$	8.35665e9	0.527025	0.263512	+	0.964656i	$0.415119\pi$
$822$	0	0
$823$	2.54233e10i	1.58977i	0.606763	+	0.794883i	$0.292468\pi$
$823$	2.54233e10i	1.58977i	−0.606763	+	0.794883i	$0.707532\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	2.71179e10i	1.66720i	0.552371	+	0.833599i	$0.313723\pi$
$827$	2.71179e10i	1.66720i	−0.552371	+	0.833599i	$0.686277\pi$
$828$	0	0
$829$	1.95618e10i	1.19252i	0.802790	+	0.596262i	$0.203348\pi$
$829$	1.95618e10i	1.19252i	−0.802790	+	0.596262i	$0.796652\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 3.56806e9i	− 0.213882i
$834$	0	0
$835$	−6.17089e9	−0.366813
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1.86844e10	1.09223	0.546113	−	0.837711i	$-0.316107\pi$
$839$	1.86844e10	1.09223	0.546113	+	0.837711i	$0.316107\pi$
$840$	0	0
$841$	−7.44381e9	−0.431528
$842$	0	0
$843$	0	0
$844$	0	0
$845$	2.84047e9	0.161954
$846$	0	0
$847$	1.75268e10i	0.991085i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 1.01124e10i	− 0.562473i
$852$	0	0
$853$	− 3.55557e10i	− 1.96150i	−0.195273	−	0.980749i	$-0.562559\pi$
$853$	− 3.55557e10i	− 1.96150i	0.195273	−	0.980749i	$-0.437441\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 4.79883e6i	0 0.000260437i	−1.00000		0.000130218i	$-0.999959\pi$
$857$	− 4.79883e6i	0 0.000260437i	1.00000		0.000130218i	$-4.14498e-5\pi$
$858$	0	0
$859$	1.84728e10	0.994388	0.497194	−	0.867639i	$-0.334364\pi$
$859$	1.84728e10	0.994388	0.497194	+	0.867639i	$0.334364\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	6.22171e9	0.329513	0.164756	−	0.986334i	$-0.447316\pi$
$863$	6.22171e9	0.329513	0.164756	+	0.986334i	$0.447316\pi$
$864$	0	0
$865$	5.58414e9	0.293359
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−1.17533e10	−0.607560
$870$	0	0
$871$	− 2.63950e9i	− 0.135350i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 1.40820e10i	− 0.710619i
$876$	0	0
$877$	− 1.16634e10i	− 0.583884i	−0.956436	−	0.291942i	$-0.905699\pi$
$877$	− 1.16634e10i	− 0.583884i	0.956436	−	0.291942i	$-0.0943014\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 9.09495e9i	− 0.448110i	−0.974577	−	0.224055i	$-0.928070\pi$
$881$	− 9.09495e9i	− 0.448110i	0.974577	−	0.224055i	$-0.0719295\pi$
$882$	0	0
$883$	−1.53416e10	−0.749907	−0.374954	−	0.927044i	$-0.622341\pi$
$883$	−1.53416e10	−0.749907	−0.374954	+	0.927044i	$0.622341\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−2.84449e10	−1.36858	−0.684291	−	0.729209i	$-0.739888\pi$
$887$	−2.84449e10	−1.36858	−0.684291	+	0.729209i	$0.739888\pi$
$888$	0	0
$889$	1.01225e10	0.483205
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−1.36439e10	−0.641148
$894$	0	0
$895$	− 7.45683e9i	− 0.347675i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 1.51296e10i	− 0.694494i
$900$	0	0
$901$	− 1.04032e10i	− 0.473837i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	3.57104e9i	0.160149i
$906$	0	0
$907$	1.63965e10	0.729666	0.364833	−	0.931073i	$-0.381126\pi$
$907$	1.63965e10	0.729666	0.364833	+	0.931073i	$0.381126\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1.21999e10	0.534617	0.267309	−	0.963611i	$-0.413866\pi$
$911$	1.21999e10	0.534617	0.267309	+	0.963611i	$0.413866\pi$
$912$	0	0
$913$	1.25360e10	0.545142
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2.22114e10	0.951226
$918$	0	0
$919$	− 3.87196e10i	− 1.64561i	−0.568326	−	0.822803i	$-0.692409\pi$
$919$	− 3.87196e10i	− 1.64561i	0.568326	−	0.822803i	$-0.307591\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	2.00912e10i	0.841009i
$924$	0	0
$925$	2.33607e10i	0.970487i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 3.34318e10i	− 1.36806i	−0.729455	−	0.684029i	$-0.760226\pi$
$929$	− 3.34318e10i	− 1.36806i	0.729455	−	0.684029i	$-0.239774\pi$
$930$	0	0
$931$	7.71138e9	0.313190
$932$	0	0
$933$	0	0
$934$	0	0
$935$	2.25928e9	0.0903917
$936$	0	0
$937$	−1.13884e8	−0.00452247	−0.00226123	−	0.999997i	$-0.500720\pi$
$937$	−1.13884e8	−0.00452247	−0.00226123	+	0.999997i	$0.500720\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−3.30760e10	−1.29404	−0.647022	−	0.762472i	$-0.723986\pi$
$941$	−3.30760e10	−1.29404	−0.647022	+	0.762472i	$0.723986\pi$
$942$	0	0
$943$	− 1.99487e10i	− 0.774682i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 4.30694e10i	− 1.64795i	−0.566627	−	0.823974i	$-0.691752\pi$
$947$	− 4.30694e10i	− 1.64795i	0.566627	−	0.823974i	$-0.308248\pi$
$948$	0	0
$949$	1.94020e10i	0.736912i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	2.45613e10i	0.919233i	0.888118	+	0.459616i	$0.152013\pi$
$953$	2.45613e10i	0.919233i	−0.888118	+	0.459616i	$0.847987\pi$
$954$	0	0
$955$	−1.17514e10	−0.436594
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−5.25221e10	−1.92299
$960$	0	0
$961$	4.16944e9	0.151546
$962$	0	0
$963$	0	0
$964$	0	0
$965$	1.11952e10	0.401037
$966$	0	0
$967$	− 2.28479e10i	− 0.812556i	−0.913750	−	0.406278i	$-0.866826\pi$
$967$	− 2.28479e10i	− 0.812556i	0.913750	−	0.406278i	$-0.133174\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	4.59295e10i	1.60999i	0.593279	+	0.804997i	$0.297833\pi$
$971$	4.59295e10i	1.60999i	−0.593279	+	0.804997i	$0.702167\pi$
$972$	0	0
$973$	− 4.01854e10i	− 1.39853i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	4.53672e10i	1.55636i	0.628038	+	0.778182i	$0.283858\pi$
$977$	4.53672e10i	1.55636i	−0.628038	+	0.778182i	$0.716142\pi$
$978$	0	0
$979$	−1.55988e10	−0.531314
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−2.77122e10	−0.930536	−0.465268	−	0.885170i	$-0.654042\pi$
$983$	−2.77122e10	−0.930536	−0.465268	+	0.885170i	$0.654042\pi$
$984$	0	0
$985$	−1.15333e10	−0.384527
$986$	0	0
$987$	0	0
$988$	0	0
$989$	2.12846e10	0.699647
$990$	0	0
$991$	− 7.33226e9i	− 0.239321i	−0.992815	−	0.119660i	$-0.961819\pi$
$991$	− 7.33226e9i	− 0.239321i	0.992815	−	0.119660i	$-0.0381806\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 1.56405e10i	− 0.503350i
$996$	0	0
$997$	− 2.27024e10i	− 0.725502i	−0.931886	−	0.362751i	$-0.881838\pi$
$997$	− 2.27024e10i	− 0.725502i	0.931886	−	0.362751i	$-0.118162\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	288.8.f.a.143.16		28
3.2	odd	2	inner	288.8.f.a.143.14		28
4.3	odd	2		72.8.f.a.35.1	✓	28
8.3	odd	2	inner	288.8.f.a.143.13		28
8.5	even	2		72.8.f.a.35.27	yes	28
12.11	even	2		72.8.f.a.35.28	yes	28
24.5	odd	2		72.8.f.a.35.2	yes	28
24.11	even	2	inner	288.8.f.a.143.15		28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.8.f.a.35.1	✓	28	4.3	odd	2
72.8.f.a.35.2	yes	28	24.5	odd	2
72.8.f.a.35.27	yes	28	8.5	even	2
72.8.f.a.35.28	yes	28	12.11	even	2
288.8.f.a.143.13		28	8.3	odd	2	inner
288.8.f.a.143.14		28	3.2	odd	2	inner
288.8.f.a.143.15		28	24.11	even	2	inner
288.8.f.a.143.16		28	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 288 = 2^{5} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	288.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+93.0830 q^{5} +1025.06i q^{7} +O(q^{10})\)	\(q+93.0830 q^{5} +1025.06i q^{7} -1545.61i q^{11} -5677.41i q^{13} +15703.6i q^{17} -33939.0 q^{19} +30068.2 q^{23} -69460.6 q^{25} -99025.6 q^{29} +152785. i q^{31} +95416.0i q^{35} -336316. i q^{37} -663449. i q^{41} +707879. q^{43} +402012. q^{47} -227213. q^{49} -662469. q^{53} -143870. i q^{55} +1.74805e6i q^{59} -321803. i q^{61} -528470. i q^{65} +464913. q^{67} -3.53880e6 q^{71} -3.41741e6 q^{73} +1.58435e6 q^{77} -7.60430e6i q^{79} +8.11070e6i q^{83} +1.46174e6i q^{85} -1.00923e7i q^{89} +5.81971e6 q^{91} -3.15915e6 q^{95} -5.88229e6 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q+O(q^{10}) \) 28 * q	\( 28 q + 121168 q^{19} + 437500 q^{25} - 1505696 q^{43} - 2272076 q^{49} + 776272 q^{67} - 2534128 q^{73} + 3406992 q^{91} - 26311456 q^{97}+O(q^{100}) \) 28 * q + 121168 * q^19 + 437500 * q^25 - 1505696 * q^43 - 2272076 * q^49 + 776272 * q^67 - 2534128 * q^73 + 3406992 * q^91 - 26311456 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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