LMFDB - Embedded newform 288.3.h.a.17.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [288,3,Mod(17,288)]

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([0, 1, 1]))

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

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

chi := DirichletCharacter("288.17");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 288 = 2^{5} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	288.h (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:	$7.84743161358$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.33808912384.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + 11x^{6} - 18x^{5} + 47x^{4} - 28x^{3} - 44x^{2} + 48x + 36 $ x^8 - 2x^7 + 11x^6 - 18x^5 + 47x^4 - 28x^3 - 44x^2 + 48*x + 36
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{14}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 72)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			17.6
Root			$-0.651388 - 0.158947i$ of defining polynomial
Character	$\chi$	$=$	288.17
Dual form			288.3.h.a.17.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.07498 q^{5} -7.21110 q^{7} +O(q^{10})$	$q+1.07498 q^{5} -7.21110 q^{7} -16.3517 q^{11} +21.6045i q^{13} -18.9819i q^{17} +17.0438i q^{19} +1.11567i q^{23} -23.8444 q^{25} -29.4784 q^{29} -5.63331 q^{31} -7.75182 q^{35} +17.0438i q^{37} +27.4671i q^{41} +52.3306i q^{43} -64.5352i q^{47} +3.00000 q^{49} -35.9283 q^{53} -17.5778 q^{55} +56.8069 q^{59} -69.3743i q^{61} +23.2245i q^{65} -69.3743i q^{67} +98.4764i q^{71} +37.6888 q^{73} +117.914 q^{77} +127.322 q^{79} +7.75182 q^{83} -20.4052i q^{85} -76.1475i q^{89} -155.792i q^{91} +18.3218i q^{95} +4.84441 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 40 q^{25} + 128 q^{31} + 24 q^{49} - 256 q^{55} - 160 q^{73} + 384 q^{79} - 192 q^{97}+O(q^{100}) $ 8 * q + 40 * q^25 + 128 * q^31 + 24 * q^49 - 256 * q^55 - 160 * q^73 + 384 * q^79 - 192 * 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
$5$	1.07498	0.214997	0.107498	−	0.994205i	$-0.465716\pi$
$5$	1.07498	0.214997	0.107498	+	0.994205i	$0.465716\pi$
$6$	0	0
$7$	−7.21110	−1.03016	−0.515079	−	0.857143i	$-0.672237\pi$
$7$	−7.21110	−1.03016	−0.515079	+	0.857143i	$0.672237\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−16.3517	−1.48652	−0.743258	−	0.669004i	$-0.766721\pi$
$11$	−16.3517	−1.48652	−0.743258	+	0.669004i	$0.766721\pi$
$12$	0	0
$13$	21.6045i	1.66189i	0.556357	+	0.830943i	$0.312199\pi$
$13$	21.6045i	1.66189i	−0.556357	+	0.830943i	$0.687801\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 18.9819i	− 1.11658i	−0.829646	−	0.558290i	$-0.811458\pi$
$17$	− 18.9819i	− 1.11658i	0.829646	−	0.558290i	$-0.188542\pi$
$18$	0	0
$19$	17.0438i	0.897040i	0.893773	+	0.448520i	$0.148049\pi$
$19$	17.0438i	0.897040i	−0.893773	+	0.448520i	$0.851951\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.11567i	0.0485074i	0.999706	+	0.0242537i	$0.00772094\pi$
$23$	1.11567i	0.0485074i	−0.999706	+	0.0242537i	$0.992279\pi$
$24$	0	0
$25$	−23.8444	−0.953776
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−29.4784	−1.01650	−0.508249	−	0.861210i	$-0.669707\pi$
$29$	−29.4784	−1.01650	−0.508249	+	0.861210i	$0.669707\pi$
$30$	0	0
$31$	−5.63331	−0.181720	−0.0908598	−	0.995864i	$-0.528962\pi$
$31$	−5.63331	−0.181720	−0.0908598	+	0.995864i	$0.528962\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−7.75182	−0.221480
$36$	0	0
$37$	17.0438i	0.460642i	0.973115	+	0.230321i	$0.0739776\pi$
$37$	17.0438i	0.460642i	−0.973115	+	0.230321i	$0.926022\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	27.4671i	0.669930i	0.942231	+	0.334965i	$0.108725\pi$
$41$	27.4671i	0.669930i	−0.942231	+	0.334965i	$0.891275\pi$
$42$	0	0
$43$	52.3306i	1.21699i	0.793558	+	0.608495i	$0.208226\pi$
$43$	52.3306i	1.21699i	−0.793558	+	0.608495i	$0.791774\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 64.5352i	− 1.37309i	−0.727087	−	0.686545i	$-0.759126\pi$
$47$	− 64.5352i	− 1.37309i	0.727087	−	0.686545i	$-0.240874\pi$
$48$	0	0
$49$	3.00000	0.0612245
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−35.9283	−0.677893	−0.338946	−	0.940806i	$-0.610071\pi$
$53$	−35.9283	−0.677893	−0.338946	+	0.940806i	$0.610071\pi$
$54$	0	0
$55$	−17.5778	−0.319596
$56$	0	0
$57$	0	0
$58$	0	0
$59$	56.8069	0.962828	0.481414	−	0.876493i	$-0.340123\pi$
$59$	56.8069	0.962828	0.481414	+	0.876493i	$0.340123\pi$
$60$	0	0
$61$	− 69.3743i	− 1.13728i	−0.822585	−	0.568642i	$-0.807469\pi$
$61$	− 69.3743i	− 1.13728i	0.822585	−	0.568642i	$-0.192531\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	23.2245i	0.357300i
$66$	0	0
$67$	− 69.3743i	− 1.03544i	−0.855551	−	0.517719i	$-0.826781\pi$
$67$	− 69.3743i	− 1.03544i	0.855551	−	0.517719i	$-0.173219\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	98.4764i	1.38699i	0.720461	+	0.693496i	$0.243930\pi$
$71$	98.4764i	1.38699i	−0.720461	+	0.693496i	$0.756070\pi$
$72$	0	0
$73$	37.6888	0.516285	0.258143	−	0.966107i	$-0.416890\pi$
$73$	37.6888	0.516285	0.258143	+	0.966107i	$0.416890\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	117.914	1.53135
$78$	0	0
$79$	127.322	1.61167	0.805836	−	0.592138i	$-0.201716\pi$
$79$	127.322	1.61167	0.805836	+	0.592138i	$0.201716\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	7.75182	0.0933954	0.0466977	−	0.998909i	$-0.485130\pi$
$83$	7.75182	0.0933954	0.0466977	+	0.998909i	$0.485130\pi$
$84$	0	0
$85$	− 20.4052i	− 0.240061i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 76.1475i	− 0.855590i	−0.903876	−	0.427795i	$-0.859291\pi$
$89$	− 76.1475i	− 0.855590i	0.903876	−	0.427795i	$-0.140709\pi$
$90$	0	0
$91$	− 155.792i	− 1.71200i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	18.3218i	0.192861i
$96$	0	0
$97$	4.84441	0.0499424	0.0249712	−	0.999688i	$-0.492051\pi$
$97$	4.84441	0.0499424	0.0249712	+	0.999688i	$0.492051\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	105.635	1.04589	0.522946	−	0.852366i	$-0.324833\pi$
$101$	105.635	1.04589	0.522946	+	0.852366i	$0.324833\pi$
$102$	0	0
$103$	−104.789	−1.01737	−0.508684	−	0.860953i	$-0.669868\pi$
$103$	−104.789	−1.01737	−0.508684	+	0.860953i	$0.669868\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−65.4067	−0.611278	−0.305639	−	0.952147i	$-0.598870\pi$
$107$	−65.4067	−0.611278	−0.305639	+	0.952147i	$0.598870\pi$
$108$	0	0
$109$	− 3.36144i	− 0.0308389i	−0.999881	−	0.0154195i	$-0.995092\pi$
$109$	− 3.36144i	− 0.0308389i	0.999881	−	0.0154195i	$-0.00490836\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	84.1927i	0.745068i	0.928019	+	0.372534i	$0.121511\pi$
$113$	84.1927i	0.745068i	−0.928019	+	0.372534i	$0.878489\pi$
$114$	0	0
$115$	1.19933i	0.0104289i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	136.880i	1.15025i
$120$	0	0
$121$	146.378	1.20973
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−52.5069	−0.420056
$126$	0	0
$127$	45.5223	0.358443	0.179222	−	0.983809i	$-0.442642\pi$
$127$	45.5223	0.358443	0.179222	+	0.983809i	$0.442642\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	129.117	0.985629	0.492814	−	0.870134i	$-0.335968\pi$
$131$	129.117	0.985629	0.492814	+	0.870134i	$0.335968\pi$
$132$	0	0
$133$	− 122.904i	− 0.924092i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	10.9367i	0.0798296i	0.999203	+	0.0399148i	$0.0127087\pi$
$137$	10.9367i	0.0798296i	−0.999203	+	0.0399148i	$0.987291\pi$
$138$	0	0
$139$	− 1.19933i	− 0.00862825i	−0.999991	−	0.00431412i	$-0.998627\pi$
$139$	− 1.19933i	− 0.00862825i	0.999991	−	0.00431412i	$-0.00137323\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 353.270i	− 2.47042i
$144$	0	0
$145$	−31.6888	−0.218544
$146$	0	0
$147$	0	0
$148$	0	0
$149$	29.9323	0.200888	0.100444	−	0.994943i	$-0.467974\pi$
$149$	29.9323	0.200888	0.100444	+	0.994943i	$0.467974\pi$
$150$	0	0
$151$	−61.5223	−0.407432	−0.203716	−	0.979030i	$-0.565302\pi$
$151$	−61.5223	−0.407432	−0.203716	+	0.979030i	$0.565302\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−6.05571	−0.0390691
$156$	0	0
$157$	137.549i	0.876110i	0.898948	+	0.438055i	$0.144333\pi$
$157$	137.549i	0.876110i	−0.898948	+	0.438055i	$0.855667\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 8.04521i	− 0.0499702i
$162$	0	0
$163$	191.079i	1.17227i	0.810215	+	0.586133i	$0.199350\pi$
$163$	191.079i	1.17227i	−0.810215	+	0.586133i	$0.800650\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	233.125i	1.39596i	0.716118	+	0.697980i	$0.245917\pi$
$167$	233.125i	1.39596i	−0.716118	+	0.697980i	$0.754083\pi$
$168$	0	0
$169$	−297.755	−1.76187
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−323.809	−1.87173	−0.935864	−	0.352363i	$-0.885378\pi$
$173$	−323.809	−1.87173	−0.935864	+	0.352363i	$0.885378\pi$
$174$	0	0
$175$	171.944	0.982540
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−185.924	−1.03868	−0.519342	−	0.854567i	$-0.673823\pi$
$179$	−185.924	−1.03868	−0.519342	+	0.854567i	$0.673823\pi$
$180$	0	0
$181$	126.266i	0.697600i	0.937197	+	0.348800i	$0.113411\pi$
$181$	126.266i	0.697600i	−0.937197	+	0.348800i	$0.886589\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	18.3218i	0.0990365i
$186$	0	0
$187$	310.385i	1.65982i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 233.125i	− 1.22055i	−0.792189	−	0.610275i	$-0.791059\pi$
$191$	− 233.125i	− 1.22055i	0.792189	−	0.610275i	$-0.208941\pi$
$192$	0	0
$193$	−117.378	−0.608174	−0.304087	−	0.952644i	$-0.598351\pi$
$193$	−117.378	−0.608174	−0.304087	+	0.952644i	$0.598351\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−296.313	−1.50413	−0.752064	−	0.659091i	$-0.770941\pi$
$197$	−296.313	−1.50413	−0.752064	+	0.659091i	$0.770941\pi$
$198$	0	0
$199$	−233.011	−1.17091	−0.585455	−	0.810705i	$-0.699084\pi$
$199$	−233.011	−1.17091	−0.585455	+	0.810705i	$0.699084\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	212.572	1.04715
$204$	0	0
$205$	29.5267i	0.144033i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 278.694i	− 1.33346i
$210$	0	0
$211$	− 35.2868i	− 0.167236i	−0.996498	−	0.0836181i	$-0.973352\pi$
$211$	− 35.2868i	− 0.167236i	0.996498	−	0.0836181i	$-0.0266476\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	56.2545i	0.261649i
$216$	0	0
$217$	40.6224	0.187200
$218$	0	0
$219$	0	0
$220$	0	0
$221$	410.094	1.85563
$222$	0	0
$223$	51.8335	0.232437	0.116219	−	0.993224i	$-0.462923\pi$
$223$	51.8335	0.232437	0.116219	+	0.993224i	$0.462923\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−334.786	−1.47483	−0.737413	−	0.675442i	$-0.763953\pi$
$227$	−334.786	−1.47483	−0.737413	+	0.675442i	$0.763953\pi$
$228$	0	0
$229$	219.407i	0.958108i	0.877786	+	0.479054i	$0.159020\pi$
$229$	219.407i	0.958108i	−0.877786	+	0.479054i	$0.840980\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	203.867i	0.874965i	0.899227	+	0.437482i	$0.144130\pi$
$233$	203.867i	0.874965i	−0.899227	+	0.437482i	$0.855870\pi$
$234$	0	0
$235$	− 69.3743i	− 0.295210i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 54.0232i	− 0.226038i	−0.993593	−	0.113019i	$-0.963948\pi$
$239$	− 54.0232i	− 0.226038i	0.993593	−	0.113019i	$-0.0360522\pi$
$240$	0	0
$241$	327.600	1.35933	0.679667	−	0.733520i	$-0.262124\pi$
$241$	327.600	1.35933	0.679667	+	0.733520i	$0.262124\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	3.22495	0.0131631
$246$	0	0
$247$	−368.222	−1.49078
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−362.281	−1.44335	−0.721676	−	0.692231i	$-0.756628\pi$
$251$	−362.281	−1.44335	−0.721676	+	0.692231i	$0.756628\pi$
$252$	0	0
$253$	− 18.2431i	− 0.0721070i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	21.2132i	0.0825416i	0.999148	+	0.0412708i	$0.0131406\pi$
$257$	21.2132i	0.0825416i	−0.999148	+	0.0412708i	$0.986859\pi$
$258$	0	0
$259$	− 122.904i	− 0.474534i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	205.878i	0.782806i	0.920219	+	0.391403i	$0.128010\pi$
$263$	205.878i	0.782806i	−0.920219	+	0.391403i	$0.871990\pi$
$264$	0	0
$265$	−38.6224	−0.145745
$266$	0	0
$267$	0	0
$268$	0	0
$269$	2.77109	0.0103014	0.00515072	−	0.999987i	$-0.498360\pi$
$269$	2.77109	0.0103014	0.00515072	+	0.999987i	$0.498360\pi$
$270$	0	0
$271$	125.744	0.464001	0.232001	−	0.972716i	$-0.425473\pi$
$271$	125.744	0.464001	0.232001	+	0.972716i	$0.425473\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	389.896	1.41780
$276$	0	0
$277$	162.752i	0.587552i	0.955874	+	0.293776i	$0.0949119\pi$
$277$	162.752i	0.587552i	−0.955874	+	0.293776i	$0.905088\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 301.227i	− 1.07198i	−0.844223	−	0.535992i	$-0.819938\pi$
$281$	− 301.227i	− 1.07198i	0.844223	−	0.535992i	$-0.180062\pi$
$282$	0	0
$283$	− 31.6888i	− 0.111975i	−0.998431	−	0.0559874i	$-0.982169\pi$
$283$	− 31.6888i	− 0.111975i	0.998431	−	0.0559874i	$-0.0178307\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 198.068i	− 0.690134i
$288$	0	0
$289$	−71.3112	−0.246751
$290$	0	0
$291$	0	0
$292$	0	0
$293$	304.913	1.04066	0.520329	−	0.853966i	$-0.325809\pi$
$293$	304.913	1.04066	0.520329	+	0.853966i	$0.325809\pi$
$294$	0	0
$295$	61.0665	0.207005
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−24.1035	−0.0806137
$300$	0	0
$301$	− 377.361i	− 1.25369i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 74.5763i	− 0.244512i
$306$	0	0
$307$	− 105.860i	− 0.344822i	−0.985025	−	0.172411i	$-0.944844\pi$
$307$	− 105.860i	− 0.344822i	0.985025	−	0.172411i	$-0.0551558\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	555.157i	1.78507i	0.450978	+	0.892535i	$0.351075\pi$
$311$	555.157i	1.78507i	−0.450978	+	0.892535i	$0.648925\pi$
$312$	0	0
$313$	158.000	0.504792	0.252396	−	0.967624i	$-0.418781\pi$
$313$	158.000	0.504792	0.252396	+	0.967624i	$0.418781\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−97.0351	−0.306105	−0.153052	−	0.988218i	$-0.548910\pi$
$317$	−97.0351	−0.306105	−0.153052	+	0.988218i	$0.548910\pi$
$318$	0	0
$319$	482.022	1.51104
$320$	0	0
$321$	0	0
$322$	0	0
$323$	323.522	1.00162
$324$	0	0
$325$	− 515.147i	− 1.58507i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	465.370i	1.41450i
$330$	0	0
$331$	− 619.572i	− 1.87182i	−0.352242	−	0.935909i	$-0.614581\pi$
$331$	− 619.572i	− 1.87182i	0.352242	−	0.935909i	$-0.385419\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 74.5763i	− 0.222616i
$336$	0	0
$337$	−170.755	−0.506692	−0.253346	−	0.967376i	$-0.581531\pi$
$337$	−170.755	−0.506692	−0.253346	+	0.967376i	$0.581531\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	92.1141	0.270129
$342$	0	0
$343$	331.711	0.967087
$344$	0	0
$345$	0	0
$346$	0	0
$347$	300.386	0.865666	0.432833	−	0.901474i	$-0.357514\pi$
$347$	300.386	0.865666	0.432833	+	0.901474i	$0.357514\pi$
$348$	0	0
$349$	53.5299i	0.153381i	0.997055	+	0.0766904i	$0.0244353\pi$
$349$	53.5299i	0.153381i	−0.997055	+	0.0766904i	$0.975565\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	600.475i	1.70106i	0.525925	+	0.850531i	$0.323719\pi$
$353$	600.475i	1.70106i	−0.525925	+	0.850531i	$0.676281\pi$
$354$	0	0
$355$	105.860i	0.298199i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 148.508i	− 0.413671i	−0.978376	−	0.206836i	$-0.933683\pi$
$359$	− 148.508i	− 0.413671i	0.978376	−	0.206836i	$-0.0663165\pi$
$360$	0	0
$361$	70.5106	0.195320
$362$	0	0
$363$	0	0
$364$	0	0
$365$	40.5149	0.111000
$366$	0	0
$367$	−316.389	−0.862094	−0.431047	−	0.902329i	$-0.641856\pi$
$367$	−316.389	−0.862094	−0.431047	+	0.902329i	$0.641856\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	259.083	0.698336
$372$	0	0
$373$	− 626.768i	− 1.68034i	−0.542321	−	0.840171i	$-0.682455\pi$
$373$	− 626.768i	− 1.68034i	0.542321	−	0.840171i	$-0.317545\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 636.867i	− 1.68930i
$378$	0	0
$379$	331.027i	0.873423i	0.899602	+	0.436711i	$0.143857\pi$
$379$	331.027i	0.873423i	−0.899602	+	0.436711i	$0.856143\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 407.765i	− 1.06466i	−0.846537	−	0.532330i	$-0.821317\pi$
$383$	− 407.765i	− 1.06466i	0.846537	−	0.532330i	$-0.178683\pi$
$384$	0	0
$385$	126.755	0.329234
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−53.5819	−0.137743	−0.0688714	−	0.997626i	$-0.521940\pi$
$389$	−53.5819	−0.137743	−0.0688714	+	0.997626i	$0.521940\pi$
$390$	0	0
$391$	21.1775	0.0541624
$392$	0	0
$393$	0	0
$394$	0	0
$395$	136.869	0.346504
$396$	0	0
$397$	289.744i	0.729833i	0.931040	+	0.364917i	$0.118902\pi$
$397$	289.744i	0.729833i	−0.931040	+	0.364917i	$0.881098\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	203.458i	0.507376i	0.967286	+	0.253688i	$0.0816436\pi$
$401$	203.458i	0.507376i	−0.967286	+	0.253688i	$0.918356\pi$
$402$	0	0
$403$	− 121.705i	− 0.301997i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 278.694i	− 0.684752i
$408$	0	0
$409$	412.844	1.00940	0.504700	−	0.863295i	$-0.331603\pi$
$409$	412.844	1.00940	0.504700	+	0.863295i	$0.331603\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−409.640	−0.991865
$414$	0	0
$415$	8.33308	0.0200797
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−336.482	−0.803059	−0.401529	−	0.915846i	$-0.631521\pi$
$419$	−336.482	−0.803059	−0.401529	+	0.915846i	$0.631521\pi$
$420$	0	0
$421$	217.481i	0.516582i	0.966067	+	0.258291i	$0.0831594\pi$
$421$	217.481i	0.516582i	−0.966067	+	0.258291i	$0.916841\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	452.611i	1.06497i
$426$	0	0
$427$	500.265i	1.17158i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	750.994i	1.74245i	0.490888	+	0.871223i	$0.336672\pi$
$431$	750.994i	1.74245i	−0.490888	+	0.871223i	$0.663328\pi$
$432$	0	0
$433$	−176.133	−0.406773	−0.203387	−	0.979098i	$-0.565195\pi$
$433$	−176.133	−0.406773	−0.203387	+	0.979098i	$0.565195\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−19.0152	−0.0435130
$438$	0	0
$439$	−417.788	−0.951681	−0.475841	−	0.879531i	$-0.657856\pi$
$439$	−417.788	−0.951681	−0.475841	+	0.879531i	$0.657856\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	61.0471	0.137804	0.0689020	−	0.997623i	$-0.478050\pi$
$443$	61.0471	0.137804	0.0689020	+	0.997623i	$0.478050\pi$
$444$	0	0
$445$	− 81.8573i	− 0.183949i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 182.905i	− 0.407360i	−0.979038	−	0.203680i	$-0.934710\pi$
$449$	− 182.905i	− 0.407360i	0.979038	−	0.203680i	$-0.0652902\pi$
$450$	0	0
$451$	− 449.134i	− 0.995863i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 167.474i	− 0.368075i
$456$	0	0
$457$	−443.600	−0.970678	−0.485339	−	0.874326i	$-0.661304\pi$
$457$	−443.600	−0.970678	−0.485339	+	0.874326i	$0.661304\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	728.241	1.57970	0.789850	−	0.613301i	$-0.210159\pi$
$461$	728.241	1.57970	0.789850	+	0.613301i	$0.210159\pi$
$462$	0	0
$463$	107.722	0.232662	0.116331	−	0.993211i	$-0.462887\pi$
$463$	107.722	0.232662	0.116331	+	0.993211i	$0.462887\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−820.248	−1.75642	−0.878210	−	0.478276i	$-0.841262\pi$
$467$	−820.248	−1.75642	−0.878210	+	0.478276i	$0.841262\pi$
$468$	0	0
$469$	500.265i	1.06666i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 855.693i	− 1.80908i
$474$	0	0
$475$	− 406.398i	− 0.855575i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	120.790i	0.252171i	0.992019	+	0.126085i	$0.0402413\pi$
$479$	120.790i	0.252171i	−0.992019	+	0.126085i	$0.959759\pi$
$480$	0	0
$481$	−368.222	−0.765534
$482$	0	0
$483$	0	0
$484$	0	0
$485$	5.20766	0.0107374
$486$	0	0
$487$	128.234	0.263314	0.131657	−	0.991295i	$-0.457970\pi$
$487$	128.234	0.263314	0.131657	+	0.991295i	$0.457970\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−382.144	−0.778298	−0.389149	−	0.921175i	$-0.627231\pi$
$491$	−382.144	−0.778298	−0.389149	+	0.921175i	$0.627231\pi$
$492$	0	0
$493$	559.555i	1.13500i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 710.123i	− 1.42882i
$498$	0	0
$499$	554.995i	1.11221i	0.831111	+	0.556107i	$0.187705\pi$
$499$	554.995i	1.11221i	−0.831111	+	0.556107i	$0.812295\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	78.3943i	0.155854i	0.996959	+	0.0779268i	$0.0248300\pi$
$503$	78.3943i	0.155854i	−0.996959	+	0.0779268i	$0.975170\pi$
$504$	0	0
$505$	113.556	0.224863
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−514.941	−1.01167	−0.505836	−	0.862630i	$-0.668816\pi$
$509$	−514.941	−1.01167	−0.505836	+	0.862630i	$0.668816\pi$
$510$	0	0
$511$	−271.778	−0.531855
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−112.646	−0.218731
$516$	0	0
$517$	1055.26i	2.04112i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	489.695i	0.939914i	0.882689	+	0.469957i	$0.155730\pi$
$521$	489.695i	0.939914i	−0.882689	+	0.469957i	$0.844270\pi$
$522$	0	0
$523$	774.165i	1.48024i	0.672476	+	0.740119i	$0.265231\pi$
$523$	774.165i	1.48024i	−0.672476	+	0.740119i	$0.734769\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	106.931i	0.202905i
$528$	0	0
$529$	527.755	0.997647
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−593.415	−1.11335
$534$	0	0
$535$	−70.3112	−0.131423
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−49.0551	−0.0910112
$540$	0	0
$541$	− 590.045i	− 1.09066i	−0.838223	−	0.545328i	$-0.816405\pi$
$541$	− 590.045i	− 1.09066i	0.838223	−	0.545328i	$-0.183595\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 3.61350i	− 0.00663027i
$546$	0	0
$547$	707.189i	1.29285i	0.762977	+	0.646425i	$0.223737\pi$
$547$	707.189i	1.29285i	−0.762977	+	0.646425i	$0.776263\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 502.423i	− 0.911838i
$552$	0	0
$553$	−918.133	−1.66028
$554$	0	0
$555$	0	0
$556$	0	0
$557$	75.0816	0.134796	0.0673982	−	0.997726i	$-0.478530\pi$
$557$	75.0816	0.134796	0.0673982	+	0.997726i	$0.478530\pi$
$558$	0	0
$559$	−1130.58	−2.02250
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−114.462	−0.203307	−0.101653	−	0.994820i	$-0.532413\pi$
$563$	−114.462	−0.203307	−0.101653	+	0.994820i	$0.532413\pi$
$564$	0	0
$565$	90.5058i	0.160187i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	45.8199i	0.0805270i	0.999189	+	0.0402635i	$0.0128197\pi$
$569$	45.8199i	0.0805270i	−0.999189	+	0.0402635i	$0.987180\pi$
$570$	0	0
$571$	− 830.093i	− 1.45375i	−0.686768	−	0.726877i	$-0.740971\pi$
$571$	− 830.093i	− 1.45375i	0.686768	−	0.726877i	$-0.259029\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 26.6025i	− 0.0462652i
$576$	0	0
$577$	−69.3776	−0.120239	−0.0601193	−	0.998191i	$-0.519148\pi$
$577$	−69.3776	−0.120239	−0.0601193	+	0.998191i	$0.519148\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−55.8991	−0.0962120
$582$	0	0
$583$	587.489	1.00770
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−428.655	−0.730248	−0.365124	−	0.930959i	$-0.618973\pi$
$587$	−428.655	−0.730248	−0.365124	+	0.930959i	$0.618973\pi$
$588$	0	0
$589$	− 96.0127i	− 0.163010i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 77.0276i	− 0.129895i	−0.997889	−	0.0649474i	$-0.979312\pi$
$593$	− 77.0276i	− 0.129895i	0.997889	−	0.0649474i	$-0.0206880\pi$
$594$	0	0
$595$	147.144i	0.247301i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 963.566i	− 1.60862i	−0.594207	−	0.804312i	$-0.702534\pi$
$599$	− 963.566i	− 1.60862i	0.594207	−	0.804312i	$-0.297466\pi$
$600$	0	0
$601$	840.133	1.39789	0.698946	−	0.715175i	$-0.253653\pi$
$601$	840.133	1.39789	0.698946	+	0.715175i	$0.253653\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	157.354	0.260089
$606$	0	0
$607$	−156.611	−0.258008	−0.129004	−	0.991644i	$-0.541178\pi$
$607$	−156.611	−0.258008	−0.129004	+	0.991644i	$0.541178\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1394.25	2.28192
$612$	0	0
$613$	− 426.094i	− 0.695096i	−0.937662	−	0.347548i	$-0.887014\pi$
$613$	− 426.094i	− 0.695096i	0.937662	−	0.347548i	$-0.112986\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	963.520i	1.56162i	0.624769	+	0.780810i	$0.285193\pi$
$617$	963.520i	1.56162i	−0.624769	+	0.780810i	$0.714807\pi$
$618$	0	0
$619$	175.235i	0.283093i	0.989932	+	0.141547i	$0.0452075\pi$
$619$	175.235i	0.283093i	−0.989932	+	0.141547i	$0.954792\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	549.107i	0.881392i
$624$	0	0
$625$	539.666	0.863466
$626$	0	0
$627$	0	0
$628$	0	0
$629$	323.522	0.514344
$630$	0	0
$631$	1140.72	1.80780	0.903900	−	0.427744i	$-0.140692\pi$
$631$	1140.72	1.80780	0.903900	+	0.427744i	$0.140692\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	48.9357	0.0770641
$636$	0	0
$637$	64.8136i	0.101748i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	351.730i	0.548721i	0.961627	+	0.274360i	$0.0884661\pi$
$641$	351.730i	0.548721i	−0.961627	+	0.274360i	$0.911534\pi$
$642$	0	0
$643$	− 86.4181i	− 0.134398i	−0.997740	−	0.0671991i	$-0.978594\pi$
$643$	− 86.4181i	− 0.134398i	0.997740	−	0.0671991i	$-0.0214063\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 430.723i	− 0.665723i	−0.942976	−	0.332861i	$-0.891986\pi$
$647$	− 430.723i	− 0.665723i	0.942976	−	0.332861i	$-0.108014\pi$
$648$	0	0
$649$	−928.888	−1.43126
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−886.670	−1.35784	−0.678920	−	0.734212i	$-0.737552\pi$
$653$	−886.670	−1.35784	−0.678920	+	0.734212i	$0.737552\pi$
$654$	0	0
$655$	138.799	0.211907
$656$	0	0
$657$	0	0
$658$	0	0
$659$	352.833	0.535407	0.267704	−	0.963501i	$-0.413735\pi$
$659$	352.833	0.535407	0.267704	+	0.963501i	$0.413735\pi$
$660$	0	0
$661$	− 1206.26i	− 1.82489i	−0.409195	−	0.912447i	$-0.634190\pi$
$661$	− 1206.26i	− 1.82489i	0.409195	−	0.912447i	$-0.365810\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 132.120i	− 0.198677i
$666$	0	0
$667$	− 32.8882i	− 0.0493076i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	1134.39i	1.69059i
$672$	0	0
$673$	−956.133	−1.42070	−0.710351	−	0.703847i	$-0.751464\pi$
$673$	−956.133	−1.42070	−0.710351	+	0.703847i	$0.751464\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−420.557	−0.621207	−0.310604	−	0.950540i	$-0.600531\pi$
$677$	−420.557	−0.621207	−0.310604	+	0.950540i	$0.600531\pi$
$678$	0	0
$679$	−34.9335	−0.0514485
$680$	0	0
$681$	0	0
$682$	0	0
$683$	539.725	0.790227	0.395113	−	0.918632i	$-0.370705\pi$
$683$	539.725	0.790227	0.395113	+	0.918632i	$0.370705\pi$
$684$	0	0
$685$	11.7567i	0.0171631i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 776.214i	− 1.12658i
$690$	0	0
$691$	− 432.090i	− 0.625312i	−0.949866	−	0.312656i	$-0.898781\pi$
$691$	− 432.090i	− 0.625312i	0.949866	−	0.312656i	$-0.101219\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 1.28926i	− 0.00185504i
$696$	0	0
$697$	521.378	0.748031
$698$	0	0
$699$	0	0
$700$	0	0
$701$	880.339	1.25583	0.627917	−	0.778280i	$-0.283908\pi$
$701$	880.339	1.25583	0.627917	+	0.778280i	$0.283908\pi$
$702$	0	0
$703$	−290.489	−0.413214
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−761.745	−1.07743
$708$	0	0
$709$	− 799.367i	− 1.12746i	−0.825960	−	0.563729i	$-0.809366\pi$
$709$	− 799.367i	− 1.12746i	0.825960	−	0.563729i	$-0.190634\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 6.28491i	− 0.00881474i
$714$	0	0
$715$	− 379.760i	− 0.531133i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	258.786i	0.359924i	0.983674	+	0.179962i	$0.0575975\pi$
$719$	258.786i	0.359924i	−0.983674	+	0.179962i	$0.942402\pi$
$720$	0	0
$721$	755.643	1.04805
$722$	0	0
$723$	0	0
$724$	0	0
$725$	702.896	0.969511
$726$	0	0
$727$	663.211	0.912257	0.456129	−	0.889914i	$-0.349236\pi$
$727$	663.211	0.912257	0.456129	+	0.889914i	$0.349236\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	993.332	1.35887
$732$	0	0
$733$	908.826i	1.23987i	0.784653	+	0.619936i	$0.212841\pi$
$733$	908.826i	1.23987i	−0.784653	+	0.619936i	$0.787159\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1134.39i	1.53920i
$738$	0	0
$739$	− 1112.39i	− 1.50526i	−0.658443	−	0.752631i	$-0.728785\pi$
$739$	− 1112.39i	− 1.50526i	0.658443	−	0.752631i	$-0.271215\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 541.298i	− 0.728530i	−0.931295	−	0.364265i	$-0.881320\pi$
$743$	− 541.298i	− 0.728530i	0.931295	−	0.364265i	$-0.118680\pi$
$744$	0	0
$745$	32.1767	0.0431902
$746$	0	0
$747$	0	0
$748$	0	0
$749$	471.655	0.629713
$750$	0	0
$751$	763.699	1.01691	0.508455	−	0.861089i	$-0.330217\pi$
$751$	763.699	1.01691	0.508455	+	0.861089i	$0.330217\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−66.1354	−0.0875966
$756$	0	0
$757$	419.134i	0.553678i	0.960916	+	0.276839i	$0.0892869\pi$
$757$	419.134i	0.553678i	−0.960916	+	0.276839i	$0.910713\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 1330.24i	− 1.74802i	−0.485912	−	0.874008i	$-0.661512\pi$
$761$	− 1330.24i	− 1.74802i	0.485912	−	0.874008i	$-0.338488\pi$
$762$	0	0
$763$	24.2397i	0.0317690i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1227.29i	1.60011i
$768$	0	0
$769$	671.511	0.873226	0.436613	−	0.899649i	$-0.356178\pi$
$769$	671.511	0.873226	0.436613	+	0.899649i	$0.356178\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	1421.13	1.83846	0.919229	−	0.393724i	$-0.128813\pi$
$773$	1421.13	1.83846	0.919229	+	0.393724i	$0.128813\pi$
$774$	0	0
$775$	134.323	0.173320
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−468.143	−0.600954
$780$	0	0
$781$	− 1610.25i	− 2.06179i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	147.863i	0.188361i
$786$	0	0
$787$	− 649.808i	− 0.825677i	−0.910804	−	0.412839i	$-0.864537\pi$
$787$	− 649.808i	− 0.825677i	0.910804	−	0.412839i	$-0.135463\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 607.122i	− 0.767538i
$792$	0	0
$793$	1498.80	1.89004
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−422.946	−0.530672	−0.265336	−	0.964156i	$-0.585483\pi$
$797$	−422.946	−0.530672	−0.265336	+	0.964156i	$0.585483\pi$
$798$	0	0
$799$	−1225.00	−1.53317
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−616.276	−0.767467
$804$	0	0
$805$	− 8.64847i	− 0.0107434i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1353.87i	1.67351i	0.547574	+	0.836757i	$0.315552\pi$
$809$	1353.87i	1.67351i	−0.547574	+	0.836757i	$0.684448\pi$
$810$	0	0
$811$	− 842.340i	− 1.03864i	−0.854579	−	0.519322i	$-0.826185\pi$
$811$	− 842.340i	− 1.03864i	0.854579	−	0.519322i	$-0.173815\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	205.407i	0.252033i
$816$	0	0
$817$	−891.909	−1.09169
$818$	0	0
$819$	0	0
$820$	0	0
$821$	616.897	0.751397	0.375698	−	0.926742i	$-0.377403\pi$
$821$	616.897	0.751397	0.375698	+	0.926742i	$0.377403\pi$
$822$	0	0
$823$	−980.500	−1.19137	−0.595686	−	0.803217i	$-0.703120\pi$
$823$	−980.500	−1.19137	−0.595686	+	0.803217i	$0.703120\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−70.4951	−0.0852419	−0.0426210	−	0.999091i	$-0.513571\pi$
$827$	−70.4951	−0.0852419	−0.0426210	+	0.999091i	$0.513571\pi$
$828$	0	0
$829$	515.147i	0.621408i	0.950507	+	0.310704i	$0.100565\pi$
$829$	515.147i	0.621408i	−0.950507	+	0.310704i	$0.899435\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 56.9456i	− 0.0683621i
$834$	0	0
$835$	250.606i	0.300127i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	47.1556i	0.0562045i	0.999605	+	0.0281022i	$0.00894640\pi$
$839$	47.1556i	0.0562045i	−0.999605	+	0.0281022i	$0.991054\pi$
$840$	0	0
$841$	27.9773	0.0332667
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−320.082	−0.378795
$846$	0	0
$847$	−1055.54	−1.24622
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−19.0152	−0.0223445
$852$	0	0
$853$	763.118i	0.894628i	0.894377	+	0.447314i	$0.147619\pi$
$853$	763.118i	0.894628i	−0.894377	+	0.447314i	$0.852381\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 1641.05i	− 1.91488i	−0.288631	−	0.957440i	$-0.593200\pi$
$857$	− 1641.05i	− 1.91488i	0.288631	−	0.957440i	$-0.406800\pi$
$858$	0	0
$859$	1366.84i	1.59120i	0.605819	+	0.795602i	$0.292845\pi$
$859$	1366.84i	1.59120i	−0.605819	+	0.795602i	$0.707155\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	836.082i	0.968809i	0.874844	+	0.484405i	$0.160964\pi$
$863$	836.082i	0.968809i	−0.874844	+	0.484405i	$0.839036\pi$
$864$	0	0
$865$	−348.089	−0.402415
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−2081.93	−2.39578
$870$	0	0
$871$	1498.80	1.72078
$872$	0	0
$873$	0	0
$874$	0	0
$875$	378.633	0.432723
$876$	0	0
$877$	158.191i	0.180377i	0.995925	+	0.0901887i	$0.0287470\pi$
$877$	158.191i	0.180377i	−0.995925	+	0.0901887i	$0.971253\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	237.430i	0.269500i	0.990880	+	0.134750i	$0.0430232\pi$
$881$	237.430i	0.269500i	−0.990880	+	0.134750i	$0.956977\pi$
$882$	0	0
$883$	592.933i	0.671499i	0.941951	+	0.335749i	$0.108989\pi$
$883$	592.933i	0.671499i	−0.941951	+	0.335749i	$0.891011\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 1420.72i	− 1.60171i	−0.598858	−	0.800855i	$-0.704379\pi$
$887$	− 1420.72i	− 1.60171i	0.598858	−	0.800855i	$-0.295621\pi$
$888$	0	0
$889$	−328.266	−0.369253
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1099.92	1.23172
$894$	0	0
$895$	−199.866	−0.223313
$896$	0	0
$897$	0	0
$898$	0	0
$899$	166.061	0.184717
$900$	0	0
$901$	681.987i	0.756922i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	135.734i	0.149982i
$906$	0	0
$907$	816.648i	0.900383i	0.892932	+	0.450192i	$0.148644\pi$
$907$	816.648i	0.900383i	−0.892932	+	0.450192i	$0.851356\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 469.895i	− 0.515801i	−0.966171	−	0.257901i	$-0.916969\pi$
$911$	− 469.895i	− 0.515801i	0.966171	−	0.257901i	$-0.0830307\pi$
$912$	0	0
$913$	−126.755	−0.138834
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−931.079	−1.01535
$918$	0	0
$919$	1648.21	1.79348	0.896741	−	0.442555i	$-0.145928\pi$
$919$	1648.21	1.79348	0.896741	+	0.442555i	$0.145928\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−2127.53	−2.30502
$924$	0	0
$925$	− 406.398i	− 0.439349i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	421.311i	0.453510i	0.973952	+	0.226755i	$0.0728116\pi$
$929$	421.311i	0.453510i	−0.973952	+	0.226755i	$0.927188\pi$
$930$	0	0
$931$	51.1313i	0.0549208i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	333.659i	0.356855i
$936$	0	0
$937$	−1434.27	−1.53070	−0.765350	−	0.643614i	$-0.777434\pi$
$937$	−1434.27	−1.53070	−0.765350	+	0.643614i	$0.777434\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−732.876	−0.778827	−0.389413	−	0.921063i	$-0.627322\pi$
$941$	−732.876	−0.778827	−0.389413	+	0.921063i	$0.627322\pi$
$942$	0	0
$943$	−30.6443	−0.0324966
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1385.53	1.46308	0.731538	−	0.681800i	$-0.238803\pi$
$947$	1385.53	1.46308	0.731538	+	0.681800i	$0.238803\pi$
$948$	0	0
$949$	814.249i	0.858007i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 957.328i	− 1.00454i	−0.864711	−	0.502270i	$-0.832498\pi$
$953$	− 957.328i	− 1.00454i	0.864711	−	0.502270i	$-0.167502\pi$
$954$	0	0
$955$	− 250.606i	− 0.262414i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 78.8654i	− 0.0822371i
$960$	0	0
$961$	−929.266	−0.966978
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−126.179	−0.130755
$966$	0	0
$967$	−1225.90	−1.26773	−0.633867	−	0.773442i	$-0.718533\pi$
$967$	−1225.90	−1.26773	−0.633867	+	0.773442i	$0.718533\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	752.906	0.775393	0.387696	−	0.921787i	$-0.373271\pi$
$971$	752.906	0.775393	0.387696	+	0.921787i	$0.373271\pi$
$972$	0	0
$973$	8.64847i	0.00888845i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 642.932i	− 0.658068i	−0.944318	−	0.329034i	$-0.893277\pi$
$977$	− 642.932i	− 0.658068i	0.944318	−	0.329034i	$-0.106723\pi$
$978$	0	0
$979$	1245.14i	1.27185i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 161.251i	− 0.164040i	−0.996631	−	0.0820200i	$-0.973863\pi$
$983$	− 161.251i	− 0.164040i	0.996631	−	0.0820200i	$-0.0261371\pi$
$984$	0	0
$985$	−318.532	−0.323382
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−58.3836	−0.0590330
$990$	0	0
$991$	−811.500	−0.818870	−0.409435	−	0.912339i	$-0.634274\pi$
$991$	−811.500	−0.818870	−0.409435	+	0.912339i	$0.634274\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−250.483	−0.251742
$996$	0	0
$997$	− 390.554i	− 0.391729i	−0.980631	−	0.195864i	$-0.937249\pi$
$997$	− 390.554i	− 0.391729i	0.980631	−	0.195864i	$-0.0627513\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	288.3.h.a.17.6		8
3.2	odd	2	inner	288.3.h.a.17.4		8
4.3	odd	2		72.3.h.a.53.2	yes	8
8.3	odd	2		72.3.h.a.53.8	yes	8
8.5	even	2	inner	288.3.h.a.17.3		8
12.11	even	2		72.3.h.a.53.7	yes	8
16.3	odd	4		2304.3.e.n.1025.3		8
16.5	even	4		2304.3.e.o.1025.6		8
16.11	odd	4		2304.3.e.n.1025.6		8
16.13	even	4		2304.3.e.o.1025.3		8
24.5	odd	2	inner	288.3.h.a.17.5		8
24.11	even	2		72.3.h.a.53.1	✓	8
48.5	odd	4		2304.3.e.o.1025.4		8
48.11	even	4		2304.3.e.n.1025.4		8
48.29	odd	4		2304.3.e.o.1025.5		8
48.35	even	4		2304.3.e.n.1025.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.3.h.a.53.1	✓	8	24.11	even	2
72.3.h.a.53.2	yes	8	4.3	odd	2
72.3.h.a.53.7	yes	8	12.11	even	2
72.3.h.a.53.8	yes	8	8.3	odd	2
288.3.h.a.17.3		8	8.5	even	2	inner
288.3.h.a.17.4		8	3.2	odd	2	inner
288.3.h.a.17.5		8	24.5	odd	2	inner
288.3.h.a.17.6		8	1.1	even	1	trivial
2304.3.e.n.1025.3		8	16.3	odd	4
2304.3.e.n.1025.4		8	48.11	even	4
2304.3.e.n.1025.5		8	48.35	even	4
2304.3.e.n.1025.6		8	16.11	odd	4
2304.3.e.o.1025.3		8	16.13	even	4
2304.3.e.o.1025.4		8	48.5	odd	4
2304.3.e.o.1025.5		8	48.29	odd	4
2304.3.e.o.1025.6		8	16.5	even	4

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:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	288.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.07498 q^{5} -7.21110 q^{7} +O(q^{10})\)	\(q+1.07498 q^{5} -7.21110 q^{7} -16.3517 q^{11} +21.6045i q^{13} -18.9819i q^{17} +17.0438i q^{19} +1.11567i q^{23} -23.8444 q^{25} -29.4784 q^{29} -5.63331 q^{31} -7.75182 q^{35} +17.0438i q^{37} +27.4671i q^{41} +52.3306i q^{43} -64.5352i q^{47} +3.00000 q^{49} -35.9283 q^{53} -17.5778 q^{55} +56.8069 q^{59} -69.3743i q^{61} +23.2245i q^{65} -69.3743i q^{67} +98.4764i q^{71} +37.6888 q^{73} +117.914 q^{77} +127.322 q^{79} +7.75182 q^{83} -20.4052i q^{85} -76.1475i q^{89} -155.792i q^{91} +18.3218i q^{95} +4.84441 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 40 q^{25} + 128 q^{31} + 24 q^{49} - 256 q^{55} - 160 q^{73} + 384 q^{79} - 192 q^{97}+O(q^{100}) \) 8 * q + 40 * q^25 + 128 * q^31 + 24 * q^49 - 256 * q^55 - 160 * q^73 + 384 * q^79 - 192 * q^97

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