LMFDB - Embedded newform 8280.2.a.bn.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8280,2,Mod(1,8280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8280, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("8280.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8280 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8280.a (trivial)

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:	yes
Analytic conductor:	$66.1161328736$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.1436.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 11x - 12 $ x^3 - 11*x - 12
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2760)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.48361$ of defining polynomial
Character	$\chi$	$=$	8280.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} +3.13555 q^{7} +O(q^{10})$	$q+1.00000 q^{5} +3.13555 q^{7} -2.65194 q^{11} +0.651938 q^{13} -4.48361 q^{17} -0.651938 q^{19} -1.00000 q^{23} +1.00000 q^{25} -2.48361 q^{29} -3.13555 q^{31} +3.13555 q^{35} -6.10277 q^{37} -0.820265 q^{41} +0.696123 q^{43} +11.8903 q^{47} +2.83167 q^{49} -9.45083 q^{53} -2.65194 q^{55} -12.7547 q^{59} +2.65194 q^{61} +0.651938 q^{65} -6.10277 q^{67} -8.48361 q^{71} +6.31528 q^{73} -8.31528 q^{77} -8.00000 q^{79} -5.51639 q^{83} -4.48361 q^{85} +11.2383 q^{89} +2.04419 q^{91} -0.651938 q^{95} -5.93445 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{5} - 2 q^{7}+O(q^{10}) $ 3 * q + 3 * q^5 - 2 * q^7	$ 3 q + 3 q^{5} - 2 q^{7} - 4 q^{11} - 2 q^{13} - 6 q^{17} + 2 q^{19} - 3 q^{23} + 3 q^{25} + 2 q^{31} - 2 q^{35} + 8 q^{37} - 2 q^{41} + 10 q^{43} - 6 q^{47} + 5 q^{49} - 6 q^{53} - 4 q^{55} - 8 q^{59} + 4 q^{61} - 2 q^{65} + 8 q^{67} - 18 q^{71} + 8 q^{73} - 14 q^{77} - 24 q^{79} - 24 q^{83} - 6 q^{85} - 4 q^{89} + 18 q^{91} + 2 q^{95} + 12 q^{97}+O(q^{100}) $ 3 * q + 3 * q^5 - 2 * q^7 - 4 * q^11 - 2 * q^13 - 6 * q^17 + 2 * q^19 - 3 * q^23 + 3 * q^25 + 2 * q^31 - 2 * q^35 + 8 * q^37 - 2 * q^41 + 10 * q^43 - 6 * q^47 + 5 * q^49 - 6 * q^53 - 4 * q^55 - 8 * q^59 + 4 * q^61 - 2 * q^65 + 8 * q^67 - 18 * q^71 + 8 * q^73 - 14 * q^77 - 24 * q^79 - 24 * q^83 - 6 * q^85 - 4 * q^89 + 18 * q^91 + 2 * q^95 + 12 * q^97

Display 100 coefficients 10 coefficients

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
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	3.13555	1.18513	0.592563	−	0.805524i	$-0.298116\pi$
$7$	3.13555	1.18513	0.592563	+	0.805524i	$0.298116\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−2.65194	−0.799589	−0.399795	−	0.916605i	$-0.630919\pi$
$11$	−2.65194	−0.799589	−0.399795	+	0.916605i	$0.630919\pi$
$12$	0	0
$13$	0.651938	0.180815	0.0904076	−	0.995905i	$-0.471183\pi$
$13$	0.651938	0.180815	0.0904076	+	0.995905i	$0.471183\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.48361	−1.08744	−0.543718	−	0.839268i	$-0.682984\pi$
$17$	−4.48361	−1.08744	−0.543718	+	0.839268i	$0.682984\pi$
$18$	0	0
$19$	−0.651938	−0.149565	−0.0747825	−	0.997200i	$-0.523826\pi$
$19$	−0.651938	−0.149565	−0.0747825	+	0.997200i	$0.523826\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−2.48361	−0.461195	−0.230598	−	0.973049i	$-0.574068\pi$
$29$	−2.48361	−0.461195	−0.230598	+	0.973049i	$0.574068\pi$
$30$	0	0
$31$	−3.13555	−0.563161	−0.281581	−	0.959538i	$-0.590859\pi$
$31$	−3.13555	−0.563161	−0.281581	+	0.959538i	$0.590859\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	3.13555	0.530005
$36$	0	0
$37$	−6.10277	−1.00329	−0.501645	−	0.865074i	$-0.667272\pi$
$37$	−6.10277	−1.00329	−0.501645	+	0.865074i	$0.667272\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−0.820265	−0.128104	−0.0640519	−	0.997947i	$-0.520402\pi$
$41$	−0.820265	−0.128104	−0.0640519	+	0.997947i	$0.520402\pi$
$42$	0	0
$43$	0.696123	0.106158	0.0530789	−	0.998590i	$-0.483097\pi$
$43$	0.696123	0.106158	0.0530789	+	0.998590i	$0.483097\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	11.8903	1.73437	0.867186	−	0.497984i	$-0.165926\pi$
$47$	11.8903	1.73437	0.867186	+	0.497984i	$0.165926\pi$
$48$	0	0
$49$	2.83167	0.404525
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−9.45083	−1.29817	−0.649086	−	0.760715i	$-0.724848\pi$
$53$	−9.45083	−1.29817	−0.649086	+	0.760715i	$0.724848\pi$
$54$	0	0
$55$	−2.65194	−0.357587
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−12.7547	−1.66052	−0.830261	−	0.557375i	$-0.811809\pi$
$59$	−12.7547	−1.66052	−0.830261	+	0.557375i	$0.811809\pi$
$60$	0	0
$61$	2.65194	0.339546	0.169773	−	0.985483i	$-0.445697\pi$
$61$	2.65194	0.339546	0.169773	+	0.985483i	$0.445697\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0.651938	0.0808630
$66$	0	0
$67$	−6.10277	−0.745572	−0.372786	−	0.927917i	$-0.621598\pi$
$67$	−6.10277	−0.745572	−0.372786	+	0.927917i	$0.621598\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−8.48361	−1.00682	−0.503410	−	0.864048i	$-0.667921\pi$
$71$	−8.48361	−1.00682	−0.503410	+	0.864048i	$0.667921\pi$
$72$	0	0
$73$	6.31528	0.739148	0.369574	−	0.929201i	$-0.379504\pi$
$73$	6.31528	0.739148	0.369574	+	0.929201i	$0.379504\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−8.31528	−0.947615
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−5.51639	−0.605502	−0.302751	−	0.953070i	$-0.597905\pi$
$83$	−5.51639	−0.605502	−0.302751	+	0.953070i	$0.597905\pi$
$84$	0	0
$85$	−4.48361	−0.486316
$86$	0	0
$87$	0	0
$88$	0	0
$89$	11.2383	1.19126	0.595630	−	0.803259i	$-0.296902\pi$
$89$	11.2383	1.19126	0.595630	+	0.803259i	$0.296902\pi$
$90$	0	0
$91$	2.04419	0.214289
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−0.651938	−0.0668875
$96$	0	0
$97$	−5.93445	−0.602552	−0.301276	−	0.953537i	$-0.597413\pi$
$97$	−5.93445	−0.602552	−0.301276	+	0.953537i	$0.597413\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−8.75471	−0.871126	−0.435563	−	0.900158i	$-0.643451\pi$
$101$	−8.75471	−0.871126	−0.435563	+	0.900158i	$0.643451\pi$
$102$	0	0
$103$	−3.66335	−0.360960	−0.180480	−	0.983579i	$-0.557765\pi$
$103$	−3.66335	−0.360960	−0.180480	+	0.983579i	$0.557765\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−3.24529	−0.313734	−0.156867	−	0.987620i	$-0.550139\pi$
$107$	−3.24529	−0.313734	−0.156867	+	0.987620i	$0.550139\pi$
$108$	0	0
$109$	3.61916	0.346653	0.173326	−	0.984864i	$-0.444548\pi$
$109$	3.61916	0.346653	0.173326	+	0.984864i	$0.444548\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−3.51639	−0.330794	−0.165397	−	0.986227i	$-0.552891\pi$
$113$	−3.51639	−0.330794	−0.165397	+	0.986227i	$0.552891\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−14.0586	−1.28875
$120$	0	0
$121$	−3.96722	−0.360657
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	9.89026	0.877619	0.438809	−	0.898580i	$-0.355400\pi$
$127$	9.89026	0.877619	0.438809	+	0.898580i	$0.355400\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−21.9344	−1.91642	−0.958211	−	0.286063i	$-0.907653\pi$
$131$	−21.9344	−1.91642	−0.958211	+	0.286063i	$0.907653\pi$
$132$	0	0
$133$	−2.04419	−0.177253
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−6.33665	−0.541377	−0.270688	−	0.962667i	$-0.587251\pi$
$137$	−6.33665	−0.541377	−0.270688	+	0.962667i	$0.587251\pi$
$138$	0	0
$139$	−7.13555	−0.605229	−0.302615	−	0.953113i	$-0.597860\pi$
$139$	−7.13555	−0.605229	−0.302615	+	0.953113i	$0.597860\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−1.72890	−0.144578
$144$	0	0
$145$	−2.48361	−0.206253
$146$	0	0
$147$	0	0
$148$	0	0
$149$	14.3153	1.17275	0.586377	−	0.810038i	$-0.300554\pi$
$149$	14.3153	1.17275	0.586377	+	0.810038i	$0.300554\pi$
$150$	0	0
$151$	11.5750	0.941958	0.470979	−	0.882144i	$-0.343901\pi$
$151$	11.5750	0.941958	0.470979	+	0.882144i	$0.343901\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−3.13555	−0.251853
$156$	0	0
$157$	13.7661	1.09866	0.549328	−	0.835607i	$-0.314884\pi$
$157$	13.7661	1.09866	0.549328	+	0.835607i	$0.314884\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−3.13555	−0.247116
$162$	0	0
$163$	6.27110	0.491190	0.245595	−	0.969372i	$-0.421017\pi$
$163$	6.27110	0.491190	0.245595	+	0.969372i	$0.421017\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−13.2825	−1.02783	−0.513916	−	0.857841i	$-0.671806\pi$
$167$	−13.2825	−1.02783	−0.513916	+	0.857841i	$0.671806\pi$
$168$	0	0
$169$	−12.5750	−0.967306
$170$	0	0
$171$	0	0
$172$	0	0
$173$	8.27110	0.628840	0.314420	−	0.949284i	$-0.398190\pi$
$173$	8.27110	0.628840	0.314420	+	0.949284i	$0.398190\pi$
$174$	0	0
$175$	3.13555	0.237025
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−5.64053	−0.421593	−0.210797	−	0.977530i	$-0.567606\pi$
$179$	−5.64053	−0.421593	−0.210797	+	0.977530i	$0.567606\pi$
$180$	0	0
$181$	0.359470	0.0267192	0.0133596	−	0.999911i	$-0.495747\pi$
$181$	0.359470	0.0267192	0.0133596	+	0.999911i	$0.495747\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−6.10277	−0.448685
$186$	0	0
$187$	11.8903	0.869502
$188$	0	0
$189$	0	0
$190$	0	0
$191$	19.1286	1.38410	0.692048	−	0.721852i	$-0.256709\pi$
$191$	19.1286	1.38410	0.692048	+	0.721852i	$0.256709\pi$
$192$	0	0
$193$	−10.9672	−0.789438	−0.394719	−	0.918802i	$-0.629158\pi$
$193$	−10.9672	−0.789438	−0.394719	+	0.918802i	$0.629158\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	13.2383	0.943191	0.471596	−	0.881815i	$-0.343678\pi$
$197$	13.2383	0.943191	0.471596	+	0.881815i	$0.343678\pi$
$198$	0	0
$199$	−2.35947	−0.167258	−0.0836292	−	0.996497i	$-0.526651\pi$
$199$	−2.35947	−0.167258	−0.0836292	+	0.996497i	$0.526651\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−7.78749	−0.546575
$204$	0	0
$205$	−0.820265	−0.0572898
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1.72890	0.119591
$210$	0	0
$211$	15.1355	1.04197	0.520987	−	0.853565i	$-0.325564\pi$
$211$	15.1355	1.04197	0.520987	+	0.853565i	$0.325564\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0.696123	0.0474752
$216$	0	0
$217$	−9.83167	−0.667417
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−2.92304	−0.196625
$222$	0	0
$223$	−12.6078	−0.844278	−0.422139	−	0.906531i	$-0.638721\pi$
$223$	−12.6078	−0.844278	−0.422139	+	0.906531i	$0.638721\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4.54220	0.301476	0.150738	−	0.988574i	$-0.451835\pi$
$227$	4.54220	0.301476	0.150738	+	0.988574i	$0.451835\pi$
$228$	0	0
$229$	1.57498	0.104077	0.0520387	−	0.998645i	$-0.483428\pi$
$229$	1.57498	0.104077	0.0520387	+	0.998645i	$0.483428\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6.33665	0.415128	0.207564	−	0.978221i	$-0.433446\pi$
$233$	6.33665	0.415128	0.207564	+	0.978221i	$0.433446\pi$
$234$	0	0
$235$	11.8903	0.775635
$236$	0	0
$237$	0	0
$238$	0	0
$239$	7.09136	0.458702	0.229351	−	0.973344i	$-0.426340\pi$
$239$	7.09136	0.458702	0.229351	+	0.973344i	$0.426340\pi$
$240$	0	0
$241$	−5.68472	−0.366185	−0.183092	−	0.983096i	$-0.558611\pi$
$241$	−5.68472	−0.366185	−0.183092	+	0.983096i	$0.558611\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2.83167	0.180909
$246$	0	0
$247$	−0.425024	−0.0270436
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−27.9344	−1.76321	−0.881603	−	0.471991i	$-0.843535\pi$
$251$	−27.9344	−1.76321	−0.881603	+	0.471991i	$0.843535\pi$
$252$	0	0
$253$	2.65194	0.166726
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−7.61916	−0.475270	−0.237635	−	0.971354i	$-0.576372\pi$
$257$	−7.61916	−0.475270	−0.237635	+	0.971354i	$0.576372\pi$
$258$	0	0
$259$	−19.1355	−1.18903
$260$	0	0
$261$	0	0
$262$	0	0
$263$	5.85304	0.360914	0.180457	−	0.983583i	$-0.442242\pi$
$263$	5.85304	0.360914	0.180457	+	0.983583i	$0.442242\pi$
$264$	0	0
$265$	−9.45083	−0.580560
$266$	0	0
$267$	0	0
$268$	0	0
$269$	15.7875	0.962580	0.481290	−	0.876561i	$-0.340168\pi$
$269$	15.7875	0.962580	0.481290	+	0.876561i	$0.340168\pi$
$270$	0	0
$271$	3.56057	0.216289	0.108145	−	0.994135i	$-0.465509\pi$
$271$	3.56057	0.216289	0.108145	+	0.994135i	$0.465509\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−2.65194	−0.159918
$276$	0	0
$277$	−20.8789	−1.25449	−0.627244	−	0.778823i	$-0.715817\pi$
$277$	−20.8789	−1.25449	−0.627244	+	0.778823i	$0.715817\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−11.5536	−0.689230	−0.344615	−	0.938744i	$-0.611991\pi$
$281$	−11.5536	−0.689230	−0.344615	+	0.938744i	$0.611991\pi$
$282$	0	0
$283$	7.83167	0.465545	0.232772	−	0.972531i	$-0.425220\pi$
$283$	7.83167	0.465545	0.232772	+	0.972531i	$0.425220\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2.57198	−0.151819
$288$	0	0
$289$	3.10277	0.182516
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−3.42802	−0.200267	−0.100133	−	0.994974i	$-0.531927\pi$
$293$	−3.42802	−0.200267	−0.100133	+	0.994974i	$0.531927\pi$
$294$	0	0
$295$	−12.7547	−0.742608
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−0.651938	−0.0377026
$300$	0	0
$301$	2.18273	0.125810
$302$	0	0
$303$	0	0
$304$	0	0
$305$	2.65194	0.151850
$306$	0	0
$307$	−6.67475	−0.380948	−0.190474	−	0.981692i	$-0.561003\pi$
$307$	−6.67475	−0.380948	−0.190474	+	0.981692i	$0.561003\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	14.2055	0.805523	0.402761	−	0.915305i	$-0.368050\pi$
$311$	14.2055	0.805523	0.402761	+	0.915305i	$0.368050\pi$
$312$	0	0
$313$	1.83167	0.103532	0.0517661	−	0.998659i	$-0.483515\pi$
$313$	1.83167	0.103532	0.0517661	+	0.998659i	$0.483515\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	16.9230	0.950493	0.475246	−	0.879853i	$-0.342359\pi$
$317$	16.9230	0.950493	0.475246	+	0.879853i	$0.342359\pi$
$318$	0	0
$319$	6.58638	0.368767
$320$	0	0
$321$	0	0
$322$	0	0
$323$	2.92304	0.162642
$324$	0	0
$325$	0.651938	0.0361630
$326$	0	0
$327$	0	0
$328$	0	0
$329$	37.2825	2.05545
$330$	0	0
$331$	−18.3739	−1.00992	−0.504960	−	0.863143i	$-0.668493\pi$
$331$	−18.3739	−1.00992	−0.504960	+	0.863143i	$0.668493\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−6.10277	−0.333430
$336$	0	0
$337$	26.8133	1.46061	0.730307	−	0.683119i	$-0.239377\pi$
$337$	26.8133	1.46061	0.730307	+	0.683119i	$0.239377\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	8.31528	0.450298
$342$	0	0
$343$	−13.0700	−0.705713
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−28.4538	−1.52748	−0.763741	−	0.645523i	$-0.776639\pi$
$347$	−28.4538	−1.52748	−0.763741	+	0.645523i	$0.776639\pi$
$348$	0	0
$349$	−4.79890	−0.256879	−0.128440	−	0.991717i	$-0.540997\pi$
$349$	−4.79890	−0.256879	−0.128440	+	0.991717i	$0.540997\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	20.7919	1.10664	0.553321	−	0.832968i	$-0.313360\pi$
$353$	20.7919	1.10664	0.553321	+	0.832968i	$0.313360\pi$
$354$	0	0
$355$	−8.48361	−0.450263
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−2.04419	−0.107888	−0.0539440	−	0.998544i	$-0.517179\pi$
$359$	−2.04419	−0.107888	−0.0539440	+	0.998544i	$0.517179\pi$
$360$	0	0
$361$	−18.5750	−0.977630
$362$	0	0
$363$	0	0
$364$	0	0
$365$	6.31528	0.330557
$366$	0	0
$367$	0.527797	0.0275508	0.0137754	−	0.999905i	$-0.495615\pi$
$367$	0.527797	0.0275508	0.0137754	+	0.999905i	$0.495615\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−29.6336	−1.53850
$372$	0	0
$373$	29.4439	1.52455	0.762273	−	0.647256i	$-0.224083\pi$
$373$	29.4439	1.52455	0.762273	+	0.647256i	$0.224083\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−1.61916	−0.0833911
$378$	0	0
$379$	7.32669	0.376347	0.188173	−	0.982136i	$-0.439743\pi$
$379$	7.32669	0.376347	0.188173	+	0.982136i	$0.439743\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	5.42802	0.277359	0.138679	−	0.990337i	$-0.455714\pi$
$383$	5.42802	0.277359	0.138679	+	0.990337i	$0.455714\pi$
$384$	0	0
$385$	−8.31528	−0.423786
$386$	0	0
$387$	0	0
$388$	0	0
$389$	28.3883	1.43934	0.719671	−	0.694315i	$-0.244292\pi$
$389$	28.3883	1.43934	0.719671	+	0.694315i	$0.244292\pi$
$390$	0	0
$391$	4.48361	0.226746
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−8.00000	−0.402524
$396$	0	0
$397$	−14.6078	−0.733142	−0.366571	−	0.930390i	$-0.619468\pi$
$397$	−14.6078	−0.733142	−0.366571	+	0.930390i	$0.619468\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	19.5750	0.977528	0.488764	−	0.872416i	$-0.337448\pi$
$401$	19.5750	0.977528	0.488764	+	0.872416i	$0.337448\pi$
$402$	0	0
$403$	−2.04419	−0.101828
$404$	0	0
$405$	0	0
$406$	0	0
$407$	16.1842	0.802220
$408$	0	0
$409$	−10.5278	−0.520566	−0.260283	−	0.965532i	$-0.583816\pi$
$409$	−10.5278	−0.520566	−0.260283	+	0.965532i	$0.583816\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−39.9930	−1.96793
$414$	0	0
$415$	−5.51639	−0.270789
$416$	0	0
$417$	0	0
$418$	0	0
$419$	11.1941	0.546869	0.273435	−	0.961891i	$-0.411840\pi$
$419$	11.1941	0.546869	0.273435	+	0.961891i	$0.411840\pi$
$420$	0	0
$421$	−23.5308	−1.14682	−0.573410	−	0.819268i	$-0.694380\pi$
$421$	−23.5308	−1.14682	−0.573410	+	0.819268i	$0.694380\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−4.48361	−0.217487
$426$	0	0
$427$	8.31528	0.402405
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−27.2383	−1.31202	−0.656012	−	0.754751i	$-0.727758\pi$
$431$	−27.2383	−1.31202	−0.656012	+	0.754751i	$0.727758\pi$
$432$	0	0
$433$	13.8317	0.664708	0.332354	−	0.943155i	$-0.392157\pi$
$433$	13.8317	0.664708	0.332354	+	0.943155i	$0.392157\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0.651938	0.0311864
$438$	0	0
$439$	−33.4211	−1.59510	−0.797550	−	0.603253i	$-0.793871\pi$
$439$	−33.4211	−1.59510	−0.797550	+	0.603253i	$0.793871\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	31.1286	1.47896	0.739482	−	0.673177i	$-0.235071\pi$
$443$	31.1286	1.47896	0.739482	+	0.673177i	$0.235071\pi$
$444$	0	0
$445$	11.2383	0.532748
$446$	0	0
$447$	0	0
$448$	0	0
$449$	23.0030	1.08558	0.542789	−	0.839869i	$-0.317368\pi$
$449$	23.0030	1.08558	0.542789	+	0.839869i	$0.317368\pi$
$450$	0	0
$451$	2.17529	0.102431
$452$	0	0
$453$	0	0
$454$	0	0
$455$	2.04419	0.0958329
$456$	0	0
$457$	−0.233880	−0.0109405	−0.00547023	−	0.999985i	$-0.501741\pi$
$457$	−0.233880	−0.0109405	−0.00547023	+	0.999985i	$0.501741\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−13.5094	−0.629197	−0.314598	−	0.949225i	$-0.601870\pi$
$461$	−13.5094	−0.629197	−0.314598	+	0.949225i	$0.601870\pi$
$462$	0	0
$463$	−27.3997	−1.27337	−0.636686	−	0.771123i	$-0.719695\pi$
$463$	−27.3997	−1.27337	−0.636686	+	0.771123i	$0.719695\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−9.17974	−0.424787	−0.212394	−	0.977184i	$-0.568126\pi$
$467$	−9.17974	−0.424787	−0.212394	+	0.977184i	$0.568126\pi$
$468$	0	0
$469$	−19.1355	−0.883598
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−1.84608	−0.0848827
$474$	0	0
$475$	−0.651938	−0.0299130
$476$	0	0
$477$	0	0
$478$	0	0
$479$	2.04419	0.0934012	0.0467006	−	0.998909i	$-0.485129\pi$
$479$	2.04419	0.0934012	0.0467006	+	0.998909i	$0.485129\pi$
$480$	0	0
$481$	−3.97863	−0.181410
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−5.93445	−0.269469
$486$	0	0
$487$	29.0402	1.31594	0.657969	−	0.753045i	$-0.271416\pi$
$487$	29.0402	1.31594	0.657969	+	0.753045i	$0.271416\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−37.9275	−1.71164	−0.855822	−	0.517271i	$-0.826948\pi$
$491$	−37.9275	−1.71164	−0.855822	+	0.517271i	$0.826948\pi$
$492$	0	0
$493$	11.1355	0.501520
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−26.6008	−1.19321
$498$	0	0
$499$	−12.7761	−0.571936	−0.285968	−	0.958239i	$-0.592315\pi$
$499$	−12.7761	−0.571936	−0.285968	+	0.958239i	$0.592315\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−38.2641	−1.70611	−0.853057	−	0.521818i	$-0.825254\pi$
$503$	−38.2641	−1.70611	−0.853057	+	0.521818i	$0.825254\pi$
$504$	0	0
$505$	−8.75471	−0.389580
$506$	0	0
$507$	0	0
$508$	0	0
$509$	28.0000	1.24108	0.620539	−	0.784176i	$-0.286914\pi$
$509$	28.0000	1.24108	0.620539	+	0.784176i	$0.286914\pi$
$510$	0	0
$511$	19.8019	0.875984
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−3.66335	−0.161426
$516$	0	0
$517$	−31.5322	−1.38679
$518$	0	0
$519$	0	0
$520$	0	0
$521$	24.7691	1.08515	0.542577	−	0.840006i	$-0.317449\pi$
$521$	24.7691	1.08515	0.542577	+	0.840006i	$0.317449\pi$
$522$	0	0
$523$	−0.271100	−0.0118544	−0.00592718	−	0.999982i	$-0.501887\pi$
$523$	−0.271100	−0.0118544	−0.00592718	+	0.999982i	$0.501887\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	14.0586	0.612402
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−0.534762	−0.0231631
$534$	0	0
$535$	−3.24529	−0.140306
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−7.50942	−0.323454
$540$	0	0
$541$	−40.1400	−1.72575	−0.862877	−	0.505415i	$-0.831340\pi$
$541$	−40.1400	−1.72575	−0.862877	+	0.505415i	$0.831340\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	3.61916	0.155028
$546$	0	0
$547$	−4.67331	−0.199816	−0.0999081	−	0.994997i	$-0.531855\pi$
$547$	−4.67331	−0.199816	−0.0999081	+	0.994997i	$0.531855\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1.61916	0.0689786
$552$	0	0
$553$	−25.0844	−1.06670
$554$	0	0
$555$	0	0
$556$	0	0
$557$	21.5680	0.913866	0.456933	−	0.889501i	$-0.348948\pi$
$557$	21.5680	0.913866	0.456933	+	0.889501i	$0.348948\pi$
$558$	0	0
$559$	0.453830	0.0191949
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−31.9047	−1.34462	−0.672311	−	0.740269i	$-0.734698\pi$
$563$	−31.9047	−1.34462	−0.672311	+	0.740269i	$0.734698\pi$
$564$	0	0
$565$	−3.51639	−0.147936
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−15.4439	−0.647441	−0.323720	−	0.946153i	$-0.604934\pi$
$569$	−15.4439	−0.647441	−0.323720	+	0.946153i	$0.604934\pi$
$570$	0	0
$571$	−28.6380	−1.19846	−0.599232	−	0.800576i	$-0.704527\pi$
$571$	−28.6380	−1.19846	−0.599232	+	0.800576i	$0.704527\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	−1.12115	−0.0466740	−0.0233370	−	0.999728i	$-0.507429\pi$
$577$	−1.12115	−0.0466740	−0.0233370	+	0.999728i	$0.507429\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−17.2969	−0.717597
$582$	0	0
$583$	25.0630	1.03800
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−28.2055	−1.16417	−0.582084	−	0.813129i	$-0.697762\pi$
$587$	−28.2055	−1.16417	−0.582084	+	0.813129i	$0.697762\pi$
$588$	0	0
$589$	2.04419	0.0842292
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−19.5308	−0.802033	−0.401017	−	0.916071i	$-0.631343\pi$
$593$	−19.5308	−0.802033	−0.401017	+	0.916071i	$0.631343\pi$
$594$	0	0
$595$	−14.0586	−0.576346
$596$	0	0
$597$	0	0
$598$	0	0
$599$	32.9017	1.34433	0.672163	−	0.740403i	$-0.265365\pi$
$599$	32.9017	1.34433	0.672163	+	0.740403i	$0.265365\pi$
$600$	0	0
$601$	−19.2011	−0.783229	−0.391615	−	0.920129i	$-0.628083\pi$
$601$	−19.2011	−0.783229	−0.391615	+	0.920129i	$0.628083\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−3.96722	−0.161291
$606$	0	0
$607$	−23.2825	−0.945008	−0.472504	−	0.881329i	$-0.656650\pi$
$607$	−23.2825	−0.945008	−0.472504	+	0.881329i	$0.656650\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	7.75172	0.313601
$612$	0	0
$613$	−5.86889	−0.237042	−0.118521	−	0.992952i	$-0.537815\pi$
$613$	−5.86889	−0.237042	−0.118521	+	0.992952i	$0.537815\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−33.2314	−1.33784	−0.668922	−	0.743333i	$-0.733244\pi$
$617$	−33.2314	−1.33784	−0.668922	+	0.743333i	$0.733244\pi$
$618$	0	0
$619$	−19.3267	−0.776805	−0.388403	−	0.921490i	$-0.626973\pi$
$619$	−19.3267	−0.776805	−0.388403	+	0.921490i	$0.626973\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	35.2383	1.41179
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	27.3625	1.09101
$630$	0	0
$631$	9.39968	0.374196	0.187098	−	0.982341i	$-0.440092\pi$
$631$	9.39968	0.374196	0.187098	+	0.982341i	$0.440092\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	9.89026	0.392483
$636$	0	0
$637$	1.84608	0.0731442
$638$	0	0
$639$	0	0
$640$	0	0
$641$	42.0303	1.66010	0.830048	−	0.557693i	$-0.188313\pi$
$641$	42.0303	1.66010	0.830048	+	0.557693i	$0.188313\pi$
$642$	0	0
$643$	−31.9488	−1.25994	−0.629970	−	0.776620i	$-0.716933\pi$
$643$	−31.9488	−1.25994	−0.629970	+	0.776620i	$0.716933\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−20.7691	−0.816518	−0.408259	−	0.912866i	$-0.633864\pi$
$647$	−20.7691	−0.816518	−0.408259	+	0.912866i	$0.633864\pi$
$648$	0	0
$649$	33.8247	1.32774
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−13.3481	−0.522350	−0.261175	−	0.965291i	$-0.584110\pi$
$653$	−13.3481	−0.522350	−0.261175	+	0.965291i	$0.584110\pi$
$654$	0	0
$655$	−21.9344	−0.857050
$656$	0	0
$657$	0	0
$658$	0	0
$659$	18.5636	0.723134	0.361567	−	0.932346i	$-0.382242\pi$
$659$	18.5636	0.723134	0.361567	+	0.932346i	$0.382242\pi$
$660$	0	0
$661$	23.1728	0.901316	0.450658	−	0.892697i	$-0.351189\pi$
$661$	23.1728	0.901316	0.450658	+	0.892697i	$0.351189\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−2.04419	−0.0792701
$666$	0	0
$667$	2.48361	0.0961658
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−7.03278	−0.271497
$672$	0	0
$673$	51.0630	1.96834	0.984168	−	0.177240i	$-0.0567170\pi$
$673$	51.0630	1.96834	0.984168	+	0.177240i	$0.0567170\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−5.87586	−0.225828	−0.112914	−	0.993605i	$-0.536018\pi$
$677$	−5.87586	−0.225828	−0.112914	+	0.993605i	$0.536018\pi$
$678$	0	0
$679$	−18.6078	−0.714100
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−19.2597	−0.736952	−0.368476	−	0.929637i	$-0.620120\pi$
$683$	−19.2597	−0.736952	−0.368476	+	0.929637i	$0.620120\pi$
$684$	0	0
$685$	−6.33665	−0.242111
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−6.16136	−0.234729
$690$	0	0
$691$	8.11718	0.308792	0.154396	−	0.988009i	$-0.450657\pi$
$691$	8.11718	0.308792	0.154396	+	0.988009i	$0.450657\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−7.13555	−0.270667
$696$	0	0
$697$	3.67775	0.139305
$698$	0	0
$699$	0	0
$700$	0	0
$701$	18.9459	0.715575	0.357788	−	0.933803i	$-0.383531\pi$
$701$	18.9459	0.715575	0.357788	+	0.933803i	$0.383531\pi$
$702$	0	0
$703$	3.97863	0.150057
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−27.4508	−1.03239
$708$	0	0
$709$	−6.06700	−0.227851	−0.113926	−	0.993489i	$-0.536343\pi$
$709$	−6.06700	−0.227851	−0.113926	+	0.993489i	$0.536343\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	3.13555	0.117427
$714$	0	0
$715$	−1.72890	−0.0646572
$716$	0	0
$717$	0	0
$718$	0	0
$719$	19.4737	0.726245	0.363122	−	0.931741i	$-0.381711\pi$
$719$	19.4737	0.726245	0.363122	+	0.931741i	$0.381711\pi$
$720$	0	0
$721$	−11.4866	−0.427784
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−2.48361	−0.0922390
$726$	0	0
$727$	−41.5238	−1.54003	−0.770017	−	0.638024i	$-0.779752\pi$
$727$	−41.5238	−1.54003	−0.770017	+	0.638024i	$0.779752\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−3.12115	−0.115440
$732$	0	0
$733$	−13.8833	−0.512791	−0.256396	−	0.966572i	$-0.582535\pi$
$733$	−13.8833	−0.512791	−0.256396	+	0.966572i	$0.582535\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	16.1842	0.596152
$738$	0	0
$739$	−23.7661	−0.874251	−0.437125	−	0.899401i	$-0.644003\pi$
$739$	−23.7661	−0.874251	−0.437125	+	0.899401i	$0.644003\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	41.9344	1.53843	0.769213	−	0.638993i	$-0.220649\pi$
$743$	41.9344	1.53843	0.769213	+	0.638993i	$0.220649\pi$
$744$	0	0
$745$	14.3153	0.524471
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−10.1758	−0.371814
$750$	0	0
$751$	7.89026	0.287920	0.143960	−	0.989584i	$-0.454016\pi$
$751$	7.89026	0.287920	0.143960	+	0.989584i	$0.454016\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	11.5750	0.421257
$756$	0	0
$757$	−26.5566	−0.965216	−0.482608	−	0.875836i	$-0.660310\pi$
$757$	−26.5566	−0.965216	−0.482608	+	0.875836i	$0.660310\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	18.1241	0.657000	0.328500	−	0.944504i	$-0.393457\pi$
$761$	18.1241	0.657000	0.328500	+	0.944504i	$0.393457\pi$
$762$	0	0
$763$	11.3481	0.410827
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−8.31528	−0.300248
$768$	0	0
$769$	7.86744	0.283707	0.141854	−	0.989888i	$-0.454694\pi$
$769$	7.86744	0.283707	0.141854	+	0.989888i	$0.454694\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	7.17277	0.257987	0.128993	−	0.991645i	$-0.458825\pi$
$773$	7.17277	0.257987	0.128993	+	0.991645i	$0.458825\pi$
$774$	0	0
$775$	−3.13555	−0.112632
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0.534762	0.0191598
$780$	0	0
$781$	22.4980	0.805042
$782$	0	0
$783$	0	0
$784$	0	0
$785$	13.7661	0.491334
$786$	0	0
$787$	−6.01440	−0.214390	−0.107195	−	0.994238i	$-0.534187\pi$
$787$	−6.01440	−0.214390	−0.107195	+	0.994238i	$0.534187\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−11.0258	−0.392033
$792$	0	0
$793$	1.72890	0.0613950
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.45083	0.0513912	0.0256956	−	0.999670i	$-0.491820\pi$
$797$	1.45083	0.0513912	0.0256956	+	0.999670i	$0.491820\pi$
$798$	0	0
$799$	−53.3113	−1.88602
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−16.7477	−0.591015
$804$	0	0
$805$	−3.13555	−0.110514
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−9.11418	−0.320438	−0.160219	−	0.987082i	$-0.551220\pi$
$809$	−9.11418	−0.320438	−0.160219	+	0.987082i	$0.551220\pi$
$810$	0	0
$811$	19.0044	0.667336	0.333668	−	0.942691i	$-0.391714\pi$
$811$	19.0044	0.667336	0.333668	+	0.942691i	$0.391714\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	6.27110	0.219667
$816$	0	0
$817$	−0.453830	−0.0158775
$818$	0	0
$819$	0	0
$820$	0	0
$821$	55.5610	1.93909	0.969547	−	0.244906i	$-0.0787570\pi$
$821$	55.5610	1.93909	0.969547	+	0.244906i	$0.0787570\pi$
$822$	0	0
$823$	5.66335	0.197412	0.0987059	−	0.995117i	$-0.468530\pi$
$823$	5.66335	0.197412	0.0987059	+	0.995117i	$0.468530\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−46.0586	−1.60161	−0.800807	−	0.598922i	$-0.795596\pi$
$827$	−46.0586	−1.60161	−0.800807	+	0.598922i	$0.795596\pi$
$828$	0	0
$829$	−0.710526	−0.0246776	−0.0123388	−	0.999924i	$-0.503928\pi$
$829$	−0.710526	−0.0246776	−0.0123388	+	0.999924i	$0.503928\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−12.6961	−0.439895
$834$	0	0
$835$	−13.2825	−0.459660
$836$	0	0
$837$	0	0
$838$	0	0
$839$	10.9444	0.377843	0.188921	−	0.981992i	$-0.439501\pi$
$839$	10.9444	0.377843	0.188921	+	0.981992i	$0.439501\pi$
$840$	0	0
$841$	−22.8317	−0.787299
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−12.5750	−0.432592
$846$	0	0
$847$	−12.4394	−0.427424
$848$	0	0
$849$	0	0
$850$	0	0
$851$	6.10277	0.209200
$852$	0	0
$853$	−1.81727	−0.0622222	−0.0311111	−	0.999516i	$-0.509905\pi$
$853$	−1.81727	−0.0622222	−0.0311111	+	0.999516i	$0.509905\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	27.8033	0.949744	0.474872	−	0.880055i	$-0.342494\pi$
$857$	27.8033	0.949744	0.474872	+	0.880055i	$0.342494\pi$
$858$	0	0
$859$	−21.4066	−0.730385	−0.365193	−	0.930932i	$-0.618997\pi$
$859$	−21.4066	−0.730385	−0.365193	+	0.930932i	$0.618997\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	18.2284	0.620501	0.310250	−	0.950655i	$-0.399587\pi$
$863$	18.2284	0.620501	0.310250	+	0.950655i	$0.399587\pi$
$864$	0	0
$865$	8.27110	0.281226
$866$	0	0
$867$	0	0
$868$	0	0
$869$	21.2155	0.719687
$870$	0	0
$871$	−3.97863	−0.134811
$872$	0	0
$873$	0	0
$874$	0	0
$875$	3.13555	0.106001
$876$	0	0
$877$	−47.2671	−1.59610	−0.798049	−	0.602593i	$-0.794134\pi$
$877$	−47.2671	−1.59610	−0.798049	+	0.602593i	$0.794134\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	2.71749	0.0915546	0.0457773	−	0.998952i	$-0.485424\pi$
$881$	2.71749	0.0915546	0.0457773	+	0.998952i	$0.485424\pi$
$882$	0	0
$883$	20.6380	0.694524	0.347262	−	0.937768i	$-0.387111\pi$
$883$	20.6380	0.694524	0.347262	+	0.937768i	$0.387111\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	48.1172	1.61562	0.807808	−	0.589445i	$-0.200654\pi$
$887$	48.1172	1.61562	0.807808	+	0.589445i	$0.200654\pi$
$888$	0	0
$889$	31.0114	1.04009
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−7.75172	−0.259401
$894$	0	0
$895$	−5.64053	−0.188542
$896$	0	0
$897$	0	0
$898$	0	0
$899$	7.78749	0.259727
$900$	0	0
$901$	42.3739	1.41168
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0.359470	0.0119492
$906$	0	0
$907$	2.30832	0.0766465	0.0383232	−	0.999265i	$-0.487798\pi$
$907$	2.30832	0.0766465	0.0383232	+	0.999265i	$0.487798\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−50.4766	−1.67237	−0.836183	−	0.548451i	$-0.815218\pi$
$911$	−50.4766	−1.67237	−0.836183	+	0.548451i	$0.815218\pi$
$912$	0	0
$913$	14.6291	0.484153
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−68.7766	−2.27120
$918$	0	0
$919$	−32.0000	−1.05558	−0.527791	−	0.849374i	$-0.676980\pi$
$919$	−32.0000	−1.05558	−0.527791	+	0.849374i	$0.676980\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−5.53079	−0.182048
$924$	0	0
$925$	−6.10277	−0.200658
$926$	0	0
$927$	0	0
$928$	0	0
$929$	39.6336	1.30034	0.650168	−	0.759791i	$-0.274699\pi$
$929$	39.6336	1.30034	0.650168	+	0.759791i	$0.274699\pi$
$930$	0	0
$931$	−1.84608	−0.0605027
$932$	0	0
$933$	0	0
$934$	0	0
$935$	11.8903	0.388853
$936$	0	0
$937$	−42.4766	−1.38765	−0.693826	−	0.720143i	$-0.744076\pi$
$937$	−42.4766	−1.38765	−0.693826	+	0.720143i	$0.744076\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−12.2497	−0.399330	−0.199665	−	0.979864i	$-0.563985\pi$
$941$	−12.2497	−0.399330	−0.199665	+	0.979864i	$0.563985\pi$
$942$	0	0
$943$	0.820265	0.0267115
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−29.3039	−0.952248	−0.476124	−	0.879378i	$-0.657959\pi$
$947$	−29.3039	−0.952248	−0.476124	+	0.879378i	$0.657959\pi$
$948$	0	0
$949$	4.11718	0.133649
$950$	0	0
$951$	0	0
$952$	0	0
$953$	5.32669	0.172549	0.0862743	−	0.996271i	$-0.472504\pi$
$953$	5.32669	0.172549	0.0862743	+	0.996271i	$0.472504\pi$
$954$	0	0
$955$	19.1286	0.618986
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−19.8689	−0.641600
$960$	0	0
$961$	−21.1683	−0.682849
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−10.9672	−0.353047
$966$	0	0
$967$	−10.1097	−0.325107	−0.162554	−	0.986700i	$-0.551973\pi$
$967$	−10.1097	−0.325107	−0.162554	+	0.986700i	$0.551973\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	20.7249	0.665095	0.332547	−	0.943087i	$-0.392092\pi$
$971$	20.7249	0.665095	0.332547	+	0.943087i	$0.392092\pi$
$972$	0	0
$973$	−22.3739	−0.717273
$974$	0	0
$975$	0	0
$976$	0	0
$977$	6.37242	0.203872	0.101936	−	0.994791i	$-0.467496\pi$
$977$	6.37242	0.203872	0.101936	+	0.994791i	$0.467496\pi$
$978$	0	0
$979$	−29.8033	−0.952519
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−26.6008	−0.848433	−0.424217	−	0.905561i	$-0.639451\pi$
$983$	−26.6008	−0.848433	−0.424217	+	0.905561i	$0.639451\pi$
$984$	0	0
$985$	13.2383	0.421808
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−0.696123	−0.0221354
$990$	0	0
$991$	−51.0332	−1.62112	−0.810562	−	0.585652i	$-0.800838\pi$
$991$	−51.0332	−1.62112	−0.810562	+	0.585652i	$0.800838\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−2.35947	−0.0748002
$996$	0	0
$997$	0.0883702	0.00279871	0.00139936	−	0.999999i	$-0.499555\pi$
$997$	0.0883702	0.00279871	0.00139936	+	0.999999i	$0.499555\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8280.2.a.bn.1.3		3
3.2	odd	2		2760.2.a.s.1.3	✓	3
12.11	even	2		5520.2.a.bw.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2760.2.a.s.1.3	✓	3	3.2	odd	2
5520.2.a.bw.1.1		3	12.11	even	2
8280.2.a.bn.1.3		3	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:	\( N \)	\(=\)	\( 8280 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8280.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} +3.13555 q^{7} +O(q^{10})\)	\(q+1.00000 q^{5} +3.13555 q^{7} -2.65194 q^{11} +0.651938 q^{13} -4.48361 q^{17} -0.651938 q^{19} -1.00000 q^{23} +1.00000 q^{25} -2.48361 q^{29} -3.13555 q^{31} +3.13555 q^{35} -6.10277 q^{37} -0.820265 q^{41} +0.696123 q^{43} +11.8903 q^{47} +2.83167 q^{49} -9.45083 q^{53} -2.65194 q^{55} -12.7547 q^{59} +2.65194 q^{61} +0.651938 q^{65} -6.10277 q^{67} -8.48361 q^{71} +6.31528 q^{73} -8.31528 q^{77} -8.00000 q^{79} -5.51639 q^{83} -4.48361 q^{85} +11.2383 q^{89} +2.04419 q^{91} -0.651938 q^{95} -5.93445 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{5} - 2 q^{7}+O(q^{10}) \) 3 * q + 3 * q^5 - 2 * q^7	\( 3 q + 3 q^{5} - 2 q^{7} - 4 q^{11} - 2 q^{13} - 6 q^{17} + 2 q^{19} - 3 q^{23} + 3 q^{25} + 2 q^{31} - 2 q^{35} + 8 q^{37} - 2 q^{41} + 10 q^{43} - 6 q^{47} + 5 q^{49} - 6 q^{53} - 4 q^{55} - 8 q^{59} + 4 q^{61} - 2 q^{65} + 8 q^{67} - 18 q^{71} + 8 q^{73} - 14 q^{77} - 24 q^{79} - 24 q^{83} - 6 q^{85} - 4 q^{89} + 18 q^{91} + 2 q^{95} + 12 q^{97}+O(q^{100}) \) 3 * q + 3 * q^5 - 2 * q^7 - 4 * q^11 - 2 * q^13 - 6 * q^17 + 2 * q^19 - 3 * q^23 + 3 * q^25 + 2 * q^31 - 2 * q^35 + 8 * q^37 - 2 * q^41 + 10 * q^43 - 6 * q^47 + 5 * q^49 - 6 * q^53 - 4 * q^55 - 8 * q^59 + 4 * q^61 - 2 * q^65 + 8 * q^67 - 18 * q^71 + 8 * q^73 - 14 * q^77 - 24 * q^79 - 24 * q^83 - 6 * q^85 - 4 * q^89 + 18 * q^91 + 2 * q^95 + 12 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more